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Mad about

Modern Physics

Mad about

Modern Physics

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Mad about Modern Physics

Mad about Modern PhysicsBraintwisters, Paradoxes, and Curiosities

Franklin Potter

and

Christopher Jargodzki

John Wiley & Sons, Inc.

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This book is printed on acid-free paper.

Copyright © 2005 by Franklin Potter and Christopher Jargodzki. All rights reserved

Illustrations on pages 2, 4, 9, 26, 31, 134, and 161 copyright © 2005 by Tina Cash-Walsh.

Published by John Wiley & Sons, Inc., Hoboken, New JerseyPublished simultaneously in Canada

Design and production by Navta Associates, Inc.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in anyform or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise,except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, withouteither the prior written permission of the Publisher, or authorization through payment of the appro-priate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923,(978) 750-8400, fax (978) 646-8600, or on the web at www.copyright.com. Requests to the Pub-lisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc.,111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008.

Limit of Liability/Disclaimer of Warranty: While the publisher and the author have used their bestefforts in preparing this book, they make no representations or warranties with respect to the accu-racy or completeness of the contents of this book and specifically disclaim any implied warrantiesof merchantability or fitness for a particular purpose. No warranty may be created or extended bysales representatives or written sales materials. The advice and strategies contained herein may notbe suitable for your situation. You should consult with a professional where appropriate. Neitherthe publisher nor the author shall be liable for any loss of profit or any other commercial damages,including but not limited to special, incidental, consequential, or other damages.

For general information about our other products and services, please contact our Customer CareDepartment within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in printmay not be available in electronic books. For more information about Wiley products, visit our web

Library of Congress Cataloging-in-Publication Data:

Potter, Frank, dateMad about modern physics : braintwisters, paradoxes and curiosities / Franklin Potter and

Christopher Jargodzki.p. cm.

Includes index.ISBN 0-471-44855-9

1. Physics--Popular works. I. Jargodzki, Christopher II. TitleQC24.5.P68 2004530—dc22

2004014941

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

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site at www.wiley.com.

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To my late parents, who nourished my formative years andhave now crossed that portal to another world.

F. P.

To my late grandmother—Zofia Lesinska, who instilled in me the idea that the visible world

owes its being to the invisible one.

C. J.

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vii

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Acknowledgments. . . . . . . . . . . . . . . . . . xii

To the Reader . . . . . . . . . . . . . . . . . . . . . xiii

Chapter 1 The Heat Is On . . . . . . . . . . . . . . . . . . . . 1

Chapter 2 Does Anybody Really Know What

Time It Is? . . . . . . . . . . . . . . . . . . . . . . . . 11

Chapter 3 Crazy Circles . . . . . . . . . . . . . . . . . . . . . 19

Chapter 4 Fly Me to the Moon. . . . . . . . . . . . . . . . . 29

Chapter 5 Go Ask Alice . . . . . . . . . . . . . . . . . . . . . . 39

Chapter 6 Start Me Up . . . . . . . . . . . . . . . . . . . . . . 49

Chapter 7 A Whole New World. . . . . . . . . . . . . . . . . 63

Chapter 8 Chances Are . . . . . . . . . . . . . . . . . . . . . . 75

Chapter 9 Can This Be Real? . . . . . . . . . . . . . . . . . 9 1

Chapter 10 Over My Head. . . . . . . . . . . . . . . . . . . . . 105

Chapter 1 1 Crystal Blue Persuasion . . . . . . . . . . . . . 1 1 7

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viii Contents

Answers

The Heat Is On . . . . . . . . . . . . . . . . . . . . . . 125

Does Anybody Really Know What

Time It Is? . . . . . . . . . . . . . . . . . . . . . . . 139

Crazy Circles . . . . . . . . . . . . . . . . . . . . . . . 15 1

Fly Me to the Moon . . . . . . . . . . . . . . . . . . 164

Go Ask Alice . . . . . . . . . . . . . . . . . . . . . . . 18 1

Start Me Up . . . . . . . . . . . . . . . . . . . . . . . . 192

A Whole New World . . . . . . . . . . . . . . . . . . 206

Chances Are. . . . . . . . . . . . . . . . . . . . . . . . 224

Can This Be Real? . . . . . . . . . . . . . . . . . . . 241

Over My Head . . . . . . . . . . . . . . . . . . . . . . 257

Crystal Blue Persuasion . . . . . . . . . . . . . . 27 7

Index . . . . . . . . . . . . . . . . . . . . . . . . . . 287

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ix

Preface

This book of almost 250 puzzles begins where our first book, MadAbout Physics: Braintwisters, Paradoxes, and Curiosities (2001)ended—with the physics of the late nineteenth and early twenti-

eth centuries. The Michelson-Morley experiment of 1887, the challenges posed by atomic spectra and blackbody radiation, theunexpected discoveries of X-rays in 1895, radioactivity in 1896, andthe electron in 1897 all loosened the protective belt of ad hoc hypothe-ses around the mechanistic physics the nineteenth century had so laboriously built. Anomalies and paradoxes abounded, ultimatelynecessitating a radical rethinking of the very foundations of physicsand culminating in the theory of relativity and quantum mechanics.Numerous applications of these new and strange concepts followedvery quickly as atomic and nuclear physics led to semiconductordevices on the small scale and nuclear energy on the large scale. There-fore we have developed a whole new set of challenges to tickle theminds of our scientifically literate readers, from science students toengineers to professionals in the sciences.

The challenges begin with the classical problem of getting a cookedegg into a bottle through a narrow bottleneck and back out again andprogress gradually to the famous aging-twin paradox of the theory ofspecial relativity and eventually reach problems dealing with the large-scale universe. In between, we explore the nature of time and of spaceas well as how the world of films and television tends to sacrificephysics for the sake of entertainment. We also consider some of themore startling questions in relativity. For example, we ask whether aperson can go on a space journey out to a star 7,000 light-years distantand return while aging only 40 years! And we certainly want toemphasize the practical applications of microphysics through an exam-ination of some properties of exotic fluids, unusual motors running onair or on random motion, as well as thermal, electrical, and photonicproperties of materials in a challenging journey into the atomic world.

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x Preface

Particularly important microworld challenges include: What happenedto Schrödinger’s cat? Can a cup of coffee be the ultimate quantumcomputer? Why is a Bose-Einstein condensate a new state of matter?Why is quantum mechanical coherent scattering so important in devel-oping new detectors for neutrinos and gravitational waves? When wereach the nucleus, there are challenges about the accuracy of carbon-14dating, the reason for neutron decay, and the amount of humanradioactivity. Then our journey reverses as we reach for the stars to con-sider Olbers’ paradox about why the night sky is dark instead of burst-ing with light, how gravitational lensing by galaxies works, and whatthe total energy in the universe might be. This book finishes with a pot-pourri of challenges from all categories that ranges from using bicycletracks in the mud to determine the direction of travel, to analyzingwater-spouting alligators, and ending with a space-crawling mechanicalinvention that seems to defy the laws of physics.

The puzzles range in difficulty from simple questions (e.g., “Willan old mechanical watch run faster or slower when taken to themountains?”) to subtle problems requiring more analysis (e.g., “Is theBragg scattering of X-rays from an ideal crystal a coherent scatteringprocess?”) Solutions and more than 300 references are provided, andthey constitute about two-thirds of the book.

As these examples demonstrate, most of the puzzles contain an ele-ment of surprise. Indeed, one finds that commonsense conjecture andproper physical reasoning often clash throughout this volume. Ein-stein characterized common sense as the collection of prejudicesacquired by age eighteen, and we agree: at least in science, commonsense is to be refined and often transcended rather than venerated.Many of the challenges were devised to undermine physical precon-ceptions by employing paradoxes (from the Greek para and doxos,meaning “beyond belief”) to create cognitive dissonance. Far frombeing simply amusing, paradoxes are uniquely effective in addressingspecific deficiencies in understanding. Usually the contradictionbetween gut instinct and physical reasoning for some people will be sopainful that they will go to great lengths to escape it even if it meanshaving to learn some physics in the process.

Philosopher Ludwig Wittgenstein considered paradoxes to be anembodiment of disquietude, and as we have learned, these disqui-etudes often foreshadow revolutionary developments in our thinking

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Preface xi

about the natural world. The counterintuitive upheavals resultingfrom relativity theory and quantum mechanics in the twentieth cen-tury only enhanced the reputation of the paradox as an agent forchange in our understanding of physical reality.

Such disquietudes, rather than unexplained experimental facts,writes Gerald Holton in Thematic Origins of Scientific Thought, werewhat led Einstein to rethink the foundations of physics in his threepapers of 1905. Each begins with the statement of formal asymmetriesof a predominantly aesthetic nature, then proposes a general postu-late, not derivable directly from experience, that removes the asym-metries. For example, in the paper on the quantum theory of light,formal asymmetry existed between the discontinuous nature of parti-cles and the continuous functions used to describe electromagneticradiation. As Holton notes, “The discussion of the photoelectriceffect, for which this paper is mostly remembered, occurs toward theend, in a little over two pages out of the total sixteen.” Consistentwith this approach is Einstein’s statement in Physics and Reality(1936), “We now realize . . . how much in error are those theoristswho believe that theory comes inductively from experience,” and laterin The Evolution of Physics (1938), coauthored with the Polish physi-cist Leopold Infeld, “Physical concepts are free creations of the humanmind, and are not, however it may seem, uniquely determined by theexternal world.”

As another sore point, the term “quantum mechanics” is really amisnomer: quantum systems cannot be regarded as made up of sepa-rate building blocks. In the helium atom, for instance, we do not haveelectron A and electron B but simply a two-electron pattern in whichall separate identity is lost. This indivisible unity of the quantum worldis paralleled by another kind of unity—between subject and object. Islight a wave or a particle? The answer seems to depend on the experi-mental setup. In the double-slit experiment, the observations of lightyield characteristics of the box and its slits as much as of light itself. Is reality then observer-dependent? And would this justify Einstein’sinsistence on the power of pure thought in the construction of physi-cal reality? Modern physics seems particularly adept at generating suchdisquietudes. If that’s the case, then perhaps the word Mad in the titleof our book should not be construed as a mere metaphor!

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Acknowledgments

We all “stand on the shoulders of giants” as we develop ourminds to become individuals living today on our planet Earth.And we owe so much to so many people that we cannot

acknowledge all of them.Franklin Potter would like to express appreciation to his wife,

Patricia, and their two sons, David and Steven, for their love andinspiration through many wonderful years of family adventures. Healso treasures the numerous inspiring physics discussions over thedecades with many friends and colleagues: Howard G. Preston, Gregory Endo, Fletcher Goldin, David M. Scott, John Priest, LowellWood, Julius S. Miller, George E. Miller, Leigh H. Palmer, Charles W.Peck, Myron Bander, Joseph Weber, Richard Feynman, Willard Libby,Edward Teller, and Kamal Das Gupta.

Christopher Jargodzki would like to express appreciation toMyron Bander of the University of California at Irvine; Stephen Reu-croft of Northeastern University in Boston; and James H. Taylor ofCentral Missouri State University in Warrensburg. His interactionswith close to twenty thousand students (and counting!) in his classesat UC Irvine, Northeastern University, and CMSU have been, over theyears, never-ending sources of stimulation, as well as occasional exas-peration. In fact, the present volume got its start in 1975 when one ofus (C. J.), still a graduate student at UC Irvine, put together a proposalfor a book of paradoxes in modern physics, partly to allay his ownexasperation with the koanlike conundrums that abound in modernphysics. Alas, the project had to wait several decades for the author tomature and join forces with Franklin Potter in our joint inquiry intothe nature of physical reality. The authors hope that physical reality isduly impressed with their efforts.

Both authors sincerely thank Kate C. Bradford, senior editor atJohn Wiley & Sons, Inc., who continues to support our paradoxicaladventures into the world of physics.

xii

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xiii

To the Reader

These puzzles are meant to be fun. How many puzzles you solve isnot as important as how many you enjoy thinking about. Someof them are even challenging to research physicists, and some

were generated by research articles that have appeared only recently inphysics journals, so these topics may not have been part of physics just10 years ago! It would be a rare reader who could provide detailedsolutions to all the puzzles. Indeed, sometimes you may need to thinka bit to even understand the answer. If we included all the steps, thisbook would double its present size. We offer no apologies, but we dotry to provide all the key steps to make each answer complete on itsown. If you find the puzzles perplexing and intriguing, we have suc-ceeded in our mission.

Mad about Modern Physics can be read with profit by anyone whohas had some exposure to a year of introductory physics and is eagerto learn more about its applications and its more recent discoveries.Most puzzles are nonmathematical in character and require only aqualitative application of fundamental physics principles. Manyphysics concepts are defined directly or indirectly in the questions orin the answers, so they can be found with the aid of the index. How-ever, even someone who knows the subject will quickly realize that theapplication of physics to the real world can be quite challenging, andin this sense this is not an elementary book.

More than three hundred follow-up references provide furtherresources for interested readers. These references—to journal researchpapers, books, and magazine articles—are included with only some ofthe puzzles, typically those that are either controversial or that involverelatively new concepts. There was no space to include a more com-plete list of references. Consequently we had to make choices, and weapologize to the authors whose work may have been left out or inad-vertently overlooked.

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Any errors are solely those of the authors, and we would appre-ciate your communications via e-mail to Franklin Potter (seewww.sciencegems.com) with regard to the puzzles and theiranswers.

xiv To the Reader

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The Heat Is On

1

The Heat Is On

S CIENCE IN THE HOME CONTRIBUTES IMMENSELY

to our everyday repertoire of activities, although

most of us are unaware of exactly how science does so.

Physics, in particular, is all around us and plays a crucial role

in determining what we can and cannot do. One enjoyable

activity for many people is cooking, which is an application

of physics and chemistry to satisfy our gastronomical tastes.

Or are physics and chemistry just other modes of cooking?

We’ll let you decide. Most of the challenges in this chapter

involve physics from a high-school-level course. But be care-

ful. Quick responses may be correct occasionally, but you

should not rely on your intuition very much, for Nature,

particularly in the kitchen, is nonintuitive for the most part.

Anyone who has tried to make a soufflé can attest to how

limited a recipe can be!

1

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1. Egg into a Bottle

Perhaps the most intriguingphysics-in-the-kitchen demon-stration for all ages is get-ting a hard-boiled egg withthe shell removed into abottle that has an openingdiameter smaller than theminimum diameter of theegg. One solution is to verycarefully drop some bits ofburning paper intothe upright bottle andthen place the egg atthe opening. Soon, ifthe sequence is donewith the correct tim-ing, the egg will have the urge to go inside. What is thecorrect timing, and why does the egg have this urge?

2. Egg out of a Bottle

Perhaps the most challenging physics-in-the-kitchendemonstration for all ages is getting a hard-boiled eggwith the shell removed out of a bottle that has an open-ing diameter smaller than the minimum diameter ofthe egg. Of course, one could cut up the egg with aknife inserted into the bottle and then pour out thepieces. However, we want the egg out whole andundamaged.

Long ago, physics professor Julius Sumner Miller,(who was Professor Wonderful on the early MickeyMouse Club shows) was on the Tonight Show withhost Johnny Carson and showed first how to get theegg into the bottle and then, taking no more than three

2 Mad about Modern Physics

We can detect five

basic tastes—four are

very familiar: sweet,

sour, bitter, and salty.

The fifth, while familiar

in East Asia, is less well

known in Western

cuisine—it is called

and is the taste

of monosodium gluta-

mate, MSG. MSG is

used widely in Eastern

cooking and that is

probably why it is rec-

ognized as a separate

taste sensation more

readily by those familiar

with that cuisine. How-

ever, many common

western foods contain

large amounts of MSG,

notably tomatoes and

parmesan cheese.

—PETER BARHAM,THE SCIENCE OF COOKING

I was raised in Alabama

and Florida a Southern

Baptist, a lad given

simple answers to pro-

found questions. At the

same time I came to

love science, which

seeks profound answers

to simple questions.

—EDWARD O. WILSON

Umami

Egg

Burningpaper

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The Heat Is On 3

seconds, had the same egg back in his hand. What is theprocedure? (Hint: the same physics principles that putthe egg into the bottle can get the egg out.)

3. Sugar

Add two cups of sugar to one cup of water in asaucepan and stir while heating slightly. All the sugarwill dissolve. About how much total sugar will dissolvein one cup of water? What is the physics?

4. Kneading Bread

Bread made with yeast is usually kneaded—that is,drawn out and pressed together to create a distributionof the ingredients. Then the bread dough is set aside to“rise.” Why is some bread then kneaded a second timeand sometimes even a third time before baking?

5. Measuring Out Butter

Suppose you have a solid chunk of butter and a meas-uring cup in the kitchen. You desire to accuratelymeasure one-half cup of butter chunks without meltingthem. What is a quick, easy way to do so? Often oneencounters the statement in cookbooks that Archi-medes’ principle is being used. What is this principle,and why is the statement erroneous?

6. Milk and Cream

You are given two identical bottles, one with milk andthe other with cream, both filled to the top. Quick now,which is heavier? And is light cream lighter than heavycream?

Why is it that tea made

with microwave-heated

water doesn’t taste as

good as tea made with

teakettle water? The

main reason is that

microwaves heat only

the outer inch or so of

the water all around the

cup, because that’s as

far as they can pene-

trate. The water in the

middle of the cup gets

hot more slowly, through

contact with the outer

portions. When the

outer portions of the

water have reached boil-

ing temperature and

start to bubble, you can

be tricked into thinking

that all the water in the

cup is that hot. But the

average temperature

may be much lower, and

your tea will be short-

changed of good flavor.

—ROBERT L. WOLKE, WHAT

EINSTEIN TOLD HIS COOK:KITCHEN SCIENCE EXPLAINED

If there were one drop

of water less in the

universe,

the whole world would

thirst.

—UGO BETTI,ITALIAN PLAYWRIGHT

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7. Straw and Potato

A paper or plastic drinking straw can be pushedthrough an uncooked potato. Explain the physics. Ifyou plan to try this demonstration, be sure that youtake appropriate safety precautions—keep your handsand body out of harm’s way.

8. Blueberry Muffins

Marion loves to bake warm, fresh blueberry muffins,with the blueberries almost uniformly distributedthroughout the muffin. She knows that if one simplyprepares the batter and mixes in the blueberries, theymay be uniformly distributed before entering the oven,but upon baking they will gravitate to lodge in thelower part of the muffin. How does she prevent thisnatural downward drift?

9. Can of Soup

Some people buy canned soup and store the cans in thecupboard. Some people even turn these soup cansupside down for storage. If we open a can of soup thatwas stored in the upright position by removing the top,quite often all the concentrated ingredients are on thebottom and must be scooped out with a spoon. Even

4 Mad about Modern Physics

CALORIC REQUIREMENT

BASED ON BODY WEIGHT

The basal calorie

requirement of the

average adult is ten

times the ideal weight

in pounds (e.g., 1,270

for 127 lbs.), plus 30

percent for light activity

(i.e., 1,650 kcal), 50

percent for moderate

activity (1,905 kcal),

and 100 percent for

heavy activity (2,540

kcal). Expressed slightly

differently, the basal

energy requirement is

about 1 kilocalorie per

hour for every kilogram

(2.2 lbs.) of ideal body

weight. Of course, any

estimate of calorie

requirements based on

such formulas is just

that—an .

Individual requirements

vary widely with age,

health, body size,

and environmental

temperature.

When men reach their

sixties and retire, they

go to pieces. Women go

right on cooking.

—GAIL SHEEHY, AMERICAN

JOURNALIST

estimate

Straw

Potato

Push down

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The Heat Is On 5

then, not all the concentrate is removed. Suppose,instead, we turn the same can upside down and openthe bottom. Upon turning the can over, the soup simplyrushes out into the pot. Why so?

10. Salt and Sugar

Salts have been used for thousands of years to preservemeats, and sugar has been used to preserve fruits andberries. How do they work?

11. Defrosting Tray

In catalogs and cookware stores one can buy a “mira-cle” defrosting tray advertised as made of an“advanced, space-age super-conductive alloy” that“takes heat right out of the air.” How does this defrost-ing tray work?

12. Ice Cream Delight

Most of us have made ice cream or seen ice cream beingmade. Milk, eggs, sugar, and flavorings are slowlychilled. Terri likes to make ice cream in a simpler andmore efficient way. Practicing proper safety precautions,she pours liquid nitrogen directly into the ingredients ina metal bowl. About equal volumes of liquid nitrogenand the mixture are used for ice cream or sorbet, andshe stirs while adding the coolant until the ice cream isnicely stiff. Why does this method produce absolutelymarvelous ice cream, and what is the physics here?

13. Cooking a Roast

For many types of meat—beef, pork, lamb, etc.—onecan buy a roast from the butcher with or without thebone inside. Suppose we have two beef roasts of the same

The boiling temperature

of water decreases

about 1.9°F for every

1,000 feet above sea

level. So in Denver, the

mile-high city, water will

boil at 202°F—that is,

at 94.4°C. Tempera-

tures above 165°F are

generally thought to be

high enough to kill most

germs, so there is no

danger on this account

until you get to about

25,000 feet.

On the average we get

about 9 (food) calories

(kcal) of energy from

each gram of fat and

4 calories from each

gram of protein or

carbohydrate. To lose a

pound (454 g) of fat,

we have to cut the food

intake by 3,500 calo-

ries. The discrepancy in

numbers is due to the

fact that body fat is

only about 85 percent

actual fat, the rest

coming from connective

tissue, blood vessels,

and other things.

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weight of 4.4 pounds (2 kg) and cook them in identicalovens at the same temperature. One roast has the bone inand the other does not. Which roast cooks faster? Why?

14. Cooking Chinese Style

Estimates of Chinese meals include more than 3,000varieties, possibly more meal types than the total num-ber of meals by all other cultures combined. Many ofthe Chinese dishes use meats cut into small cubes orother small volumes. Certainly, these small volumes aremuch easier to eat with chopsticks. Are there any sig-nificant scientific reasons for cutting up the meats intosmall volumes?

15. Baked Beans

If you buy dry beans in bulk, they must be soaked inwater overnight in a covered container before they areready to be baked. To bake them without soakingwould require an enormous amount of cooking time.An alternative preparation is to “parboil” them in acooking pot—that is, simmer them. Simmer means “tobe on the verge of boiling.”

How does one know that the beans have simmeredenough? The test involves good physics. Take up a fewbeans in a spoon and, after making sure that no liquidis in the spoon, blow a stream of air gently with pursedlips against the beans. If the bean skin cracks, the beansare ready for baking. Why must the lips be pursed, andwhy do the bean skins then crack open?

16. Ice Water

Normally, to cool a pitcher of water quickly, one addsice. The ice floats at the top. Suppose one could add thesame amount of ice so it could be held in the water at

6 Mad about Modern Physics

Light bounces off mir-

rors; microwaves bounce

off metal. If what you

put in the microwave

oven reflects too many

microwaves back

instead of absorbing

them, the magnetron

tube that generates the

microwaves can be

damaged. There must

always be something in

the oven to absorb

microwaves. That’s why

you should never run it

empty.

Metals in microwave

ovens can behave

unpredictably. Micro-

waves set up electrical

currents in metals, and

if the metal object is too

thin it may not be able

to support the current

and will turn red hot and

melt. And if it has sharp

points, it may even act

like a lightning rod and

concentrate so much

microwave energy at the

points that it will send

off lightning-like

sparks.

—ROBERT L. WOLKE, WHAT

EINSTEIN TOLD HIS COOK:KITCHEN SCIENCE EXPLAINED

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The Heat Is On 7

the bottom of the pitcher. Which technique would leadto faster cooling of the water?

17. Peeling Vegetables

A friend of ours peels ripe tomatoes by impaling thetomato on a fork, then holding it over a gas flame androtating gently. If you try this procedure, use appropri-ate safety procedures to protect your eyes and body.

Peeling fresh beets is also a messy chore. Their col-ored liquid stains everything, including your fingers.Another friend of ours peels fresh beets by first boilingthem, then immediately holding them under cold waterwith a fork. What is the physics in both of these meth-ods used for preparing vegetables for peeling?

18. Igniting a Sugar Cube

Sugar burns in air. But ignitinga sugar cube is much more dif-ficult than expected. Put asugar cube on the end of atoothpick and bring a lightedmatch flame under a remotecorner. The sugar meltsinstead of burning, and thebrown, gooey stuff is caramel.

However, we wish to burnthe sugar, not melt it! We want to see it on fire with aflame of its own. Why is this process so difficult toachieve? How can we succeed in lighting the sugar cubewith the burning match?

19. Water Boiling

An open pot of water is boiling on the kitchen stove.Sprinkle some room-temperature table salt (which

A standard 12-ounce

aluminum can, whose

wall surfaces are

thinner than two pages

from this book (about

0.00762 cm), with-

stands more than 90

pounds of pressure per

square inch—three

times the pressure in an

automobile tire.

—WILLIAM HOSFORD AND

JOHN DUNCAN, “THE

ALUMINUM BEVERAGE CAN,”SCIENTIFIC AMERICAN,

SEPTEMBER 1994

Decaffeinated coffee

still contains caffeine!

A regular cup of

coffee has 80 to 135

milligrams of caffeine.

For a coffee to be con-

sidered decaffeinated,

at least 97 percent of

the coffee’s caffeine

must be removed. Test-

ing shows that decafs

typically have 2 to 6

milligrams of caffeine

per cup.

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contains mostly NaCl and some KCl) into the clearboiling water, and the boiling ceases. Isn’t it amazinghow the water ceases its boiling as the salt warms up!Can you explain the physics? What is the surprise here?

20. Put the Kettle On

Bring water to a boil in a teakettle with a spout. Let itcook! Now watch the mouth of the spout carefully.What do you see? Can you see the water vapor comeout?

21. The Watched Pot

You have probably heard the expression “A watchedpot never boils.” Is this statement correct physics? Thatis, when would this statement be good physics? (Hint:One should interpret the phrase “never boils” here tomean that the cooking takes a longer time.)

22. Ice in a Microwave

The microwave oven emits microwaves that areabsorbed by water molecules in food. Microwavesmake the polar water molecules rotate or oscillate, andtheir “friction” within the material converts some ofthis kinetic energy into thermal energy to raise the tem-perature of the food.

Suppose you made an ice block that had liquidwater trapped in a large cavity inside and then youplaced the block into a microwave oven. Could thetrapped water be brought to a boil while the iceremained ice?

8 Mad about Modern Physics

An object at room tem-

perature (20°C) emits

radiation with a peak at

the wavelength 9.89

micrometers, roughly

.01 mm, in the infrared

region of the electro-

magnetic spectrum.

For an isolated water

molecule the H-O-H

angle is 104.5°. In ice

each water molecule

forms hydrogen bonds

to four nearest neigh-

bors in a tetrahedral

arrangement. The tetra-

hedral bond geometry

explains the openness

and relatively low den-

sity of ice (i.e., why

water expands upon

freezing). In ice the

H-O-H angles are

nearly the same as the

perfect tetrahedral

angle of 109.5°.

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The Heat Is On 9

23. The Glycemic Index

The glycemic index is an important number for anyoneconcerned with the conversion of food to blood sugar(sucrose), for the gylcemic index gives the measuredrate of this conversion process. The higher the glycemicindex value, the faster the conversion rate to sucrose.There are types of sugar molecules other than sucrose.Glucose, for example, is normally the standard referencefor the conversion rate to sucrose, with a value of 100.

Some sample values of the glycemic index for foodsare: brown rice, 59; white rice, 88; table sugar, 65; grapefruit, 25; spaghetti, 25 to 45; potato, boiled,55; potato, baked, 85; and dates, 103. Brown rice hasmore outer layer intact than white rice, so its lowervalue is evident. But why would a baked potato have amuch higher glycemic index than a boiled potato? Andhow could the value for dates, or any food, be higherthan 100?

24. Electric Pickle

Some specialty and novelty stores sell an electrical“appliance” that cooks hot dogs between two metalelectrodes. A protective cover with a safety interlock

Night cooling by evapo-

ration of water and heat

radiation had been per-

fected by the peoples

of Egypt and India, and

several ancient cultures

had partially investi-

gated the ability of

salts to lower the freez-

ing temperature of

water. Both the ancient

Greeks and Romans

had figured out that

previously boiled water

will cool more rapidly

than unboiled water, but

they did not know why;

boiling rids the water

of carbon dioxide and

other gases that other-

wise retard the lowering

of water temperature.

—TOM SHACHTMAN, ABSOLUTE

ZERO AND THE CONQUEST

OF COLD

Interestingly, microwave

ovens are not very good

at melting ice. The

water molecules in ice

are bound pretty tightly

together into a crystal

lattice, so they can’t flip

back and forth under

the influence of micro-

waves’ oscillation.

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closes over the device before electrical energy in theform of a standard AC current can be applied. Supposethat instead of a hot dog one places a pickle between theelectrodes. When the room lights are dimmed, thepickle glows impressively, predominantly at one end.What is the physics, and what might the glow look like?

25. Space-Age Cooking

Microwave ovens were probably the first new methodfor making heat for cooking in more than a millionyears. In addition, two newer methods have becomeavailable for the kitchen. Magnetic induction cooktopshave been available for about fifteen years in Europeand Japan and are now becoming known in the UnitedStates. And for the modern chef, cooking with light ina “light oven” has been done since the mid-1990s andmay become a fad in the immediate future. How doboth of these cooking sources work?

10 Mad about Modern Physics

Although it flies in the

face of common sense,

people with more insula-

tion—fat—whose body

core is better protected

from the cold, may feel

cold more quickly than

thinner people with less

protection. The reason

is that insulation keeps

heat in the core, away

from the skin, which

gets cold. When the

skin gets cold, you feel

cold. Paradoxically,

women may feel colder

than men because

women are better

insulated.

—JAMES GORMAN, “BEYOND

BRR: THE ELUSIVE SCIENCE

OF COLD,” THE NEW YORK

TIMES, FEBRUARY 10, 2004

The main compartment

of a refrigerator should

always be below 40°F

(4.4°C). Above that

temperature, bacteria

can multiply fast

enough to be dangerous.

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Does AnybodyReally KnowWhat Time It is?

11

Does AnybodyReally KnowWhat Time It Is?

W HAT IS TIME?” ST. AUGUSTINE FAMOUSLY

wrote. “If no one asks me, I know. But if I

wanted to explain it to one who asks me, I plainly do not

know.” Time itself is a strange quantity to some people. To

many of us, time never seems to be going at the right rate—

sometimes too fast, sometimes too slow. In some parts of the

world, promptness and being on time are important aspects

of the local culture. In other regions, time is almost irrele-

vant. In this chapter, we have created a mixture of familiar

challenges and many new ones in preparation for later chap-

ters in which time shares its role with space as a major ingre-

dient of motion, chapters that look at concepts such as the

space-time of the special theory of relativity and the world of

astrophysics.

2

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26. January Summer

Contrary to the popular belief that Earth is closest to theSun on about June 23 or possibly December 22 eachyear, the date of perihelion actually falls between January2 and January 5! In the Northern Hemisphere, we expe-rience winter on this January date because the NorthPolar axis is tilted away from the Sun. The SouthernHemisphere enjoys a warm summer at this time. Will theNorthern Hemisphere ever enjoy summer in January?

27. Proximity of Winter

Solstice and Perihelion

Earth reaches perihelion—the point in its orbit whenit’s closest to the Sun—between January 2 and 5,depending on the year. That’s about two weeks afterthe December solstice, December 21 or 22. Thus win-ter begins in the Northern Hemisphere at about thetime that the Earth is nearest the Sun. Is there a reasonwhy the times of solstice and perihelion are so close, oris this a coincidence?

28. Earth’s Speed

The time interval required for Earth to travel from theautumnal equinox to the vernal equinox (approximately179 days) is less than the time interval from the vernal tothe autumnal equinox (roughly 186 days). Why?

29. The Equinox Displaced

At the time of the spring equinox (usually March 20) orthe fall equinox (September 22 or 23), night and dayare supposed to be of equal duration. But according tothe almanacs of sunrise and sunset times, on the datesof the equinoxes, daytime is longer by 8 to 10 minutes.How come?

12 Mad about Modern Physics

How did the day get to be

divided into 24 hours?

The night appears to have

been divided first, by the

ancient Egyptians.

According to Prof. Owen

Gingerich, a historian of

science at Harvard Uni-

versity, they divided the

heavens into intervals of

10 degrees of arc, making

it possible to squeeze 12

hours, each of 10

degrees, into the shortest

night. When the day also

became divided, the

hours of night and day

were of unequal length,

and the system of so-

called “unequal hours,”

12 each for night and

day, lasted well into the

Middle Ages, coexisting

with another reckoning

of the “equal hours.”

—Q & A, SCIENCE TIMES,THE NEW YORK TIMES,DECEMBER 13, 1983

You must remember this,

A kiss is still a kiss,

A sigh is just a sigh,

The fundamental things

apply,

As time goes by.

—HERMAN HUPFELD,“AS TIME GOES BY”

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Does Anybody Really Know What Time It Is? 13

30. The Dark Days of December

At latitude 40 degrees north, earliest sunset occurs onabout December 8 and latest sunrise on about January5. The shortest day of the year, the winter solstice, isDecember 21 or 22. Why are all these dates not thesame?

31. Days of the Year

The length of the year (i.e., the interval of time betweentwo successive passages of Earth through the samepoint in its orbit) is about 365.2422 days. How manyentire rotations on its own axis does Earth execute dur-ing that time?

32. Leap Years

Every four years, in years divisible by four, is a leapyear, when an extra day is added to the month of Feb-ruary, except years divisible by 100. For example,1700, 1800, and 1900 were not leap years, yet 2000was a leap year. Why?

33. Full Moons

Is the interval of time between one full Moon and thenext equal to 28 days?

34. Moon Time

Cheryl is sitting at a desk in an office and the clockshows 12:20 and the Moon is seen through the windowas a thin crescent with the open side pointing down-ward to the right. What do you make of this scene?Where could the Sun be?

The minute first

appeared as a division of

the hour about A.D. 1320

in Paris editions of the

so-called Alfonsine

Mean Motion Tables,

sponsored by King

Alfonso the Wise of

Spain. But the of

the minute was implicit

all the time in a method

of reckoning used by

early astronomers. They

employed a system of

sexagesimal fractions,

first devised by the

Babylonians, based on

successive powers of

60. Any unit could be

divided into 60 parts;

these were called in

Latin “partes minutae

primae,” or “first very

small parts,” yielding the

word “minute”. A minute

in turn was eventually

divided into 60 “partes

minutae secundae,”

hence the word “second.”

—Q & A, SCIENCE TIMES,THE NEW YORK TIMES,DECEMBER 13, 1983

When it comes to

procrastinating, I do it

right away!

—ANONYMOUS

idea

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35. Lunar Calendar

Although there have been numerous calendars over themillennia of civilizations, they fall into two basic types,solar and lunar calendars. Today, while practically every-one uses the solar calendar with 365.2422 days per trop-ical year, rice farmers in many parts of the worldcontinue to use the lunar calendar based on a 29.53-daylunar month. Can you figure out a scientific reason why?

36. The Sandglass

For a sandglass timer one could simply have a straightglass or plastic tube with equally spaced markings andthen the whole tube would be inverted to start the timemeasurement. Why do ruled sandglasses have a tapered“hourglass” shape instead?

37. Old Watch

Lenni has an old mechanical watch in pristine condi-tion that has an internal balance wheel that operatesperfectly. She takes a drive into the mountains. Will thewatch run fast or slow?

14 Mad about Modern Physics

In Wicca, February 2

(Groundhog Day) is one

of the four “greater

sabbats” that divide the

year at the midpoints

between the solstices

and equinoxes.

Sundials tell Sun time

while clocks tell mean

time. The true Sun

leads or lags the mean

Sun, crossing the

meridian from 16 min-

utes, 25 seconds ear-

lier than the mean Sun

(in early November) to

14 minutes, 20 seconds

later (in February). Only

on or about April 16,

June 14, September 2,

and December 25 are

the true and mean Suns

together as they cross

the meridian.

The angle between the

Equator and the ecliptic

(i.e., the plane of Earth’s

orbit), also known as the

tilt of the globe, was

23° 26' 32" in 2002.

Through the ages, this

value varies between

21° and 28°. At present

it goes down by 0.47"

per year.

12:20

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Does Anybody Really Know What Time It Is? 15

38. Reading a Digital Timer

Many digital timers show the elapsed time to one-hundredth of a second. What is the minimum uncer-tainty in the value? What value should be reported?

39. Eternal Clocks?

There are laser and atomic clocks in special laboratoryenvironments that are accurate to one second in 300million years! Yet their lifetimes are typically less than30 years. Some wristwatches run longer! There aremechanical clocks in development that could last about10,000 years! But they would need periodic winding.Why do the laser and atomic clocks have such shortlifetimes? How might one build a mechanical clock thatwould survive so long?

40. Room Light

Suppose there is a photodetector with a flash lamp atthe exact center of a 3 m × 3 m × 3 m dark, barrenroom with reflective walls. The flash lamp flashes forone nanosecond. For simplicity, assume that the light isemitted isotropically in all directions when the lampflashes. If the photodetector simply sums the light fromall directions, what is its recorded intensity versus time?If the photodetector is an array capable of discerningdifferent angular directions, what is the intensity versustime for several different directions? Suppose the lampflashes for one microsecond. What now?

41. Right to Left Driving Switch

Suppose you live in a country in which the driving is on the right and there is to be a change to driving on the left. If highways with on-ramps and off-ramps,

More people are born on

October 5 in the United

States than on any

other day. Not so sur-

prising, as conception

would have fallen on

New Year’s Eve.

If 23 students are in a

classroom and you pick

two at random, the

probability that their

birthdays (month and

day) match is about

1/365. The probability

that at least two of the

23 have the same birth

date, however, is a trifle

better than 1⁄2. The

reason is that now there

are 1 + 2 + 3 + . . . + 22

= 253 possible match-

ing pairs.

—MARTIN GARDNER, “MATHE-MATICAL GAMES,” SCIENTIFIC

AMERICAN (OCTOBER 1972)

There are 365 days in

the year. Note the

following:

365 = 102 + 112 + 122

= 132 + 142

Coincidence? Preestab-

lished harmony? You be

the judge!

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and so on are built for driving on the right, will theywork equally well for driving on the left? Of course, we must assume the same patterns of driving speeds as before.

42. Light Clock

Some museums and labora-tories have a light clock withtwo parallel mirrors and apulse of light bouncing backand forth repeatedly, retrac-ing the same path over andover, keeping very accuratetime as each complete tran-sit of the light pulse isdetected and counted. Themirror separation is usuallyabout a meter or less, so avery large number of reflections occur during each sec-ond of time. Suppose this light clock is moved sidewaysparallel to the mirrors at a constant velocity, andassume that the light will continue to reflect off bothmirrors during this sideward movement. Will the clockcontinue to keep accurate time?

43. Time Reversal

16 Mad about Modern Physics

Studying mid-

twentieth-century

scientists, psychologist

Bernice T. Eiduson

found a disproportionate

number who had been

confined to their beds

for large amounts of

time by childhood ill-

nesses. During these

travails, they “searched

for resources within

themselves and became

comfortable being by

themselves”; most

turned to reading, and

through reading they

developed a bent for

intellectual work. Not

very good at sports,

unfit by illness to com-

pete in childhood games,

they remained emotion-

ally fragile throughout

life, deriving satisfaction

mostly from intense

involvement in science.

—TOM SHACHTMAN, ABSOLUTE

ZERO AND THE CONQUEST

OF COLD

Mirror

Mirror

Frame 1 Frame 4

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Does Anybody Really Know What Time It Is? 17

A movie is made showing successive frames for anobject accelerating downward. If the sequence is runbackward, the object accelerates (a) upward or (b)downward. Explain.

44. Molecular Clock

Different species of organisms have enormous regionsof DNA that are the same or very similar. Humans andchimpanzees, for example, share about 98 percent oftheir DNA. We share much less of our DNA withrodents and amphibians and insects.

In a general way, the percentage of shared DNAmight be a means to establish a molecular clock—thatis, the more DNA that is shared, the more recent wasthe separation of the family tree. And, if by accident,the changes in the DNA happened to proceed at a com-mon rate, then one could set up a timeline also.

However, the genetic changes do not occur withany regularity. Why not?

45. SAD

Most animals experience dramatic seasonal cycles: theymigrate, hibernate, mate, and molt at specific times ofthe year. These cycles appear to be hardwired; theyoccur even when the temperature is held constant andthe light and dark periods are varied. But humans areamong the least seasonally sensitive creatures, havingonly a vestige of seasonal effects known as seasonalaffective disorder (SAD), an extremely mild version ofthe cyclical responses animals experience. Only about 5percent of adults overtly sense the seasonal changes andsuffer from SAD during the winter days of longer dark-ness. Amazingly, light therapy—looking into a light thatmimics sunlight—or merely sleeping until dawn helps

In a

an unexpected smell or

taste or perhaps a song

from your past can

unleash in you a raging

torrent of realistic and

graphic memory. The

phrase recalls a scene

in Marcel Proust’s

when a

madeleine cake (a

small, rich cookie-like

pastry) enables the

narrator to experience

the past completely as

a simultaneous part of

his present existence:

“And suddenly the mem-

ory revealed itself: The

taste was that of the

little piece of madeleine

which on Sunday morn-

ings at Combray

(because on those

mornings I did not go

out before mass), when

I went to say good

morning to her in her

bedroom, my aunt

Leonie used to give me,

dipping it first in her

own cup of tea or

tisane.”

Things Past

Remembrance of

Proustian moment

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the people with SAD in northern latitudes. Would thesetherapies be effective on people living at the Equator?

46. Two Metronomes

Suppose the timekeeping abilities of two identicalmetronomes are compared over several hours. Theywill drift faster or slower at different rates. When bothmetronomes are placed on a skateboard that movesfreely horizontally, their drifts change gradually as theytend to synchronize. Each metronome has been sub-jected to the driving force of the other, the result beingthe phenomenon called “phase-locking” or “mode-locking.” Suppose now that each metronome on theskateboard begins with different initial conditions, butone of the two metronomes is driven by perturbationsthat fluctuate randomly in time. Can the metronomesbecome synchronized?

47. Time Symmetry

The fundamental equations of physics—at least thosethat derive from symmetries in nature—all exhibit timesymmetry because they are second-order differentialequations. Newton’s second law and Maxwell’s equa-tions are immediate examples. However, one can con-sider time running forward or backward. Evengeneral-relativity equations formulated in tensor math-ematics exhibit time symmetry. Assuming that all theseequations are correct, must nature at its most funda-mental level obey time symmetry? (Note: Entropy rela-tions are not derived from a fundamental symmetryand therefore are excluded.)

18 Mad about Modern Physics

Now he has departed

from this strange world

a little ahead of me.

That signifies nothing.

For us believing physi-

cists, the distinction

between past, present,

and future is only a

stubbornly persistent

illusion.

—ALBERT EINSTEIN ON LIFE-LONG FRIEND MICHELE BESSO,IN A LETTER OF CONDOLENCE

TO THE BESSO FAMILY, MARCH

21, 1955, LESS THAN A MONTH

BEFORE HIS OWN DEATH. ALICE

CALAPRICE, THE EXPANDED

QUOTABLE EINSTEIN

HOW TO FIND NORTH

USING A WATCH

In the Northern Hemi-

sphere, hold the watch

horizontal and point the

hour hand at the Sun.

Bisect the angle

between the hour hand

and the 12 o’clock mark

to get the north-south

line. If your watch is set

on daylight saving time,

use the midway point

between the hour hand

and one o’clock. The

farther you are from

the Equator, the more

accurate this method

will be.

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CrazyCircles

19

CrazyCircles

S PACE IS RELATED TO POSITION, DISTANCE, AND

size and has its own paradoxes and influences. We

live in a space of three dimensions, but our ability to visual-

ize three-dimensional relationships among objects is not as

easy as judging distance. Our brain activity relies on neural

connections in a 3-D biomass that would probably become

moronic if limited to two dimensions. However, robots usu-

ally operate in our 3-D space by following computer pro-

grams that maneuver in multidimensional configuration

spaces that often far exceed three dimensions. Recent theo-

retical research in quantum physics hints that the natural

world may be as large as 11-dimensional, with seven dimen-

sions curled up too small for our senses, leaving the four

dimensions of space-time. In this chapter we have created a

mixture of familiar challenges and many new ones regarding

space in preparation for a later chapter on the space-time of

the special theory of relativity.

3

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48. Spider and Fly

On a plane the shortest distance between two points is a straight line. Suppose a spider sits on a cube andwants to catch a fly sitting on the opposite face. Howwould you determine the path of shortest distance forthe spider to crawl on the surface to catch the fly?

49. Moon Distance

In measuring the length of a 1-meter table with ameterstick to within 0.1 millimeter, the uncertainty inthe measurement is one part in ten thousand. Meter-sticks, however, are usually inconvenient for measuringthe distance to the Moon. Instead, a laser light pulsecan be reflected from a stationary corner reflector onthe Moon similar to the reflectors on bicycles, and thetotal duration of the pulse from Earth to Moon andback to Earth again is timed. What do you estimate forthe uncertainty in the measurement for the Moon’s dis-tance? Which determination would you expect to havethe greater distance uncertainty, the table length or thedistance to the Moon?

50. Ideal Billiards Table

20 Mad about Modern Physics

A Pythagorean triplet is

a set of three numbers

that describes the sides

of a right triangle.

Pythagoras invented his

theorem around 550

B.C., but the Babyloni-

ans had catalogued per-

haps hundreds of

triplets by 2000 B.C.,

long before Pythagoras.

One of the triplets the

Babylonians found is

the enormous

3,367:3,456:4,825.

—DICK TERESI, LOST

DISCOVERIES: THE ANCIENT ROOTS

OF MODERN SCIENCE—FROM THE

BABYLONIANS TO THE MAYA

In the hallowed groves

of the academe they

whisper the tale of a

physicist who spent

the first half of his life

trying to become

famous, at which he

failed; then spent the

second half of his life

trying to convince him-

self it wasn’t important

to be famous, at which

he also failed.

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Suppose you have an ideal rectangular billiards table onwhich a ball collides with any wall (called the cushion)so that the angles of incidence and reflection are equal.Let there be pockets at the corners only. Describe howto shoot a given ball into a specific corner pocket witheither zero, one, two, or three banks of the ball.

51. Wallpaper Geometry

Some of the old video games used an interesting butsimple visual technique to extend the playing field. Acharacter running off the right side of the screen thenentered the left side while the background sceneryremained fixed. That is, the right side edge is matchedto the left side edge, and the top and bottom arematched also. One could even have a rectangular arrayof video screens, each right edge matched to a left edge, etc., each screen showing the same image. Fastersystems later came along, and the scenery movedinstead, and these 2-D views were eventually replacedby 3-D views.

Consider now a 3-D regular array of cubes touch-ing face to face and top to bottom, the 3-D space ana-log to the old style 2-D video game. Let opposite cubefaces be matched and imagine that these face surfacesare invisible. You are standing in one cube inside thisspace and look to your right. Behold! You see yourself!

In the of Apollo-

nius, the words “ellipse”

(defect), “parabola”

(equality), and “hyper-

bola” (excess) were

applied to the three

curves now known by

these names because of

the relationships 2 <

, 2 = , and 2 >

, respectively, where

is the parameter

of the

curve which is so placed

upon a coordinate sys-

tem that a vertex is at

the origin, and the axis

of the curve lies along

the axis of abscissas.

Hence one can see that

Apollonius applied the

name “ellipse” to indicate

not a defective circle but

a defective parabola.

—CARL B. BOYER, “LETTERS,”SCIENTIFIC AMERICAN

(FEBRUARY 1960)

I put tape on the mirrors

in my house so I don’t

accidentally walk

through into another

dimension.

—STEVEN WRIGHT, COMEDIAN

(latus rectum)p

px

ypxypx

y

Conics

You are in this cube.

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What exactly do you see? What do you see when look-ing upward?

52. Space-Filling Geometry

Cubes can be placed next to each other in three direc-tions to fill all of 3-D space. Regular octahedrons canfill 3-D space also. Spheres of the same radius cannot.Can regular tetrahedrons fill all of 3-D space and leaveno gaps? Can regular dodecahedrons and regular icosa-hedrons?

53. Archimedes’ Gravestone

Archimedes’ gravestone is said to have a sphere insidea cylinder etched into the stone as well as the symbol π.How are the two 3-D objects related if they have thesame radius? And why are they on his gravestone?

22 Mad about Modern Physics

The Chinese mathe-

matician Liu Hui calcu-

lated a value for π(3.1416) in A.D. 200

that remained the most

accurate estimation for

a thousand years.

—DICK TERESI, LOST

DISCOVERIES: THE ANCIENT

ROOTS OF MODERN SCIENCE—FROM THE BABYLONIANS

TO THE MAYA

FOUR-DIMENSIONAL

GEOMETRY IN THE BIBLE?

St. Paul’s Letter to the

Ephesians contains the

following passage: “that

you, being rooted and

grounded in love, may

be able to comprehend

with all the saints what

is the width and length

and depth and height”

(Ephesians 3: 17–18).

—MARTIN GARDNER, “MATHE-MATICAL GAMES,” SCIENTIFIC

AMERICAN (SEPTEMBER 1975)

A SILLY SYLLOGISM

Nothing is better than

eternal life;

A salami sandwich is

better than nothing;

Therefore, a salami

sandwich is better

than eternal life!

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Crazy Circles 23

54. Brain Connections

The human brain has more than 100 billion neurons,with each neuron receiving input signals from 10 to1,000 other neurons. Schematic representations ofthese connections in the brain always show an incredi-ble web of lines representing the neurons, either as a 2-D or a 3-D image. Suppose you created a scaled-down computer model of this human brain using only1 million neurons in a 3-D space. On average, howmany input connections would each neuron have?What is the surprise here?

55. Configuration Space

Suppose we have a robotic arm that mimics the move-ments of a person’s arm. The arm exists in the familiar3-D physical space. Consider a simplification of therobotic arm that assumes just three connected parts:upper arm, forearm, and hand, all in the shape ofstraight rods that are connected. The body of the robot,including the shoulder, remains fixed in position. Wewish to have the robotic arm touch a particular point-like object in the room. How many numbers arerequired in a computer program to describe the armposition?

56. Farmer Chasing a Goose

Farmers know that to catch a stray goose one does notrun after the goose in an open field. A better strategy isto corner the goose. However, suppose the farmer andthe goose are in an open field and they both run withthe same speed, V, to provide us with some semblanceof fair play. Furthermore, restrict the farmer to chasingthe goose along the instantaneous line of sight to thegoose. When will the farmer catch the goose?

THE LATE APPEARANCE IN

ENGLISH OF THE WORD

“SCIENTIST”

In 1840 William Whewell

noted that there was no

simple and natural way

to refer to “a cultivator

of science in general.”

He was, he concluded,

inclined to call him “a

scientist.” Before

Whewell scientists

tended to refer to each

other as philosophers,

or more fully, as natural

philosophers. For this

reason Newton’s treatise

on mathematical physics

was given the title

The Indian mathematician

Srinivasa Ramanujan

(1887–1920) discovered

an approximation to πthat is remarkable for its

precision and concise-

ness: (2143/22)1/4 =

3.14159265258 . . .

(to be compared with π =

3.14159265358 . . .).

(1687).

of Natural Philosophy

Mathematical Principles

The

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57. A Spooky Refrigerator

Christina notices that food is disappearing from herrefrigerator and yet her surveillance camera shows thatno one is opening the door. Suppose our 3-D spatialworld were really a 4-D spatial world, but we did notknow anything about the existence of the fourth spatialdimension. There is still the single time dimension.Could a 4-D being remove food from her 3-D refriger-ator without opening the refrigerator door?

58. Fractional Dimensions?

A point has zero dimensions. A line has one dimension.A plane has two dimensions. Space has three dimensions.Can something have 1.585 . . . spatial dimensions?

59. Platonic Solids

There are five 3-D regular polyhedrons called thePlatonic solids: the regular tetrahedron (4 faces), theregular hexahedron (cube), the regular octahedron (8faces), the regular dodecahedron (12 faces), and the

24 Mad about Modern Physics

Stigler’s law of

eponymy, formulated by

statistician Stephen

Stigler, states that no

scientific discovery is

named after its original

discoverer. Journalist

Jim Holt points out that

Stigler’s law itself is

self-confirming, given

that Stigler admits that

it was discovered by

someone else: Robert

Merton, a sociologist of

science. The most noto-

rious example of

Stigler’s law is probably

the Pythagorean theo-

rem, widely known by

the ancient Egyptians,

Babylonians, and Indi-

ans long before

Pythagoras.

—ADAPTED FROM JIM HOLT,“MISTAKEN IDENTITY THEORY,”LINGUA FRANCA (MARCH 2000)

DIVINE MADNESS

The word “theory” comes

from the Greek word

meaning

“ecstatic contemplation

of the truth,” as exem-

plified in Plato’s belief

that “the greatest truths

are those that come to

us through divine

madness.”

theoria,

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regular icosahedron (20 faces). All these solids have atwofold rotational symmetry axis through the center ofeach edge—that is, a rotation about this axis by 180degrees leaves the object looking the same as the initialview. But the regular tetrahedron does not have inver-sion symmetry.

If we intersect two identical regular tetrahedrons sotheir centers coincide, can the composite object have atwofold rotational symmetry axis? Can it have inver-sion symmetry?

60. Intersecting Spheres

If in 2-D we intersect two circles (called one-spheres bymathematicians), the intersection is either a point, twopoints, or a circle. In 3-D the intersection of twospheres (each called a two-sphere) will be either apoint, a circle, or a sphere. What can the intersection oftwo three-spheres be? And three three-spheres?

61. Arm Contortions

Normally, the rotation of an object about a fixed axisby 360 degrees brings the object back to its initial ori-entation. However, Barbara has the agility to do thefollowing double rotation. She places a small object orbook in her right hand, holding the object horizontal

The size of the Moon

compared to the Earth

is 3:11 (with accuracy of

99.9 percent). This

Earth–Moon proportion

is also precisely invoked

by our two planetary

neighbors, Venus and

Mars. The closest : far-

thest distance ratio that

each experiences of the

other is, incredibly, 3:11

(with accuracy of 99.9

percent). Quite by

chance, 3:11 is 27.3

percent, and the Moon

orbits the Earth every

27.3 days, also the

average rotation period

of a sunspot.

—JOHN MARTINEAU,A LITTLE BOOK OF COINCIDENCE

The Roman numeral

representing “five,”

symbolized by the letter

V, derives from the

shape of the space

between the open

thumb and index finger.

The Roman numeral for

“ten,” the letter X, is

actually two V’s.

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and noting its orientation in the room. While imagininga vertical axis from floor to ceiling, the book is movedinward first and then under the upper arm, keeping thebook horizontal and rotating the object completelyaround this vertical axis back to its initial position. Herarm is now twisted. Can she untwist by rotating herarm a second time in the same direction?

26 Mad about Modern Physics

The biblical approxima-

tion of π is given in

1 Kings 7:23 and is

repeated in 2 Chroni-

cles 4:2. Both verses

speak of a circular “sea

of cast bronze” with a

diameter of 10 cubits

and a circumference of

30. The Greeks used a

more accurate value of

22/7 (error 0.04 per-

cent), and the Egyptians

used a ratio of two

squares 256/81 (error

0.6 percent).

The other day, I was

walking my dog around

my building . . . on the

ledge. Some people are

afraid of heights. Not

me, I am afraid of

widths.

—STEVEN WRIGHT, COMEDIAN

R. G. Duggleby, a bio-

chemist at the Univer-

sity of Ottawa, found

that the sum of π to the

fourth power (97.40909

. . .) and π to the fifth

power (306.01968 . . .)

is (i.e., 2.7182818 . . .)

to the sixth power

(403.42879 . . .),

correct to four decimal

places!

e

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Crazy Circles 27

62. The Rotating Cup

Place a cup with a handle on a shelf at eye height. Nowwalk in a straight line at a nearly constant speed pastthe cup, all the while rotating your head to observe theorientation of the cup. Notice what you see. The cupappears to rotate in the direction opposite your walk-ing direction, at first very slowly, then quickly, thenslowly again. Now consider yourself to be stationaryand imagine the cup itself moving past in a straight linewith constant speed. You could try to demonstrate thiswith the cup in your hand. What do you see now?

63. Space and Time Together

To explain Einstein’s 1905 special theory of relativityand Minkowski’s 1908 unification that combines threespace dimensions and one time dimension into a four-dimensional space-time continuum, most introductoryphysics textbooks use a four-dimensional coordinatesystem, with three real coordinates for space and oneimaginary coordinate for the time coordinate. Why notfour real coordinates? Why not have three imaginaryspace coordinates and one real-time coodinate?

64. Space > 3-D?

Can you provide arguments for why space has threedimensions? Hint: Are planetary orbits stable in a spaceof n dimensions, where n > 3? Is the hydrogen atomstable when n > 3?

The ancient composers

of Vedic literature in

India had to develop a

method of evaluating

square roots. The tech-

nique apparently

evolved from a need to

double the size of a

square altar. One needs

a square whose sides

are the square root of

2. In the a

collection that dictates

the shapes and areas of

altars and the location

of the sacred fires, the

square root of 2 is

stated as 1.414215 . . . ,

an amazingly accurate

value! The

were written between

800 and 500 B.C.,

making them at least as

old as the earliest

Greek mathematics.

The Greeks, however,

had no positional nota-

tion system. Hence

their approximations of

the square roots were

rather crude.

—GEORGE GHEVERGHESE

JOSEPH, THE CREST OF THE

PEACOCK: NON-EUROPEAN

ROOTS OF MATHEMATICS

Sulbasutras

Sulbasutras,

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Fly Me tothe Moon

29

Fly Me to the Moon

W E LIVE IN A WORLD OBEYING THE RULES OF

nature. But this natural world described by physics

and the other sciences can be superseded and replaced by the

imagination of the human mind. The artificial worlds cre-

ated in the many forms of literature and in audio and visual

renderings today cast a powerful influence on the minds of

everyone in the modern world. In fact, more people prefer to

live in these artificial fantasy worlds than in the real world

than are willing to admit. In these challenges we focus on

some of the “fuzzy science” prevalent in movies and televi-

sion shows. In certain ways, the awareness of the correct sci-

ence can enhance your enjoyment of the entertainment

product, just like knowing how a bee communicates to the

other bees in its hive enhances the beauty of the bee itself.

4

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65. Gunfight

Some TV programs and films have high drama scenesbased on a victim being shot by the pursuer and being“blown backward” a meter or two by the projectileimpact. Is this dramatic response Hollywood hype, oris there good physics here?

66. Body Cushion

A fall from a height of several stories onto pavement oreven onto a lawn will produce serious injuries or evendeath. Yet we have seen the movie hero going over theedge of a roof holding another human body in positionjust beneath to cushion the fall on impact. Certainly,collision with this second body is better than direct col-lision with the ground. What do you think about theadvantages here?

67. Cartoon Free Fall

So many of us in our youth learned the laws of naturefrom cartoons. Some of us are still learning fromcartoons! The cartoon character steps forward off a cliff and remains there in suspension until realizing thesituation, then the accelerationdownward begins. As yourecall the scene, what violationsof physics can you discern?

68. Silhouette

of Passage

When a cartoon charactersmashes through a solid wall orother object, we see the perfo-ration as the crisp outline of the

30 Mad about Modern Physics

The anagram of

MOON STARERS is

ASTRONOMERS.

Jonathan Swift’s

published in 1726,

describes Gulliver’s

many adventures,

including his “Voyage to

Laputa.” Gulliver learns

that the scientists there

discovered two moons

of Mars, which revolve

around the planet at

distances from its cen-

ter equal to 3 and 5

Martian diameters.

When the moons of

Mars were discovered

by Asaph Hall in 1877, it

turned out that Swift

not only had the number

of the moons right, but

he also placed them

close to the actual

distances: 1.4 and 3.5

diameters of Mars.

The two moons, named

Phobos and Deimos, are

tiny. Phobos, which

measures 27 by 19 kilo-

meters, is shaped rather

like a potato. Deimos,

too, is oddly shaped,

and measures 15 by 11

kilometers.

Gulliver’s Travels,

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Fly Me to the Moon 31

character. What would a condensed matter physicistsay about this cookie cutter type of material response?

69. Artificial Gravity

We all know that a body will tend to “float” around ina space station orbiting Earth or in a spaceship cruisingat a constant velocity with respect to the stars. Somefilms depict a dumbbell-shaped space station rotatingabout an axis through its middle perpendicular to thelong axis in order to provide artificial gravity. Whatinteresting behavior patterns might be experienced byan astronaut who walks across the axis from one endto the other?

70. Small

Wings

Space heroes whovisit other planetshave encountered alienbeings who suspend them-selves in the air with twosmall beating wings, eachabout 40 centimeters long,attached to their backs. Thesecharacters are less than a

Jules Verne’s

was

published in 1865.

Breaking with literary

tradition, which called

for recounting such a

voyage only as an imagi-

nary undertaking, Verne

based his account on an

extrapolation of contem-

porary scientific princi-

ples. The resulting

prophetic qualities of

this novel are uncanny.

For instance, Verne

chose a launch site not

far from Cape Canaveral

in Florida; he also gave

his readers the initial

velocity required for

escaping the earth’s

gravitation. In the sequel,

Verne

correctly described the

effects of weightless-

ness, and he even

pictured the space-

craft’s fiery reentry and

splashdown in the

Pacific Ocean—

amazingly, at a site just

three miles from where

landed on its

return from the Moon in

1969.

—ARTHUR EVANS AND RON

MILLER, “JULES VERNE,MISUNDERSTOOD VISIONARY,”

SCIENTIFIC AMERICAN

(APRIL 1997)

Apollo 11

Around the Moon,

Earth to the Moon

From the

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meter tall but probably have a mass of at least 20 kilo-grams. Could these wings suffice?

71. Shrunken People

Suppose someone is shrunken by some gimmick in themovies. Let’s say that you suffer this consequence andare now 100 times smaller in all dimensions. Actually,there is a lot of space between the atoms and moleculesof our bodies, but let’s ignore any increase in repulsiveforces, etc., and assume that this shrinkage can bedone. What does physics tell you will be a major prob-lem as you walk?

72. Spaceship Designs

The simple but effective spaceships of Buck Rogers andFlash Gordon have been superseded by flashy newdesigns with interesting shapes, sizes, and abilities. Theadvent of the space age in the 1950s brought about aheightened awareness of the practical physics charac-terizing a successful rocket or spaceship. Yet today,more than 50 years later, the ingenuity of the movieindustry continues to defy the laws of physics. We seethe latest nuclear-powered spaceships operating inspace coming in for a landing on Earth (or other com-parable planet) at a spaceport and then taking off forspace a little while later in the same ship from the samespaceport. Why can’t we do this feat with present-dayspace vehicles?

73. Warp Speed

Spaceships are known for their ability to turn on theirwarp drives to accelerate to speeds beyond the speed oflight. Can present-day physics conceptually explain thiscapability?

32 Mad about Modern Physics

The first public suggestion

that Abraham Lincoln be

the Republican candidate

for president is believed to

be a letter written Novem-

ber 6, 1858, and published

in the

The writer, Israel Green

(a druggist in Findlay,

Ohio), proposed a ticket

of Lincoln for president,

and for

vice president. The pro-

posed Kennedy was John

Pendleton Kennedy of

Maryland, a prominent

author and politician who

had been Millard Fillmore’s

Secretary of the Navy.

—MARTIN GARDNER,THE MAGIC NUMBERS OF

DR. MATRIX

According to NASA, there

is only one proven case of

a human who was hit by an

object from outer space,

Lannie Williams of Tulsa,

Oklahoma. In 1997, when

Ms. Williams was power

walking, she felt a tap on

the back of her shoulder.

As something rolled off her

shoulder, she heard it hit

the sidewalk with a metallic

thud. When she looked

back, she saw a mangled

lump of metal, which later

turned out to be a piece

of a U.S. rocket.

John Kennedy

Cincinnati Gazette.

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Fly Me to the Moon 33

74. North Pole Ice Melt

Environmental disasters have always been popularwith filmmakers. In recent years, the trend has beentoward disasters on a global scale because the publichas become more aware of global environmentalproblems. If there is a global warming trend, therecould be much ice melting at the poles of Earth. Somefilms have portrayed seacoasts being inundated by therising water level. What would you predict for the sealevel change if the ice only at the North Pole meltedcompletely?

75. Lightning and Thunder

We see the flash of distant lightning and hear its thun-der roll simultaneously in the movie. But we all knowthat the lightning flash arrives before the thunder in thereal world, there being about five seconds of sounddelay for each mile of distance to the lightning. Supposeyou were in charge of a battle scene in a war movie.When editing the scenes of the explosions on the bat-tlefield, how would you ensure the correct experiencefor the theater patron?

76. Explosions in Outer Space

Explosions in outer space on the big screen are magnif-icent to behold. Brilliant colors of stuff shooting outward in all directions, decreasing their density as the inverse square of the distance. The sound of theexplosion rocks the spaceship with a thundering roarjust as the light flash is first seen. Finally, bits of debrisscream past. What do you think about this spacephysics?

H. G. Wells’s 1914 novel

a

speculative history of

the future, contains the

following sentence:

“Nothing could have

been more obvious to

the people of the early

twentieth century than

the rapidity with which

war was becoming

impossible. And as cer-

tainly they did not see

it. They did not see it

until the atomic bombs

burst in their fumbling

hands.”

How would you suspend

500,000 pounds of

water in the air with no

visible means of sup-

port? (Answer: build a

cloud!)

—BOB MILLER, ARTIST

On November 10, 1907,

the magazine section of

the

included the headline

“Martians Probably

Superior to Us.”

New York Times

The World Set Free,

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77. Space Wars

One space battlecruiser after another shoots powerfullaser beams that destroy the enemy’s space battlecruiser.We see the powerful red laser beams strike the oppo-nent, and we hear the explosion as the object blowsapart. What wonderful physics can be learned here?

78. Security Lasers

Quite often the drama in a crime movie or an adventuremovie is enhanced by having crisscrossed visible laserbeams around the item to be protected from theft. Thethief must avoid intersecting these beams during theescapade to steal the item; otherwise a security alarmwill notify the appropriate authorities and the thief willbe caught. If you were the movie’s director, how wouldyou make this scene to ensure good physics?

79. Bullet Fireworks

Bullets bounce everywhere. The bad guys shoot alengthy burst of submachinegun fire as the hero runsthrough an industrial plant. The bullets impacting onsteel railings, for example, give off bright flashes oflight. This scene is a dramatic event for almost anyonewatching the hero in a time of great peril. What canyou say about the physics here?

80. Internet Gaming

For years people have been playing “live time” gamesover the Internet. If the game is checkers or poker, forexample, each player must take his or her turn inproper order, so short delays are not a problem. Even

34 Mad about Modern Physics

The American socialist

writer Edward Bellamy

(1850–1898), in his

best-selling 1888 novel

antici-

pated, by seven years,

the motif of time travel

H. G. Wells was to use

in his

(1895). Bellamy’s vision

of the future included,

among other things,

shopping malls and

“credit cards” (his own

expression).

PERVERSITY OF

INANIMATE OBJECTS?

As many a scientist

concluded with con-

sternation, there is a

reason why there is

a word in

Were the sun and moon

to doubt

They’d immediately go

out.

—WILLIAM BLAKE

demonstrations.

demon

Time Machine

2000–1887,

Looking Backward:

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Fly Me to the Moon 35

when the game is a world-domination board game withmultiple players, each player can submit moves at anytime before the deadline. But many video games requiresimultaneous play by several players, so delays canmean life or death for a player’s combatant in a shoot-’em-up type of action game. One can hear commentsby some action game players that they tried to maketheir move but the Internet was too slow. What is thepossible truth here?

81. Cartoon Stretching

Objects in cartoons are stretched and squeezed intoamazing distortions and then released. Some of thecharacters suffer the same fate. When the body materialof a cartoon character is being pulled, we often see thepart closer to the applied force stretch first and then therest follow with a small time delay. For example, a car-toon dog may pull on a character’s leg, which we seebeing stretched while the torso remains normal, untilfinally the torso stretches, the arms stretch, and thecharacter releases his or her handgrip from the door-way. Using some physics concepts, what can you sayabout the speed of sound in a cartoon character’s body?

82. Infrared Images

In crime dramas and in adventure films the result of aninfrared vision device is often reconstructed and shownin a sharp greenish or black-and-white format. We seethe infrared faces of people as if they were originallycolor images seen normally with one’s eyes, but nowthese color images have been converted to black andwhite. Is there any physics violation here in depictingthe infrared images via this process?

PERIODIC CLEANUP

OPERATION

Every time the solar

cycle peaks, it causes

Earth’s atmosphere to

expand and pull in low-

orbiting debris, which

burn up on reentry.

Jules Verne died in

1905. A memorial

sculpture placed over

his grave depicts Verne

rising from his tomb,

one arm reaching

toward the stars. Some

two decades later an

American periodical

called

—the first magazine

exclusively to feature

tales of science and

adventure—used a rep-

resentation of Verne’s

tomb as a logo. To

describe these narra-

tives, the publisher,

Hugo Gernsback, coined

the term “scientifiction,”

which was later changed

to science fiction.

—ARTHUR B. EVANS AND RON

MILLER, “JULES VERNE,MISUNDERSTOOD VISIONARY,”

SCIENTIFIC AMERICAN

(APRIL 1997)

Amazing Stories

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83. Light Sabers

Enemies dueling with light sabers have graced the silverscreen for several decades now. Isn’t this type ofweapon the most ridiculous thing you’ve ever seen?

84. Force Fields

In battle scenes of many science fiction movies we seethe baddies roll up with their giant laser guns to shootthe good guys, who are protected by a visibly transpar-ent force field. Why do the laser beams suffer deflectionat the force field?

85. Cold Silence of Space

In the “cold silence of space” begins many a descriptionof space between planets. Can this statement survive aphysics analysis?

86. Nuclear Submarine

Several movies have involved an out-of-control nuclearreactor aboard a nuclear submarine. We are told thatthe containment vessel is about to fail and that the bestaction is to move the sub several hundred metersunderwater. When the explosion occurs down there,what might happen?

87. Plutonium vs. Uranium

Suppose you find a nuclear bomb and decide to trans-port the device to a safe hiding place. Would there be any difference with regard to your safety as towhether the device is made of uranium-235 orplutonium-239?

36 Mad about Modern Physics

One of the big uncer-

tainties in calculating

the path of an asteroid

involves how much sun-

light it absorbs and then

reradiates as thermal

energy. Such radiation

can, over the centuries,

gently push the asteroid

into a different orbit,

much as a tiny rocket

would. So, if scientists

in future years should

conclude that a collision

looks ever more likely,

then they can probably

find ways to alter the

asteroid’s radiation pat-

tern by dusting its sur-

face with soot or

powdered chalk or drap-

ing it with reflective

Mylar. Such tinkering

could be enough to

nudge the asteroid

safely away.

—EDITORIAL DESK,“ENCOUNTER WITH AN

ASTEROID,” NEW YORK TIMES

(APRIL 8, 2002)

If you are a scientist,

you’re more likely to be

killed in a film than a

member of any other

profession, including a

Mafia hit man.

—CARL SAGAN

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Fly Me to the Moon 37

88. Nuclear Detonation

The threat of the detonation of a hydrogen nuclearwarhead by striking one with another object such as amissile or the shrapnel from a nearby explosion lurks ina scriptwriter’s creative mind for many war and adven-ture films. Suppose the nuclear warhead is aboard anICBM and is struck by an interceptor missile or thewarhead itself is penetrated by fast-moving BBs. Whatwill happen?

89. Fabric of Space-time

Conjectures about the “fabric of space-time” and“tears in the space-time continuum” abound in sciencefiction movies. An entertaining 2001 film involved aprotagonist who derived an equation for the time andplace of a temporary tear in the fabric of space-time.Several characters jumped off the Brooklyn Bridgethrough the temporary space-time tear, acting as aportal to another dimension to the year 1876, and thenreturned through the next temporary tear by jumpingoff the bridge again days later. In addition, film charac-ters have used the phrase “speed of gravity” in anambiguous way. What can you say about the physicshere?

In a sense, 1:Sun = Moon,

and 1:Moon = Sun!

1:365.242 = 0.0027379,

which in days is 3 min-

utes and 56 seconds, the

difference between side-

real and solar days, while

1:27.322 = 0.0366,

which in days is 52 min-

utes, the difference

between lunar and solar

days.

—ROBIN HEATH, SUN,MOON, & EARTH

On the average there is

one catalogued satellite

that falls back to Earth

uncontrolled every single

day and has been since

the early 1960s. Most of

them vaporize high in the

atmosphere.

The first sentence in

H. G. Wells’s 1908 novel

reads:

“Lower Manhattan was

soon a furnace of crimson

flames, from which there

was no escape.”

The War in the Air

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Go AskAlice

39

Go AskAlice

P ERHAPS NO OTHER ASPECT OF TWENTIETH-

century physics has captured the imagination of the

general public more than the concepts of the special theory

of relativity (STR). Absolute time and absolute space are

forever cast aside in favor of a union of space and time into

one important entity called space-time. This four-dimen-

sional world of space-time has spawned an enormous num-

ber of conjectures about the behavior of nature. Among

these conjectures are time travel, two people aging at differ-

ent rates when one remains on Earth and the other travels on

a space journey, the ability to see the back side of an

approaching cube, and the conversion of mass into energy.

As you know, the STR is based on the idea that two

observers in different inertial reference frames must each

experience physics described by the same basic laws. Even

though these two inertial reference frames are moving with

5

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a constant velocity with respect to each other, the speed

of light in a vacuum is the same for both observers. The

important quantities in STR are the invariants. For

many people, the most useful invariant is the space-

time interval τ, defined by τ2 = c2 ∆t2 – ∆x2 – ∆y2 – ∆z2.

For others, the four-momentum invariant E2 – p2c2 =

m2c4 is the most useful because E0 = mc2 can be derived

directly, where the mass m is a constant, the same at all

speeds, all places, and all times. Many challenges in this

chapter test your ability to use these invariants.

40 Mad about Modern Physics

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Go Ask Alice 41

90. Spotlight

Can a spot of light move faster than c, the speed oflight? For example, if a lighthouse light beacon spinsaround at very high speed, will the spot of light seen farfrom the beacon cut across the sky with a speed greaterthan 3 × 108 m/s?

91. Quasar Velocity

Quasars have been detected that have recessional veloc-ities greater than the speed of light c based on the cos-mological relationship for the redshift z, namely, l + z =exp[v/c]. That is, there are quasars with z > 3, forexample. Also, to explain the present state of the uni-verse, the inflationary big bang model requires a faster-than-light expansion of space in the young universe.Are these examples violations of the special theory ofrelativity?

92. Spaceship Approach

A spaceship is approaching Stephanie at the relativisticspeed of v/c = 0.98974. What does she see as the space-ship nears and then passes? Hint: for simplicity, con-sider a cube approaching in place of the spaceship.

93. Mass and Energy

A symbol of the twentieth century is the famous Ein-stein relation between mass and energy. Here are fourpossible equations: (1) E0 = mc2 (2) E = mc2 (3) E0 =m0c

2 (4) E = m0c2. In the equations c is the velocity of

light, E is the total energy of a free body, E0 its restenergy, m0 its rest mass, and m its mass.

Which of these equations expresses one of the mainconsequences of the STR? Which equation was first

Joseph Larmor in 1900,

stimulated directly by

the Michelson-Morley

experiment, gave for

the first time the full

Lorentz transformations

for coordinates and

time, as well as electric

and magnetic field com-

ponents, and showed

that the Maxwell equa-

tions remain exactly

invariant under these

transformations. It

has long appeared a

historical anomaly that

Larmor’s work, which

preceded Lorentz’s by

four years, is so little

known among physi-

cists. Earlier still, in

1897, Larmor had

discovered time dilation.

Woldemar Voigt’s paper,

published in 1887, con-

tains an early version of

the Lorentz transforma-

tions that appear to be

almost the same except

for a scale factor.

—C. KITTEL, “LARMOR AND

THE PREHISTORY OF THE

LORENTZ TRANSFORMATIONS,” AMERICAN JOURNAL OF PHYSICS

(SEPTEMBER 1974)

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written by Einstein and was considered by him a con-sequence of STR?

94. Strain Gauge

A long rectangular bar of metal sits at rest in my refer-ence frame. The strain gauge attached to its middlereads zero. Now I run in the direction parallel to thelength of the bar at an enormous constant speed V atnearly light speed. I measure the bar length to determinethat Lorentz-Fitzgerald contraction has occurred—thatis, that the bar measures shorter than before. Whatshould the strain gauge show?

95. Mass/Energy

Under certain conditions, mass can be converted intoenergy à la E0 = mc2. Under certain restricted condi-tions, energy can materialize as mass. What is wrong inthese statements?

96. System of Particles

A system of particles is composed of n freely movingparticles. Is the mass of this system equal to the sum ofthe masses of the individual particles?

97. Light Propagation

Suppose Patricia is driving her car at nearly the speedof light and turns on her headlights. For simplicity incalculations, in the rest frame of an observer on theground the light takes one second to reach the stop sign3 × 108 meters away. This ground observer then seesthe car reach the stop sign very soon after the initiallight reaches the stop sign.

42 Mad about Modern Physics

Albert Michelson was

born in Strzelno,

Poland, in 1852. The

town, about 150 miles

northwest of Warsaw,

was then under Pruss-

ian rule. His parents

were Samuel Michelson

and Rozalia Przylubska.

Four years later the

family immigrated to

California. Michelson,

famous for the 1887

Michelson-Morley ether

drift experiment, in

1907 became the first

naturalized American

citizen to win a Nobel

Prize in physics. The

Nobel committee

awarded the prize for

an investigation to

determine whether

wavelengths of light

could provide a standard

unit of length. The ether

drift experiment was not

even mentioned.

—ADAPTED FROM DOROTHY

MICHELSON LIVINGSTON,THE MASTER OF LIGHT:

A BIOGRAPHY OF

ALBERT A. MICHELSON

Are not gross bodies

and light convertible

into one another?

—ISAAC NEWTON

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Go Ask Alice 43

Patricia sees the light moving forward at 3 × 108

m/sec also, but she sees the stop sign approaching herat nearly light speed. Therefore she sees the arrival ofthe light flash at the stop sign and her arrival there inquick succession.

Call the initial arrival of the light at the stop signevent A and the car’s arrival event B. Will the elapsedtime between events A and B be the same for the driveras for the observer on the ground? No, because theground observer sees both events occur at the samelocation, at the stationary stop sign, so ∆x = 0. As seenby Patricia, these two events occur at two differentlocations separated by ∆x ≠ 0.

Who measures the longer time interval betweenevents A and B? Can you provide a conceptual argu-ment for this nonintuitive result? If the speed of the caris closer to the speed of light, how does the difference inelapsed times measured by driver and ground observerchange?

98. Sagnac Effect

Suppose two identical clocks are in motion on Earth’sEquator with constant speed v relative to Earth, onemoving east and one moving west around the Equator.Do they tick at the same rate? What do their elapsedtimes reveal when they meet again?

99. Light Flashes

Suppose that a spaceship travels at constant velocitybetween two planets, A and B. The spaceship sends outa light flash in all directions every 10 minutes by itsown clock reading. Traveling toward B, its light flashesare seen at 5-minute intervals on planet B. What is theflash interval time as seen on planet A? One of these

The notion of the

dependence of mass on

velocity according to

m/(1 – 2/c2)1/2 was

introduced by Lorentz in

1899 and then devel-

oped by him and others

in the years preceding

Einstein’s formulation

of special relativity in

1905.

—LEV B. OKUN, “THE

CONCEPT OF MASS,” PHYSICS

TODAY (JUNE 1989)

I’ll be so happy and

proud when we are

together and can bring

our work on relative

motion to a successful

conclusion!

—ALBERT EINSTEIN IN A LETTER

TO MILEVA MARIC, HIS FUTURE

WIFE, MARCH 27, 1901. JÜRGEN

RENN AND ROBERT SCHULMANN,EDS., ALBERT EINSTEIN, MILEVA

MAR I C : THE LOVE LETTERS

A student riding in a

train looks up and sees

Einstein sitting next to

him. Excited, he asks,

“Excuse me, Professor.

Does Boston stop at

this train?”

v

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possibilities is correct: 5-minute intervals; 10-minuteintervals; 15-minute intervals; 20-minute intervals.

100. Forces and Accelerations

In Newtonian physics, an applied contact force actingon a rigid object will accelerate the object in the samedirection as the applied force. Does this behavior holdfor applied contact forces in relativity physics (STR)?For example, if an applied contact force pushes on thesame rigid object in the direction perpendicular to thedirection of motion, will the resulting acceleration be inthe direction of the applied contact force?

101. Uniform Acceleration

Suppose an object starts at rest with respect to the labframe and undergoes a uniform acceleration a′ asmeasured by an observer on a spaceship moving at auniform velocity v with respect to the lab. In Newton-ian mechanics, for speeds where v << c, the velocityafter t ′ seconds in the moving frame has elapsed is

44 Mad about Modern Physics

In 1905 Einstein wrote

twenty-one reviews for

the

a

“journal about journals.”

In addition to publica-

tions written in German,

he also reviewed French

and Italian papers, being

familiar with both lan-

guages. Without his

work for the

he might have easily

missed the

on the occasion of

Ludwig Boltzmann’s

sixtieth birthday, which

included 117 contribu-

tions by prominent

authors and thus

offered an exceptionally

broad panorama of

physics at the beginning

of the twentieth cen-

tury. Einstein discussed

three papers from this

volume, and he probably

read the rest.

—ALBRECHT FÖLSING, ALBERT

EINSTEIN: A BIOGRAPHY

Festschrift

Beiblätter,

Annalen der Physik,

Beiblätter zu den

Sees flashevery ? min

Sends flashevery 10 min Sees flash

every 5 min

Academic disputes are

vicious because so little

is at stake.

—ANONYMOUS

A B

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Go Ask Alice 45

V′ = a′ t ′ as measured by the observer on the movingobject. This velocity is V = v + a′ t after the elapsed timet in the lab frame, because in Newtonian physics theclocks in the different frames run at the same rates.What is the velocity value in the lab frame when thespeed is allowed to become relativistic? Can the prod-uct at be greater than c in either reference frame?

102. Long Space Journey

Can a person go with a 1-g acceleration to a distantlocation 7,000 light-years away and return withoutaging more than 40 years? That is, the bathroom scalein the spaceship must show a person’s correct weightfor the whole journey. Is this feat within the realm ofscience or science fiction?

103. Head to Toe

Can relativitic effects make your feet age more slowlythan your head?

104. Neutrino Mass

Since their proposed existence in the 1930s, neutrinosand antineutrinos of all three lepton families have beenthought to have zero mass and travel at light speed toconserve energy and angular momentum in nucleardecays. In 1969 came the first hints that at least onetype of neutrino can become another type of neutrino,and a neutrino oscillation scheme was proposed. Wenow know that muon neutrinos created in Earth’satmosphere can oscillate into electron neutrinos andtau netrinos before reaching an underground detector.Why cannot all three neutrino types still have zeromass?

Henri Poincaré, building

on Lorentz’s work but

removing, at least for-

mally, certain inconsis-

tencies, arrived in 1905,

and more fully in 1906,

at the expressions

= c2/(1 – 2/c2)1/2

and = /(1 – 2/c2)1/2.

Einstein obtained the

same relations, at the

same time, on purely

kinematic grounds. These

are the well-tested and

familiar expressions of

today.

—J. DAVID JACKSON,“THE IMPACT OF SPECIAL

RELATIVITY ON THEORETICAL

PHYSICS,” PHYSICS TODAY

(MAY 1987)

Einstein simply

postulates what we

have deduced, with

some difficulty and not

altogether satisfactorily,

from the fundamental

equations of the electro-

magnetic field.

—H. A. LORENTZ, 1906,QUOTED IN ALBRECHT

FÖLSING, ALBERT EINSTEIN:A BIOGRAPHY

vmvp

vmE

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105. Spaceship Collision

Two spaceships, A and B, move toward one another oncourses for a head-on collision. According to anobserver at rest in an inertial reference frame, bothhave speed V along the x-axis. At the time of observa-tion, spaceship A is coincident with the observer—thatis, has the same x value. Spaceship B is at a distance Laway. One would like to know how much later the col-lision occurs according to the observer and accordingto an observer aboard spaceship A.

Let us propose a solution method. According to theobserver, the collision occurs when spaceship A or Btravels L/2, half the distance between them, whichrequires the elapsed time T = L/2V. Put into a betterformat, three events occur:

Event 1: X1 = 0 T1 = 0

Event 2: X2 = L T2 = 0

Event 3: X3 = L/2 T3 = L/2V

These same events can be specified in the inertial(primed) frame of spaceship A as:

Event 1′: X1′ = 0 T1′ = 0

Event 2′: X2′ = ? T2′ = ?

Event 3′: X3′ = ? T3′ = ?

106. Twin Paradox

On their twenty-first birthday, Peter leaves his twinbrother, Paul, behind on Earth and goes off in a straightline for 7 years on his own wristwatch time (2.2 × 108

seconds) at 0.96 c with respect to an inertial reference

46 Mad about Modern Physics

Einstein’s special rela-

tivity paper (“On the

Electrodynamics of

Moving Bodies”), pub-

lished in 1905,

attracted very little

attention, perhaps

partly because it was

one of a number of con-

tributions by many dif-

ferent authors in the

general field of the

electrodynamics of

moving bodies. In the

alone there are eight

papers from 1902 up to

1905 concerned with

this general problem.

Einstein himself always

insisted on this aspect

of continuity, “With

respect to the theory of

relativity it is not at all a

question of a revolution-

ary act, but of a natural

development of a line

which can be pursued

through centuries.”

—GERALD HOLTON, THEMATIC

ORIGINS OF SCIENTIFIC THOUGHT:KEPLER TO EINSTEIN

Annalen der Physik

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Go Ask Alice 47

frame at rest with respect to Earth, then reverses direc-tion, and in another 7 years of his time returns at thesame constant speed. Paul sees Peter’s wristwatch run-ning slower, so Peter ages √(1 –v2/c2 ) = 0.28 as much, or1.96 years for each direction. But Peter looks back tosee Paul’s clock running slower than his own wrist-watch, so Paul should be aging slower by 0.28 asmuch—that is, 1.96 years for each direction. On hisreturn, Peter is surprised: “I know that I aged 14 years,but Paul should have aged only 3.92 years. Why is Paulan old man with gray hair?”

From a historical

perspective, Einstein’s

recognition of = c2

(where is for “celeri-

tas,” from the Latin for

“swiftness”) did not

quite come “out of the

blue.” Already in 1881,

J. J. Thomson had cal-

culated that a charged

sphere behaves as if it

had an

mass of amount 4/3c–2

times the energy of its

Coulomb field. That set

off a quest for the

“electromagnetic mass”

of the electron—an

effort to explain its

inertia purely in terms

of the field energy. In

1900, Poincaré made

the simpler observation

that since the electro-

magnetic momentum of

radiation is 1/c2 times

the Poynting flux of

energy, radiation seems

to possess a mass

density 1/c2 times its

energy density.

—WOLFGANG RINDLER,RELATIVITY: SPECIAL, GENERAL,

AND COSMOLOGICAL

additional

c

mE

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Start Me Up

49

Start Me Up

E NGINEERING PHYSICS IS REALLY APPLIED PHYSICS,

but more general, with social, political, financial, and

aesthetic issues to be considered that are often beyond the

immediate concerns of the applied scientist. We have

included the ability to understand the microscopic behavior

of atoms and their components in solid and liquids, a knowl-

edge that has begun to reap huge benefits in improving the

materials and devices around us. In fact, we have entered the

era of ingenious devices and designer materials. A very small

sampling of the vast array of these advances is included in

the challenges and puzzles considered in this chapter.

6

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107. Air-Driven Automobile Engine

Can a normal four-cylinder gasoline engine actuallyoperate on compressed air instead of gasoline as itsenergy source?

108. Coin Tosses

The behavior of many systems and materials can be bet-ter understood by considering the random walk of par-ticles in the system. To get some “feeling” for a randomwalk, consider the following exercise. Divide a group ofpeople into two groups. Have each individual in onegroup toss a fair coin 256 times and write down insequence the outcome of each toss. Have each individualin the other group write down what they would imaginea typical sequence of 256 random tosses to be but notactually do the tossing. Collect all the papers and mixthem up thoroughly. Can you determine with reasonableaccuracy which sets of data were obtained experimen-tally? How accurate should your selection be?

109. More Coin Tosses

Suppose we are really ambitious about tossing a faircoin. Indeed, suppose we toss a fair coin 1000 times,and for each head we step one unit distance radiallyaway from a lamppost, and for each tail we step backradially the same unit distance. About how many timeswould you expect to be at the lamppost?

110. Brownian Motor

In his famous lectures, physicist Richard Feynmandiscussed the impossibility of violating the second law ofthermodynamics by a ratchet mechanism. The simplestmodel for a ratchet is an overdamped Brownian particlein an asymmetric but spatially periodic potential (with

50 Mad about Modern Physics

The earliest objection on

record to Aristotle’s

theory of falling bodies,

based on observing the

actual fall of two bodies,

is that of John Philo-

ponus (ca. 490–570),

also known as John of

Alexandria, a Christian

philosopher, scientist,

and theologian. He

writes, “If you let fall

from the same height

two weights of which one

is many times as heavy

as the other, you will

see that the ratio of the

times required for the

motion does not depend

on the ratio of the

weights, but that the

difference in time is a

very small one.” How-

ever, he was in no sense

a precursor of Stevin

and Galileo, believing as

he did that in a vacuum

the speeds of falling

bodies would indeed be

in the ratio of their

weights, and ascribing

the near equality of

their speeds in air

entirely to the resist-

ance provided by the

medium.

—J. L. REDDING, “ARISTOTLE’STHEORY OF FALLING BODIES,”AMERICAN JOURNAL OF PHYSICS

(JUNE 1978)

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Start Me Up 51

asymmetry and period L). Due to the fluctuating forcecaused by the pushing molecules of the surroundingfluid or gas, the Brownian particle may overcome thepotential barrier moving to the left or to the right. Theprobabilities for both directions are equal, and on aver-age the particle does not move. Hence building a motorthat turns thermal energy into mechanical work from asingle heat bath is impossible.

But the ratchet can be turned into a so-calledBrownian motor that seems to violate the second lawof thermodynamics. The idea is to turn the ratchetpotential on and off periodically. Under certain cir-cumstances, this action may yield directed motion evenagainst an applied force f. Indeed, this device doeswork. (No pun intended!)

Recall that a perpetuum mobile of the first kindviolates the law of conservation of energy, while a per-petuum mobile of the second kind uses the “free”energy around us in the form of heat—that is, randomthermal motion of molecules and atoms—to run anengine without fuel. Why isn’t a Brownian motor aperpetuum mobile of the second kind?

It’s important to learn

classical mechanics

before learning modern

physics so that you will

know how to wave your

hands correctly when

discussing things you

don’t quite understand.

—HOWARD GEORGI, U.S.PHYSICIST

DISCOVERY OF

KINETIC ENERGY

Huygens’ (and, inde-

pendently, Christopher

Wren’s) studies of rigid

colliding balls around

1655 led them to con-

clude that there was

something special about

the product of mass and

velocity-squared.

Remarkably, adding

values of 2 for each

ball prior to a collision

yielded a total that was

essentially the same

after the collision, even

though the velocities

had changed.

—EUGENE HECHT, “AN

HISTORICO-CRITICAL

ACCOUNT OF POTENTIAL

ENERGY: IS PE REALLY REAL?,”THE PHYSICS TEACHER

(NOVEMBER 2003)

mv

Movable wall

Asymmetric periodic potential

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111. Magnetocaloric Engine

A ferrofluid is a fluid containing small magneticparticles that respond to an applied magnetic field, so a ferrofluid becomes magnetized in the presence of themagnet. The diagram shows a closed tube loop con-taining a ferrofluid, a heat source, a strong magnet,and a heat sink all working together to act as an enginetransporting the ferrofluid around the closed loop. Itsthermal efficiency approaches the efficiency of a Carnotcycle, so demands for this device should increase.Exactly how does this engine maintain the fluid move-ment around the loop? Can a solar heating systemoperate in this way?

112. Magnetorheological Fluid

In a beaker is 250 milliliters of corn oil to which hasbeen added about 0.5 kilogram of iron filings about 1millimeter long. The mixture is stirred thoroughly anda strong horseshoe magnet is brought up to straddle thebeaker. The iron filings align with the magnetic field asexpected to magnetize the fluid mixture. What otherphysical property of the fluid changes?

113. Binary Fluids

The two possible phase diagrams show the miscible andimmiscible phases of a binary fluid, a mixture of two

52 Mad about Modern Physics

What is the origin of

the 260-day cycle in

the Mayan calendar?

According to Anthony

Aveni, the 260-day

cycle has meaning only

in tropical latitudes,

being connected with

the interval the noonday

sun spends north as

opposed o south of the

overhead position. These

intervals vary depending

on the latitude, but in

latitude 141⁄2°N , close to

the locations of the

great Maya city of

Copan and the pre-

Classic city of Izapa,

the annual cycle divides

up neatly into 105-

and 260-day periods.

—ADAPTED FROM ANTHONY

AVENI, EMPIRES OF TIME: CALEN-DARS, CLOCKS, AND CULTURES

If you go against the

grain of the universe,

you’re liable to get

splinters.

—ANONYMOUS

Magnet

Heat source

Heatsink

Ferrofluid

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Start Me Up 53

kinds of fluid, in a plot with axes of temperature versusconcentration. For example, coffee and cream are mis-cible at room temperature but oil and water are not.

Consider the 50 percent mixture in each phasediagram and start at a high temperature in the misciblephase. The diagram to the left reveals that the binaryfluids become immiscible upon being cooled, while the diagram to the right tells us that the fluids becomeimmiscible as the cooling proceeds but that evenfurther cooling brings back the miscible phase. Canboth phase diagrams represent a real binary fluid, or isone false?

114. Baseball Bats

Hitting a baseball well is not easy. Even professionalbaseball players have difficulty consistently makingsolid contact with a pitched baseball. Once hit, the dis-tance of flight of the ball is determined by its initialvelocity—that is, the initial speed and direction—whichdepends on how hard the ball has been hit by the bat.All other factors being held constant, the initial velocitycan be said to depend on the speed of the bat just beforecollision. A quicker swing would mean a faster bat

A technician named

Richard Woodbridge III

coined the phrase

“acoustic archaeology”

in the August 1969

issue of

Wood-

bridge theorized that

there were many occa-

sions when sound might

innocently get scooped

out of the air and pre-

served. For example,

when an ancient potter

typically held a flat

stick against a rotating

pot, he was accidentally

(and crudely) recording

into the clay the sounds

around him. Woodbridge

wrote about experi-

ments he performed

pulling basic noises off

a pot. Another experi-

ment involved setting up

a canvas and then

talking while making

different brush strokes,

hoping to record a

spoken word in an oil

portrait. In this fashion,

for instance, the word

“blue” was pulled off a

blue paint stroke.

—JACK HITT, “EAVESDROPPING

ON HISTORY,” NEW YORK TIMES

MAGAZINE (DECEMBER 3, 2000)

of the I.E.E.E.

Proceedings

Miscible

T T

% Fluid B % Fluid B

0 50 1000 50 100

Miscible

Immiscible

Immiscible

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speed during the collision to add distance to each hitand also allow the batter more time to judge the pitch.

There have been proposals to put shallow, pea-sizeddepressions—dimples—in the surface of a baseball batto allow a greater swing speed. Another sports object,the golf ball, already is made with dimples on its sur-face. How would these dimples affect the bat’s swingspeed?

115. Old Glass

In old castles and houses in Europe can be found win-dows with old glass in which many of the panes areslightly thicker on the bottom than at the top. What aresome possible reasons for this result, and what is themost likely reason?

116. Ferromagnetism

Why are so few substances ferromagnetic, yet practi-cally all materials exhibit paramagnetic behavior?

117. Coupled Flywheels

Conservation of angular momentum does not alwayshelp in understanding the behavior of rotating devices.The diagram shows two flywheels, 1 and 2, ofmoments of inertia I1 and I2, mounted on parallel hor-izontal shafts along with pulleys of diameters D1 andD2. The belt is slack at first, and the two flywheels are

54 Mad about Modern Physics

Many experts believe

that Egyptian pyramids

are aligned with true

north because the

more stationary stars

near the North Celestial

Pole represented

permanency and

eternal life.

The adult brain as a

whole consumes some

twenty-five watts of

power when in full

action.

The more I have studied

him, the more Newton

has receded from me.

—RICHARD WESTFALL,HISTORIAN OF SCIENCE

1 2

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running at angular velocities ω10 and ω20. Suddenly thebelt is tightened. One can write out the torque equa-tions and the angular momentum equation to get therelation I1 ω1 + I2 ω2 = k – (N – 1) I1 ω1. Here, k is aconstant of integration and N = D2 / D1, the ratio ofpulley diameters. When N = 1, angular momentum isconserved. If N ≠ 1 and ω1 changes, the angularmomentum is not conserved! Why not?

118. Superconductor Suspension

A popular physics demonstration since the late 1980sinvolves floating a small piece of high-temperaturesuperconductor, such as yttrium barium cuprate(YBa3Cu3O7), over a strong permanent magnet. Thelevitation is easy to see, and the suspended supercon-ductor rectangular solid spins rapidly about its longaxis. The demonstration is done by first cooling thesuperconductor in liquid nitrogen and then using tongsto place the piece in the air above the permanent mag-net. The repulsive force between the magnet and thesuperconductor is a demonstration of the Meissnereffect. Or is it?

119. Nanophase Copper

The hardness and strength of a metal are measured bystudying its deformation in response to an appliedforce. A metal is deformed when its crystalline atomicplanes slide over each other. An analogy may be thebump in a rug that can be pushed across the floor. Inother words, a dislocation in a plane of atoms is moveduntil a barrier is reached, such as a grain boundary,where the micron-sized grains are differently oriented.

One interesting advance in metal technology is theability to assemble nanometer-size clusters of atoms ingrain sizes of less than 100 nanometers in diameter

2 GAINS A

FACTOR OF 1/2

It was not until 1807

that Thomas Young, an

English physicist and

physician, spoke of 2 for the first time

as Then in a

textbook published in

1829 Gustave Coriolis,

a French physicist, was

the first to give the

exact modern definition

to kinetic energy and

work. He carried out a

calculation of the work

done in accelerating a

body and arrived at the

change in the quantity

By the end of

the 19th century, most

scientists were avoiding

Leibniz’s old phrase

(living force),

and using instead

“kinetic energy,” a term

introduced in 1849 by

Lord Kelvin to better

distinguish between

force and energy.

—EUGENE HECHT, “AN

HISTORICO-CRITICAL

ACCOUNT OF POTENTIAL

ENERGY: IS PE REALLY REAL?,”THE PHYSICS TEACHER

(NOVEMBER 2003)

vis viva

1/2mv2.

energy.

mv

MV

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56 Mad about Modern Physics

instead of having the micron-size grains found in atypical metal. A graph of hardness versus grain size isshown.

With grain sizes averaging about 10 nanometers,this nanophase copper metal has a hardness more thanthree times the hardness of normal copper metal. Why?

120. Head of a Pin

What is the smallest amount of charge that can sit onthe head of a pin? Some people say that the smallestnonvanishing amount of charge should be +e or –e,where e is the fundamental unit of electrical charge.What do you say?

121. Coulomb Blockade

The tunnel junction is a conductor-insulator-conductordevice. Suppose a very small tunnel junction is operatedat very low temperatures so that thermal fluctuationsdo not contribute to electron tunneling across the junc-tion. Now connect the tunnel junction to a source of

Jean Buridan (ca.

1295–ca. 1358), rector

of the University of

Paris in 1327, in his

impetus theory intro-

duced the prescient

notion that the true

measure of the motion

of an object was not

speed alone, but the

product of speed and

quantity of matter

In

an anticipation of New-

ton’s first law of motion,

he maintained that once

the initial impetus was

supplied, motion contin-

ued indefinitely. The

spheres of heaven, for

instance, having been

put in motion by God,

continued so and

required no constantly

working angels to keep

them moving.

—ISAAC ASIMOV, ASIMOV’SBIOGRAPHICAL ENCYCLOPEDIA

OF SCIENCE AND TECHNOLOGY,2ND REV. ED.

(quantitas materiae).

Hardness (GigaPascals)

Log of Grain Size (nm)

3

2

1

0

0 1 2 3 4 5

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Start Me Up 57

constant electrical charge. Will the flow of currentacross the junction be steady?

122. Deterministic Competition

Consider a simplified system, one that can be describedby Nt objects at time t. For example, one could considerthe number of grasshoppers on the plains of Africa, oron some small plot of land. Let there be competitionbetween the growth processes and the decay processesso that the number of objects at time t + 1 is Nt + 1 = Nt

exp[r (1 – Nt)], an exponential growth relationship.This equation is deterministic, for Nt determines Nt + 1

unambiguously. One can think of r as a measure of theratio between growth and decay. Numerous mechani-cal, hydrodynamic, chemical, and electrical systems canbe approximately modeled by this relationship.

How does the number of objects behave withelapsed time? If Nt = 1, then N remains 1 forever. In thegeneral case, we can determine Nt as t → ∞ to find outwhether N approaches the equilibrium value 1. Forinstance, let r = 1 and begin with N0 = 0.5, and calcu-late with a calculator or personal computer. Now trydifferent values for r. What behavior do you predict?

123. Two Identical Chaotic

Systems

A chaotic system exhibits a sensitivity to initial condi-tions and will evolve rapidly and deterministicallytoward different end states if begun in slightly differentstates. Although the chaos is unpredictable, each possibleoutcome is deterministic—that is, an orderly behavior.

Consider two identical chaotic systems isolatedfrom each other. They will quickly fall out of stepbecause any slight difference between them would bemagnified. Assume that these systems have several

Descartes regarded

the conservation of

momentum

as divinely

ordained. He wrote:

“[God] set in motion in

many different ways

the parts of matter

when He created them,

and since He main-

tained them with the

same behavior and with

the same laws as He

laid upon them in their

creation. He conserves

continually in this mat-

ter an equal quantity of

motion.”

Earth is gradually slow-

ing down; the day is

about 16 milliseconds

longer now than it was

1,000 years ago. This

slowing is due largely to

frictional tidal effects

of the Moon on Earth’s

oceans.

motion)

(quantity of

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58 Mad about Modern Physics

parts and that at least one of the parts is stable—that is, subjected to a perturbation, the part’s behaviorchanges a little but settles back to its normal operation.Now drive both systems with the same chaotic signalapplied to the same stable part. Can the two systems besynchronized?

124. Tilley’s Circuit

This electrical circuit near the permanent magnet hastwo ideal switches and a galvanometer. When switch Ais closed and switch B, on the right, is opened, there isa large change in the magnetic flux in the galvanometercircuit. What do you predict the galvanometer responsewill be?

125. Thermal Energy Flow

If two identical bodies at different temperatures are incontact, thermal energy will always flow from one tothe other in such a direction as to increase the totalentropy. In which direction will this flow be? Thatdepends on two factors, the amount of energy andentropy the two bodies already contain. The secondlaw of thermodynamics implies that thermal energymust flow toward the region of lower temperature—that is, each unit of thermal energy acquires greater dis-order as it moves into the cooler region. Why?

In the fall of 1915 it was

widely expected that the

Nobel Prize in Physics

was to be jointly shared

by Edison and Tesla.

Then a Reuters dispatch

from Stockholm dropped

a bombshell. The Nobel

Committee announced

that the prize for

physics would in fact be

shared by William Henry

Bragg and his son W. L.

Bragg . . . . What had

happened? The Nobel

Prize Foundation

declined to clarify. One

biographer reported

years later that the

Serbo-American had

declined the honor,

stating that as a discov-

erer he could not share

the prize with a mere

inventor. Yet another

biographer advanced the

theory that it was Edison

who objected to sharing

the prize. . . . The Nobel

Foundation said simply,

“Any rumor that a person

has not been given a

Nobel Prize because he

has made known his

intention to refuse the

award is ridiculous.”

—MARGARET CHENEY,TESLA: MAN OUT OF TIME

Magnet

A B

G

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Start Me Up 59

126. Cadmium Selenide

When atoms are arranged in nanometer-size clusters ofdiameters from less than 100 nanometers to as large as700 nanometers, interesting optical properties can bedemonstrated. For example, nanophase versions ofpure cadmium selenide can be made almost any colorin the spectrum simply by changing its cluster size.Indeed, some types of lipstick are made in many differ-ent colors even though the predominant light-scatteringmolecule is the same in all color versions. What is thephysics here?

127. Optical Solitons

A light pulse is a continuum of optical carriers of dif-ferent frequencies. Optical media are dispersive, sothese carriers in the light pulse travel at different veloc-ities, causing the energy to spread over time and dis-tance. In addition, there is the optical Kerr effect, which“instantaneously” increases the refractive index of themedium by an amount proportional to the opticalpower. Can one use these two effects—dispersion andthe Kerr effect—to ensure that a light pulse retains itsintegrity while traveling thousands of kilometersthrough an optical fiber?

128. Ceramic Light Response

Certain ceramic materials will change their shape uponexposure to light. What is the physics here?

129. Random Movements

Supposedly, research has revealed that random move-ments help explain how a tightrope walker stays aloft,for instance. If understood, robotics engineers could

Is Galileo a beneficiary of

the Matthew effect? The

latter is a term introduced in

1968 by Robert K. Merton

(1910–2003), a U.S. sociol-

ogist of science, that refers

to the disproportionately

great credit given to eminent

scientists for their contribu-

tions to science, while rela-

tively unknown ones tend to

get disproportionately little

for their occasionally com-

parable contributions. The

term derives, of course,

from the Gospel according

to Matthew (13:12 and

25:29). In the New King

James Version the passage

reads: “For whoever has, to

him more will be given, and

he will have abundance; but

whoever does not have, even

what he has will be taken

away from him.” Recognition

tends to go to those who are

already famous. In Galileo’s

case, both Philoponus in

the sixth century and the

Belgian–Dutch scientist

Simon Stevin in 1586 per-

formed the key experiment

of dropping two different

weights simultaneously and

observed that they struck

the ground at the same

time—the experiment that

today seems indissolubly, if

incorrectly, wedded to the

name of Galileo.

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60 Mad about Modern Physics

make their machines more stable by injecting a littlenoise into their systems. And persons having difficultywalking may be able to let some noisy vibrating shoesoles help them walk confidently again. What could bethe physics here?

130. Gravitational Twins

Engineering physics involves the transport of peopleand materials in space as well as practical applicationshere on Earth. So consider a pair of twins in free fall.Imagine that one twin is in circular orbit around a starand that her sister is shot out from this circular orbitlocation on a radial orbit—that is, the traveling twinwill fall back to meet up with the stay-at-home sister incircular orbit. For simplicity, let them meet after anintegral number of revolutions around the circularorbit for the one left behind.

Any clock system in a gravitational potential, suchas the clocks in the GPS system here on Earth, dependson two relativistic effects on the clock rate: (1) a clockticks slower closer to a massive object than when faraway, and (2) the faster-moving clock ticks slower thanthe slower-moving clock.

Initially, the clock rates of the twins are the samebecause they start out in the same circular orbit at thesame radial distance from the star. The traveling twinmoves away from the star along the radial line, all thewhile slowing down and eventually coming to a momen-tary stop and returning with ever-increasing speed untilrejoining her sister in orbit. So on average, the travelingtwin experiences a smaller amount of gravitational timedilation and a smaller amount of speed time dilationthan her stay-at-home sister. Therefore the travelingtwin returns home older than her sister, because herclock ticked faster on average. What do you think?

In Galileo’s time a

general impression

prevailed that a falling

body gained speed in

proportion to the dis-

tance through which it

fell, and Galileo himself

held this opinion for a

time. It appears also

that he abandoned this

idea, not because of

contrary results of

experiment, but

because deductive rea-

soning (not without a

flaw) had led him to the

impossible conclusion

that a body governed by

this law would fall

through a long distance

in the same time as

through a short one. He

therefore abandoned

the space-acceleration

idea and considered the

possibility that the gain

in speed was propor-

tional to the time of fall.

—PAUL R. HEYL,“TRANSCENDENTAL MECHANICS,”

AMERICAN JOURNAL OF PHYSICS

(MAY 1941)

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Start Me Up 61

131. Photon Engine

The ideal Carnot heat engine converts heat to work without the engine itself being a source of anywork. The reversible closed Carnot cycle consists oftwo isothermal (constant temperature) processes and two adiabatic (no external exchange of thermal energy) processes. No heat engine operating betweentwo temperatures can be more efficient than a Carnot cycle.

But Carnot could be wrong. The challenger is thenew “quantum Carnot engine,” in which the radiationpressure from photons drives a piston in an opticalcavity. The inward-facing surface of the piston ismirrored and the other cavity mirror is fixed in placewhile exchanging thermal energy with a heat sink attemperature T1. A second heat bath at a highertemperature, T2, provides the source of thermal energyfor the photons.

This source of thermal energy is a stream of hotatoms, which flows into the optical cavity andexchanges thermal energy with the photons throughemission and absorption processes. These atoms exitthe cavity at a cooler temperature and are reheated to

Nikola Tesla (1856–1943),

the Serbian-born inventor

of the first practical

alternating-current

dynamo and power trans-

mission system, was

known for unusual powers

of visualization. He was

able to construct, modify,

and even operate his

imaginary devices, purely

by visualizing them. He

wrote in “My Inventions”

1919), “It is absolutely

immaterial to me whether

I run my turbine in

thought or test it in my

shop. There is no differ-

ence whatever, the results

are the same. In this way I

am able to rapidly develop

and perfect a conception

without touching anything.

When I have gone so far

as to embody in the

invention every possible

improvement I can think

of and see no fault any-

where, I put into concrete

form this final product of

my brain. Invariably my

device works as I con-

ceived that it should,

and the experiment

comes out exactly as I

planned it. In twenty

years there has not been

a single exception.”

(Electrical Experimenter,Mirrors

Piston

Photon beam

Hot atoms

T1

T2

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T2 in a second cavity, to be reinjected into the firstcavity for the next cycle of the quantum Carnot engine.

Therefore, the quantum and classical Carnotengines operate in the same way as a closed cycle of two isothermal and two adiabatic processes. How-ever, in its simplest form, when each bath atom istreated as a two-state system, the quantum Carnotengine cannot extract work from a single heat bath.Why not? Will the engine work if each bath atom is athree-state system?

62 Mad about Modern Physics

The Earth is a somewhatirregular clock. Someyears the length of theday is found to vary byas much as one part in10 million, or three sec-onds in a year of 31.5million seconds. In addi-tion, there are also sea-sonal fluctuations of afew milliseconds peryear. In the winter theEarth slows down, and inthe summer it speedsup. Think of the Earthas a spinning skater.During the winter in thenorthern hemisphere,water evaporates fromthe ocean and accumu-lates as ice and snow onthe high mountains. Thismovement of waterfrom the oceans to themountaintops is similarto the skater’s extend-ing her arms. So theEarth slows down in winter; by the summerthe snow melts and runsback to the seas, andthe Earth speeds upagain. This effect is notcompensated by theopposite effect in thesouthern hemispherebecause most of theland mass is north ofthe equator.

—JAMES JESPERSEN AND JANE

FITZ-RANDOLPH, FROM

SUNDIALS TO ATOMIC CLOCKS

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A WholeNew World

63

A WholeNew World

A TOMIC PHYSICS BEGAN IN THE 1840S WITH

the identification of the emission lines of hydrogen

and of other atoms and ions in laboratory sources and in the

solar spectrum. In the early 1900s, the Bohr-Sommerfeld

model of the atom was the paradigm, but numerous prob-

lems with its predictions existed that were finally resolved

with the advent of quantum mechanics in 1925. The electron

in the atom occupies particular quantized energy states of

unequal energy spacing, and selection rules based on conser-

vation of energy and angular momentum dictate which

jumps between states to an available final state are allowed.

In addition to a spontaneous electron jump to a lower energy

level with the emission of a photon, external photons with

the correct energy can stimulate the atomic absorption or

emission of photons. The eventual application of quantum

mechanics to the simple molecules proved very successful, if

7

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not challenging, and today faster computers continue

to calculate the properties of atoms, inorganic and

organic molecules, and very large biomolecules such as

DNA and proteins. Enormous progress has also been

made in understanding the fundamental properties of

condensed matter of fluids and solids such as crystals,

ionically doped materials, plastics, pseudocrystals, and

so on. Our lives are becoming more dependent on the

practical devices arising from this great endeavor called

molecular design. The challenges introduced in this

chapter are but a small sample of the wide range of

possible problems.

64 Mad about Modern Physics

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A Whole New World 65

132. Grain of Sand

If the atoms in a grain of sand were laid out side by sidein a line, approximately how long would the line be?

133. Forensics

Historically, paintings could be verified with reasonableassurance of authorship by experts who knew thebrushstrokes and color and paint choices of the artist aswell as the overall style and character of the subjects.However, in some cases, fraudulent artworks have beensuccessfully passed as genuine. New techniques forassessing all types of artwork are always needed, andthe scientific community has been answering the call.One scientific technique for checking the authenticityof old paintings uses laser lights. How might this featbe accomplished?

134. Doppler Elimination?

When an atom emits or absorbs a photon, there isalways a recoil of the atom and a Doppler shift in thephoton frequency. Is it possible to have recoillessatomic emission or absorption?

Both Niels Bohr and

his wife had a similar

response to religion:

Margrethe has written

about Niels’: “There

was a period of about a

year . . . [he was] 14 or

15 . . . where he took it

all very seriously; he got

taken by it. Then sud-

denly it was all over. It

was nothing for him.”

About her own feelings,

Margrethe reported:

“You know it was often

at that age . . . that

one got very religious

and would listen to the

minister about confir-

mation. Then it all

dissolved. And for me

it was exactly the

same; it disappeared

completely.”

—LÉON ROSENFELD,“BOHR, NIELS HENRIK DAVID,”

DICTIONARY OF SCIENTIFIC

BIOGRAPHY, VOL. 2

Intensity

Frequency

Doppler shifted

Natural width

Doppler-broadened

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135. Light Tweezer

In science fiction movies we often see light beams shotout from handheld light guns supplying a tremendousimpulse to knock over an enemy storm trooperapproaching along the direction of the beam. By New-ton’s third law, the light gun itself should have experi-enced an equivalent recoil! We know that a ray of lighthas energy and linear momentum, so its impingence onany surface will produce a slight backward movementof that surface. However, we would like to knowwhether a light beam can be used to physically move atiny object, such as a small one-celled animal, in adirection perpendicular to the beam.

136. Fluorescent Lights

The gas plasma inside a fluorescent tube emits mostlyultraviolet radiation and very little visible radiation.Electrons are captured by the ions and jump down tolower energies, emitting a characteristic UV photon foreach fluorescence jump. Why are fluorescent tubes somuch more efficient in producing visible light thanincandescent lamps?

66 Mad about Modern Physics

FIELD THEORY IN THE

EIGHTEENTH CENTURY?

In many ways this major

advance had its origin in

1758 with the publication

of the immensely influen-

tial

by Roger

Joseph Boscovich

(1711–1787), a theory of

such importance that

nearly 150 years later

Lord Kelvin could describe

himself as a “true Boscov-

ichean.” Boscovich was

born Rudjer Josip

Boskovic in the Republic

of Dubrovnik. Today he is

claimed with equal vigor by

the Croats, the Serbs, and

the Dalmatians. As Mar-

garet Wertheim writes in

her

“His nationality is surely a

significant part of the

reason that this visionary

physicist isn’t more

famous today—it is diffi-

cult to imagine that any

Anglo-Saxon scientist of

such caliber would have

remained so outside the

spotlight.” Trained as a

Jesuit priest, he became a

professor of mathematics

in Rome, and over the

course of his lifetime

published over a hundred

books and papers, most of

Pythagoras’ Trousers,

Philosophy

Theory of Natural

Laser light

2

2

1

1

F2

F1

Lens

f

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A Whole New World 67

Why can the light from some fluorescent lights bedangerous to your health? Could it be that they emitsome UV? Are there types of fluorescent tubes that arebetter for human working environments? Are they bet-ter because they do not emit in the UV?

137. Phase Conjugation Mirror

Can a light wave pass through a disturbing medium, bedistorted, reflect off a special mirror, and return to thesource undisturbed?

138. Stationary States

In the Bohr model of the hydrogen atom, the angularmomentum for the orbital motion of the electron ofmass m at distance r is quantized in integral units ofPlanck’s constant h—that is, assuming the proton posi-tion to be fixed, mvr = nh/2π, where n is an integer andv the electron velocity. Using mv = h/λ, de Broglie wasable to derive Bohr’s quantization rule and nλ = 2πr. Iff1 and f2 are the frequencies of the Bohr orbital motionof the electron in energy states E1 and E2, then if anelectron jumps down from state 2 to state 1, why isn’tthe energy of the emitted photon the difference energyhf1 – hf2?

139. Angular Momentum

In classical calculations, the quantity that often appearsin the result is the square of the angular momentum J2.One can often guess at the correct quantum mechanicalformula by replacing J2 by j (j + 1) h2/4π2, where j is thez-component of the angular momentum and h isPlanck’s constant. Why is the square of the angularmomentum in quantum mechanics proportional to j ( j + 1) instead of just j 2?

which still remain

untranslated from the

Latin. To get rid of the

“spooky action at a dis-

tance,” he introduced

the proposition that

atoms have no size;

they are geometrical

“points of force” that in

turn create fields of

force, an idea later

elaborated on by Fara-

day. Moreover, he sug-

gested that all these

atomic forces along with

gravity, must be aspects

of one all-encompass-

ing universal force, an

eighteenth century ver-

sion of the “theory of

everything!”

—ADAPTED FROM LESLIE HOLLI-DAY, “EARLY VIEWS ON FORCES

BETWEEN ATOMS,” SCIENTIFIC

AMERICAN (MAY 1970)

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140. Kinetic Laser

A traditional laser involves the stimulated downwardelectron transition in an atom in a background “sea ofphotons,” with the emission of a characteristic photonmatching in fequency and momentum the stimulatingphotons. This stimulated emission process was predictedby Einstein. In 1951, J. Weber at the University of Mary-land was the first to calculate the operating principles ofthe ammonium maser and laser. However, as the storygoes, upon asking for research monies to build themaser, a few hundred thousand dollars from the univer-sity, he lost out to the athletic department’s request formoney to build up the Maryland football program.

The first operating ammonium maser was subse-quently built by C. Townes in 1954, and the first oper-ating device lasing in the optical part of the spectrumwas built in 1960 by T. H. Maiman. Laser action firstin the microwave region is no coincidence, for sponta-neous emission is proportional to the cube of the tran-sition frequency, and being extremely small in this partof the spectrum, can be neglected compared to stimu-lated emission and absorption.

Among the more exotic lasers is the kinetic laser,which is an “exploding” material that emits light andX-rays. In its simplest form, the material would be afoil of a single element such as copper that is explodedby focusing powerful laser pulses on it. How does thistype of laser produce coherent laser light?

141. Noninversion Laser

For decades, lasers have been explained as the result ofan inverted population of states with stimulated emissionof photons in a high-Q cavity. However, lasers can bemade without an inverted population. Can you explainhow this type of stimulated emission process works?

68 Mad about Modern Physics

Five Venus rounds

(synodic periods) equal

eight solar years within

about two days, and

equal also 99 lunar

months with a discrep-

ancy of less than four

days. Specifically,

5 × 583.9 = 2919.5,

8 × 365.25 = 2922.0

99 × 29.53 = 2923.5.

For ancient astronomers

this was evidence of

profound unity and even

preestablished harmony

within the cosmos.

We must be clear that,

when it comes to atoms,

language can be used

only as in poetry.

—NIELS BOHR

If all the empty space

were squeezed out of a

person, the amount of

solid matter remaining

would be no larger than

a speck of dust.

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A Whole New World 69

142. X-ray Paradox

The index of refraction n gives the ratio c/v, the speedof light in vacuum to the speed of the electromagneticwave in the material. Window glass, for example, canhave an index of about n = 1.5 for visible light, with aslight variation in the index with the color of the light.A paradox arises with X-rays because they have anindex of refraction value less than one in crystals! Whatdoes this behavior mean?

143. Benzene Ring

The benzene molecule is aring of six carbon atoms,each C atom having one Hatom attached. There is amystery about the energycontained in this molecule.The benzene ring can bebroken up into pieces, andchemists have measuredthe energies associatedwith the pieces and withthe single bonds and the double bonds by studying ethylene and so on. The expected total energy can becalculated from these data, but the actual total energyof the benzene ring is much lower, telling us that thecarbon atoms are much more tightly bound. Therefore,the bond picture would make the benzene ring easilysusceptible to chemical attack, yet the molecule is quiteresilient to breaking up.

Using the Schrödinger equation by considering eachcarbon atom on this ring as the potential home for asingle electron, one can calculate the possible energylevels for the benzene ring. Why does this method ofcalculation work?

Why do all FM radio

stations end in an odd

number? FM radio sta-

tions all transmit in a

band between 88 MHz

and 108 MHz. Inside

that band, each station

occupies a 200 kHz

slice, and all of the

slices start on odd num-

ber frequencies. This is

completely arbitrary. In

Europe, the FM stations

are spaced 100 kHz

apart, and their fre-

quencies can end on

even or odd numbers.

When Einstein regis-

tered for the draft in

Switzerland at the age

of 22, his height was

recorded as five feet

seven and a half inches.

His contemporaries

regarded him as tall. By

way of comparison,

Isaac Newton is thought

to have been about five

feet five inches tall.

—ADAPTED FROM BARRY

PARKER, EINSTEIN: THE

PASSIONS OF A SCIENTIST

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144. Graphite

Atoms in a crystal make a regular array if there are nodislocations. Most pure single-element crystals have acubic or a diamond crystal structure, with all orthogonaldirections showing the same structural spacing. Even fora pure element substance, however, the spacing may bedifferent in different directions. For example, take car-bon atoms, which probably are components of morethan 75 percent of all known compounds. In diamondthey have the same structure in all orthogonal direc-tions, but in graphite the third direction is definitelyquite different than the other two directions, whichdefine a plane of hexagonal carbon rings. How can thisthird direction be so different in an originally nonbiasedenvironment?

145. Ozone Layer

We’ve heard so much in the past few decades about theozone layer in the upper atmosphere and its possibledemise. Yet ozone is only a minor greenhouse gas, farbehind carbon dioxide, HOH vapor, and methane inoverall importance. So why is there all this fuss over theozone layer?

70 Mad about Modern Physics

Niels Bohr discovered

his ideas in the act of

enunciating them, shap-

ing thoughts as they

came out of his mouth.

Friends, colleagues,

graduate students, all

had Bohr gently entice

them into long walks in

the countryside around

Copenhagen, the heavy

clouds scudding over-

head as Bohr thrust his

hands into his overcoat

pockets and settled into

an endless, hesitant,

recondite, barely audi-

ble monologue. While he

spoke, he watched his

listeners’ reactions,

eager to establish a

bond in a shared effort

to articulate.

—ROBERT P. CREASE AND

CHARLES C. MANN, THE

SECOND CREATION: MAKERS OF

THE REVOLUTION IN

20TH-CENTURY PHYSICS

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A Whole New World 71

146. Greenhouse Gases

Why are the greenhouse gases carbon dioxide, HOHvapor, and methane important for human survival onEarth? If they are good for our existence, shouldn’thaving more carbon dioxide, etc., in the atmosphere beencouraged?

147. LED vs. LCD

An LED is a semiconductor device that emits visiblelight when an electric current passes through it. The light is not particularly bright and usually monochromatic, occurring at a single wavelength. TheLED light output range is from infrared and red toblue-violet. The LCD is a type of display used in digitalwatches and many portable computers that utilizes twosheets of polarizing material with a liquid crystal solution between them. An electric current passedthrough the liquid causes the crystals to align so thatlight cannot pass through, each crystal acting like ashutter, either allowing light to pass through or block-ing the light.

What is the difference in energy requirements in the operation of a light-emitting diode (LED) and a liq-uid crystal display (LCD)? After all, they both requireenergy to operate. And how is a plasma display differ-ent from both of them in its energy requirements?

148. Sonoluminescence

Sound energy is converted directly into light energy bya phenomenon called sonoluminescence. Discovered inthe 1800s, the process lay dormant for more than 100years, only to experience a revival in the 1990s. Howdoes one convert a small amount of sound energy intoa brief but brilliant flash of light?

The special theory of

relativity predicts that,

for an observer moving

at the speed of light,

distance traveled

shrinks to zero while

time slows to a stand-

still. Thus, as far as the

light itself is concerned,

it does not travel any

distance, and takes no

time to do so. As Gilbert

Lewis showed back in

1926 ( vol. 117),

from light’s point of view

the Universe is so “bent”

that there is no separa-

tion between the point of

emission of light and its

point of absorption. . .

If light does not expe-

rience itself to have

traveled any distance, it

does not need a vehicle

or mechanism by which

to travel. . . . It is only

in our frame of refer-

ence—the frame of

observers with mass who

move at sub-light

speeds—that light

appears to travel

through space and time;

and only in that frame

does the question of

whether it is a wave, a

particle or both arise.

—PETER RUSSELL, “HERE ISTHERE,” “LETTERS,” NEW

SCIENTIST

(NOVEMBER 23, 1991)

Nature,

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149. Siphoning Liquid Helium

Liquid helium can crawl up the wall of its containerwithout any additional help. How is this feat accom-plished?

150. Quantized Hall Effect

The Hall effect was discovered by Edwin Hall in 1879.“. . . A charged particle moving in a magnetic field feelsa ‘Lorentz’ force perpendicular to its direction ofmotion and the magnetic field. As a direct consequenceof this Lorentz force, charged particles will accumulateto one side of a wire if you send current through it andhold it still in a [perpendicular] magnetic field. . . .”When the transverse voltage is measured at a fixed cur-rent, the Hall resistance is measured and increases lin-early with an applied magnetic field.

The conduction electrons in a solid behave like agas of electrons. So the discovery of the quantized Halleffect in 1980 by von Klitzing and his research groupwhen he was investigating the conductance propertiesof two-dimensional electron gases at very low temper-atures and high magnetic fields was a surprise. What isthe physics behind this quantized Hall effect?

72 Mad about Modern Physics

Glass dewar

Liquid helium

Isidor Isaac Rabi, (Nobel

Prize in Physics in 1944)

faced a crisis of faith

while only a child.

He decided to put the

tenets of Orthodox

Judaism to the test.

Jewish law prohibited

riding streetcars on the

Sabbath, so one Sabbath

he climbed onto a street-

car, just to see what would

happen. Emboldened by

an uneventful ride, he

conducted another test:

“I remember being in the

synagogue and the priests

. . . would stand up and

with their hands out-

stretched, they would

bless the congregation.

You were not supposed to

look at their hands, you

might go blind if you did.

Well I tried . . . with one

eye.” His sight left unim-

paired, he dismissed

Jewish faith as supersti-

tion but, to please family

and friends, consented to

a bar mitzvah, at which he

ended up lecturing them

in Yiddish on the workings

of the electric light bulb.

—PAUL HOFFMAN, “THE

HIGHER TRUTH OF PHYSICS,” AREVIEW OF RABI: SCIENTIST AND

CITIZEN BY JOHN S. RIGDEN,NEW YORK TIMES BOOK

REVIEW (MAY 10, 1987)

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A Whole New World 73

151. Integrated Circuits

As integrated circuits (ICs) become crowded with more semiconductor devices and internal connections,one wonders how they will be connected to theexternal world. We know also that cosmic rays andother particle radiation from the environment alreadydisrupt some of the operations by random destruction,and these effects will become worse as the scale dimin-ishes. However, neither connection to the externalworld via gold wires of any size nor the backgroundparticle radiation is the major problem today. What is?

152. Atomic Computers?

Atoms are busy collections of electrons and nuclearparticles that are ever changing their positions in a ran-dom dance. In contrast, information storage requiresstable states over reasonable time intervals. Can infor-mation be stored on individual atoms in their restlessworld?

153. X-ray Laser?

We know that there exist free electron X-ray lasers thathave an electron moving past a rippled surface andemitting X-rays, as well as X-ray laser sources based on

Planetary atomic mod-

els were already popular

a few years before

Rutherford’s proposal.

The most elaborate

attempt was that of

Hantaro Nagaoka,

whose “Saturnian”

model was published in

1904. Nagaoka’s model

was astronomically

inspired, in the sense

that it closely relied on

Maxwell’s 1856 analysis

of the stability of Sat-

urn’s rings. The Japan-

ese physicist assumed

that the electrons were

placed uniformly on

rings moving around the

attractive center of a

positive nucleus.

Nagaoka’s calculations

led to suggestive spec-

tral formulas and a

qualitative explanation

of radioactivity. The

model was, however,

severely criticized, and

disappeared from the

scene only to reappear

in an entirely different

dressing with Ruther-

ford’s nuclear theory.

—HELGE KRAGH, QUANTUM

GENERATIONS: A HISTORY

OF PHYSICS IN THE

TWENTIETH CENTURY

Cu-W

Cu(111)

Cu X-rays

X-ray tube

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plasmas such as the kinetic laser. However, an X-raylaser with a wavelength of about 1 Å or 0.1 nanometeror less that can be operated on a tabletop would beconvenient and would be able to resolve details downto nearly 1 wavelength. The uses in physics and medi-cine are expected to be many.

A very interesting tabletop device is the workingmonochromatic X-ray source shown in the illustrationthat emits very intense, narrow beams at the Cu 1.54 Åcharacteristic emission line known as Kα1. There is aspecial bimetal X-ray tube source of Cu-W that emitsX-rays from both metals upon bombardment withhigh-energy electrons in the standard way. These X-rays exit the tube and then Bragg scatter in an exter-nal Cu crystal to produce a very narrow, intense beamof Cu Kα1 X-rays. The first surprise is the enormousline intensity at a single wavelength, and the secondsurprise is that no Cu Kα2 X-rays appear in the outputfrom the external crystal. How does the external crys-tal affect the X-ray beam? Is this device an X-ray laseror a super-radiant X-ray source?

154. Bose-Einstein Condensate

A Bose-Einstein condensate is a new form of mattermade at the coldest temperatures in the universe. Essen-tially the condensate is a collection of identical atomsbehaving as one entity. How do the individual atomslose their self-identity?

155. Quantum Dots

Quantum dots are crystals containing only a few hun-dred atoms and when illuminated with UV light, forexample, will fluoresce at only one specific wavelengthof light. Why does the dot emit only one wavelength oflight when excited?

74 Mad about Modern Physics

There was a time when

physics and philosophy

were allied disciplines.

However, Niels Bohr’s

three long historic papers

on the structure of the

hydrogen atom were

published in 1913 in

The journal, first pub-

lished in 1798, was at

that time accepting

articles from most

branches of science.

This alliance began to

break up at the end of

the nineteenth century.

Today, even though the

word “philosophical”

persists in its title, it is

devoted primarily to

condensed-matter

physics.

Things that cannot go on

forever don’t.

—ANONYMOUS

Philosophical Magazine.

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ChancesAre

75

ChancesAre

QUANTUM MECHANICS (QM) ORIGINATED IN

1925 as a theory to understand the internal

behavior of the hydrogen atom. Since then, QM has

evolved to encompass the behavior of practically everything.

In its most rudimentary version, QM is based on three

fundamental rules. The main idea of QM is not quantized

energy and quantized angular momentum, for the classical

physics of strings, tubes, drumheads, and so on, involve

quantized states of energy and angular momentum.

The heart of QM is the coherent superposition of states,

as given in rule 2 below. From The Feynman Lectures on

Physics, the three fundamental rules of QM are:

1. The probability P of an event in an ideal experiment is

given by the square of the absolute value of a complex

number ψ, which is called the probability amplitude (or

wave function):

P = |ψ|2.

8

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2. When an event can occur in several alternative

ways, the probability amplitude ψ for the event is

the sum of the probability amplitudes ψ1, ψ2, ψ3

. . . , for each way considered separately; that is,

there is superposition and interference:

ψ = ψ1 + ψ2 + ψ3 + . . .

P = | ψ1 + ψ2 + ψ3 + . . . |2

3. If an experiment is performed (or could be done)

that can determine whether one or another alterna-

tive is actually taken, the probability of the event is

the (classical) sum of the probabilities for each

alternative; that is, the interference is lost:

P = P1 + P2 + P3 + . . .

We have no knowledge about a more basic mecha-

nism from which these rules can be deduced. Numer-

ous tests have verified their fundamental validity over

and over. You will need to apply them in the challenges.

76 Mad about Modern Physics

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Chances Are 77

156. Schizophrenic Playing Card

An ideal playing card stands perfectly balanced on itsedge. According to the rules of quantum mechanics,this card will fall in both directions at once! That is, thefinal state of the card is the superposition of the twoalternative falling directions, with ψ1 for left and ψ2 forright. The card’s wave function changes smoothly andcontinuously from the balanced state to the mysteriousfinal state Ψ = ψ1 + ψ2 with two alternatives that seemto have the card in two places at once. Why haven’t weseen this happen in the everyday world around us?

157. Schrödinger’s Cat

In one version of the famous Schrödinger cat gedankenexperiment, a healthy cat is placed inside an ideal catplayroom that is isolated from the rest of the worldwhenever the door is closed. Inside is one deadly objectleft by mistake. The door is closed. After some timeelapses, one wonders whether the cat is alive or dead,the two classical possibilities. Rule 2 of QM tells us,however, that the state of the cat is Ψ = ψ1 + ψ2, whereψ1 means alive and ψ2 means dead. So QM requires usto consider the cat as being alive and dead simultane-ously! However, you are curious. You push a button

For centuries, Britain

and its colonies rang in

the New Year on March

25, Annunciation Day,

when according to the

biblical account the

angel Gabriel announced

to the Virgin Mary that

she would bear the child

of God. March 25 is

nine months before

Christmas.

—DUNCAN STEEL, MARKING

TIME: THE EPIC QUEST

TO INVENT THE PERFECT

CALENDAR

If you stood on the

moon’s near side, you

would see the Earth

suspended against the

stars more or less in

the same direction

with respect to your

horizon—never rising

or setting. But the Earth

as seen from the moon

would exhibit phases

over the course of a

month, just as the moon

does as seen from

Earth.

—MICHAEL ZEILIK AND JOHN

GAUSTAD, ASTRONOMY:THE COSMIC PERSPECTIVE

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that opens the door just enough so you could look in todetermine the status of the cat. You could peek in butyou decide not to. Now what does QM predict for Ψ?

158. Wave Functions

Wave functions can be functions of many differentphysical parameters of the system of interest. Forexample, one can define a wave function in coordinatespace, in momentum space, in spin space, and so on aslong as the unit vectors of the space are orthogonal. Fora single particle, the wave function ψ(x1,y1,z1) is theQM amplitude for finding the particle at the three-dimensional configuration space point (x1,y1,z1), whichdirectly corresponds one-to-one to position space coor-dinates x1, y1, and z1 for this one-particle system. Forthe two-particle system, the wave function ψ(x1,y1,z1;x2,y2,z2) defines a six-dimensional configuration space.Is there a direct correspondence to three-dimensionalposition space coordinates for this two-particle wavefunction as well? What about the multiparticle wavefunction?

159. Wave Function Collapse?

Consider an electron in a box. Imagine partitioning thebox into N identical cubes and assume that theamplitude Ψ for finding the electron in the box is the superposition Ψ = ψ1 + ψ2 + ψ3 + . . . , that is, thesum over all N imagined identical cubes in the box.Now use a photon to observe where the electron is byrecording the scattered photon and so on. Supposeyour incident probe photon passes right through thebox and does not interact with the electron, which youdetermine because the photon took a straight-line pathto your detector. What happens to the electron wavefunction Ψ?

78 Mad about Modern Physics

The bourgeois ambiva-lence of Werner Heisen-berg’s childhood mayhave played a role in hisown adult ambivalencetoward the sweepingclaims of every systemof thought and belief,including science. Atmiddle age and againnear the end of his life,Werner declared scienceand religion to be “com-plementary” aspects ofreality, each with its ownlanguage and symbolismand each with its ownlimited realm of validity.Different religiously orintuitively apprehendedtruths should be viewedas different sides of thesame truth, whilerational science—hisown profession—shouldbe viewed as just oneamong a variety of waysof perceiving reality.Shortly before hisdeath, Heisenbergremarked to a colleague,“If someone were to saythat I had not been aChristian, he would bewrong. But if someonewere to say that I hadbeen a Christian, hewould be saying toomuch.”

—DAVID C. CASSIDY,UNCERTAINTY: THE LIFE

AND SCIENCE OF WERNER

HEISENBERG

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Chances Are 79

160. Quantum Computer

The new quantum computers rely on quantum coher-ence. That is, the quantum computer system contains Nidentical quantum subsystems—for example, atoms, oroptical setups, or molecules, or resonant cavities. Ingeneral, each quantum subsystem can be in many pos-sible quantum states. Assume that the ψi for each quan-tum subsystem has only two states, which we label 1and 0. If N = 3, then Ψ = ψ1 + ψ2 + ψ3 is the QM stateof the system. Therefore our quantum computer repre-sents all eight states simultaneously: 000, 001, 010,011, 100, 101, 110, 111.

That is, during calculations on Ψ all eight statesparticipate in each calculation! If the quantum com-puter is actually a large molecule in a vacuum, then themolecule must be kept away from the walls of the con-tainer and away from other molecules. Why?

161. Cup of Java Quantum

Computer

One day while looking into her cup of java, Laura real-ized that this slurry of caffeine molecules could be theworld’s natural quantum computer. How could thisinherent ability in coffee be possible?

162. Bragg Scattering of X-rays

Bragg scattering of X-rays of wavelength λ in an idealcrystal satisfies Bragg’s law: 2d sin θ = m λ, where d isthe spacing between adjacent scattering planes and θ isthe angle measured from the surface of the crystal, not

QUESTION: WHAT IS “IT”?

Pascal did IT under

pressure.

Coulomb got all charged

up about IT.

Hertz did IT frequently.

Boltzmann did IT in heat.

Ampere let IT flow.

Heisenberg was never

sure whether he even

did IT.

Bohr did IT in an excited

state.

Pauli did IT but excluded

his friends.

Hubble did IT in the

dark.

Theorists do IT on paper.

Astrophysicists do IT

with young starlets.

ANSWER: IT = science, of

course!

—COPYRIGHT © 2002 BY

JUPITER SCIENTIFIC

It has been proved that

the 13th is more likely

to fall on Friday than on

any other day of the

week. For a short proof

consult the reference

below.

—JOHN WAGNER AND

ROBERT MCGINTY,“SUPERSTITIOUS?”

MATHEMATICS TEACHER 65(1972): 503–505

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the perpendicular. When this condition is met for vari-ous integer values of m, constructive interference fromthe entire family of parallel planes occurs because thepath differences are integral multiples of the X-raywavelength. One often reads that the Bragg scatteringof X-rays from an ideal crystal is a coherent scatteringprocess—that is, all the Bragg-scattered X-rays arrivein phase at the detector. Why is it not so?

163. Beautiful Faces

Why can we see a person’s face in great detail in visiblelight? Hint: think about coherent scattering versus non-coherent scattering of the light.

Why is the image of a person’s face blurry in theinfrared (IR) and in the ultraviolet (UV)? For simplicityand idealization purposes, assume that we can seeequally well in the IR, visible, and UV so that our phys-iology is not the limiting factor.

164. Gravitational Waves

In addition to telescopes for photons in the γ-ray, X-ray,UV, visible, IR, µ-wave, and radio parts of the electro-magnetic spectrum, new windows to the universe areopening up with neutrino and gravitational waveobservatories. Gravitational waves are expected to beproduced by a changing mass quadrupole—for exam-ple, two masses revolving about their commonbarycenter, such as the two stars in a binary star sys-tem. They would emit gravitational waves with wave-lengths of many kilometers that interact with allobjects—that is, they exhibit most wave phenomenasuch as scattering, reflection, and transmission throughobjects in ways similar to other types of waves. Theclassical scattering cross section of gravitational waves

80 Mad about Modern Physics

Pauli once referred to

quantum mechanics as

—boys’

physics—because so

many of the main con-

tributors were still in

their twenties. For

example, in September

1925 Heisenberg was

23 years old, Pauli 25,

Jordan 22, and Dirac

had just turned 22. In

1932 Friedrich von

Weizsäcker recalled,

“The general attitude

was one of immense

an immense

feeling of superiority,

as compared to old pro-

fessors of theoretical

physics, to every exper-

imental physicist, to

every philosopher, to

politicians, and to what-

ever sorts of people you

might find in the world,

because we had under-

stood the thing and they

didn’t know what we

were speaking about.”

—HELGE KRAGH, QUANTUM

GENERATIONS: A HISTORY

OF PHYSICS IN THE

TWENTIETH CENTURY

Hochmut,

Knabenphysik

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Chances Are 81

by a mass pair in a detector was worked out by physi-cist J. Weber about 50 years ago.

For simplicity, assume that each pair of identicalatoms in a material is a mass pair quadrupole scattererof gravitational waves. We would like to know whethergravitational waves can scatter coherently in the detec-tor—that is, whether a gravitational wave can simulta-neously scatter from many mass pairs in the detector(such as an aluminum bar) or whether a gravitationalwave must scatter from a single mass pair at a time.What is the physics here?

165. Coherent Neutrino

Scattering

Another possible window or telescope for observingthe universe is in the detection of neutrinos. The Super-Kamiokande neutrino facility in Japan and the Sud-bury Neutrino Observatory (SNO) in Canada housetwo of the largest neutrino detectors, containing thou-sands of tons of water. Already they have determinedthat the flux of solar neutrinos from the Sun agreeswith the standard solar model. In addition, researchgroups operating these neutrino detectors have verifiedneutrino oscillations in matter, the conversion of onetype of neutrino to another.

The two neutrino detectors are enormous becauseneutrinos are notorious for their extremely small prob-ability to interact with matter. Billions of neutrinos passthrough our bodies each second and do no harm! A sin-gle electron neutrino would pass through solid lead(Pb), filling space from Earth to Jupiter with only asmall chance of colliding with a Pb nucleus. However,in 1984 physicist J. Weber proposed that neutrinos ofall energies could be coherently scattered by the nucleiin large defect-free single crystals of silicon, ruby, or

FROM AN INTERVIEW WITH

THE AMERICAN PHYSICIST

ISIDOR ISAAC RABI

“The nature of discover-

ies is so remarkable, so

wonderful—if you want

to think of the goal of

the human race, there it

is. To learn more about

the Universe and our-

selves. In physics, the

newest discoveries like

relativity and the uncer-

tainty relation, uncover

new modes of thought.

They really open new

perspectives.” A sudden

sad look passed over his

face. “And I thought

that, say, fifty years

ago, that this would hap-

pen, that these revolu-

tions and advances in

science would have an

effect on mankind—on

morals, on sociology,

whatever. It hasn’t hap-

pened. We’re still up to

the same things, or,

well, I think, regressed

in values.”

—ROBERT P. CREASE AND

CHARLES C. MANN, THE

SECOND CREATION: MAKERS OF

THE REVOLUTION IN

20TH-CENTURY PHYSICS

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diamond, thereby enhancing the neutrino scatteringprobability by a factor of 1022. Therefore, in the idealcase, practically all incident neutrinos would scatter atleast once from the carbon nuclei in a perfect diamondcrystal within the first centimeter or less!

Normally, one might expect only neutrinos of wave-lengths much greater than the spacings between thenuclei in the crystal to have any chance at coherentscattering, analogous to light scattering coherently froma surface of atoms spaced much less than the wave-length of the incident light. Otherwise, when the nucleiare treated as scattering potentials, the phases con-tributed by the scattering nuclei to the QM amplitudeare random, and the scattering probability will be pro-portional to N instead of N 2, like the result for X-raysdiscussed in a previous problem. What assumption havewe made about the scatterers that Weber says leads toan incorrect conceptual argument against coherent scat-tering for the shorter-wavelength neutrinos?

166. Magnetic Resonance

Imaging (MRI)

Magnetic resonance imaging (MRI)is really the medical applicationof nuclear mag-netic resonance,which physicistshave been doingsince the 1940s.A sample of liv-ing tissue con-tains numerous hydrogen atoms bound in molecules.Each hydrogen nucleus has a spin with a magneticmoment that can be aligned by an applied magneticfield. The sample is placed in a very strong uniform

82 Mad about Modern Physics

Compared to the theory

of relativity, quantum

mechanics developed

rapidly, disseminated

very quickly, and met

almost no resistance.

Also contrary to relativ-

ity, quantum mechanics

attracted little public

interest. Eddington was

one of the few scien-

tists who wrote about

the theory for a nonsci-

entific readership.

Although quantum

mechanics was no less

counterintuitive than

relativity, there was no

quantum counterpart to

the antirelativistic liter-

ature that flourished in

the 1920s.

—HELGE KRAGH, QUANTUM

GENERATIONS: A HISTORY OF

PHYSICS IN THE TWENTIETH

CENTURY

Magnetic Field

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Chances Are 83

magnetic field to align the spins of the hydogen nuclei.A pulsed electromagnetic field is applied that would flipjust one hydrogen spin, for example. What alternativeQM interpretation can one provide that treats thenuclei as a collective whole?

167. Heisenberg Uncertainty

The Heisenberg uncertainty principle, also known asthe indeterminancy principle worldwide, states ∆px∆x≥ h/4π , where ∆x is the uncertainty in the x-positionmeasurement, ∆px is the uncertainty in the x-momen-tum measurement, and h is Planck’s constant. As somepeople say, the uncertainty principle places a limit onthe accuracy of knowing a particle’s position. What doyou think? Some people claim also that the Heisenberguncertainty principle is just an example of a more gen-eral uncertainty relationship for all waves, that theposition can be determined only at the expense of ourknowledge of its wavelength. Is this statement true?

We also know that Niels Bohr, in his discussionswith Albert Einstein over several decades on whetherquantum mechanics is a complete description ofnature, would often invoke the uncertainty principle todefend his point of view, known as the Copenhageninterpretation of QM. Bohr argued that if you pindown the particle’s position more precisely for thefamous double-slit experiment by observing with pho-tons, their interaction with the particle disturbs itsmomentum by giving it a random momentum kick.That is, without looking, the particle exhibits an inter-ference pattern on a distant screen behind the two slits.However, if you look to see which way the particlegoes through the slits, the measurement disturbs thesystem and there’s no interference pattern on the dis-tant screen. You just see a classical two-hump distribu-tion. What do you think about Bohr’s argument?

The number of photons

is in general not con-

served in particle reac-

tions and decays. I . . .

would like to note here

an ironical twist of his-

tory. The term “photon”

first appeared in the

title of a paper written

in 1926. The title: “The

conservation of pho-

tons.” The author: the

distinguished physical

chemist Gilbert Newton

Lewis (1875–1946) from

Berkeley. The subject: a

speculation that light

consists of “a new kind

of atom . . . uncreatable

and indestructible [for

which] I . . . propose the

name photon.” This idea

was soon forgotten, but

the new name almost

immediately became

part of the language.

—ABRAHAM PAIS, IN SOME

STRANGENESS IN THE

PROPORTION, EDITED BY

HARRY WOOLF

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168. Vacuum Energy?

Although the classical vacuum is a void, the quantumvacuum is a virtual “soup” of particle-antiparticle pairsthat interact with real atoms to produce the Lamb shift(slight energy shift in atomic levels) and the Casimireffect (attraction of two plates in a vacuum). Does thequantum vacuum have energy content, or does theenergy in the “soup” average out to zero?

169. Casimir Effect

When two parallel uncharged metal sheets are placed ina perfect vacuum, they attract each other with a tinyforce that is not gravitational. What is the source ofthis effect?

170. Squeezing Light

Laser light can be described in many ways. If one con-siders just the amplitude and the phase of one ray in alaser beam, there will always be shot noise—that is,random variations caused by virtual particle interac-tions in the vacuum with the beam. Yet we’ve heardthat there may be techniques to reduce the shot noise inthe amplitude, for example. Then what happens to theshot noise in the phase?

171. Electron Spin

Does the vacuum affect the spin of a particle such as anelectron?

172. Superconductivity

One quantum mechanical effect that shows itself on themacroscopic scale is superconductivity. Cooper-paired

84 Mad about Modern Physics

Quantum mechanics, in

matrix form, was born

when in July of 1925

Heisenberg, only

twenty-three years of

age, had a creative

breakthrough on the

fog-shrouded island of

Helgoland in the North

Sea. He took as his

guiding principle the

proposition that a the-

ory should not traffic

with unverifiable

abstractions. He wanted

to deal only with meas-

urable quantities. He

later told Einstein that

“the idea of observable

quantities was actually

taken from his relativ-

ity,” which had rejected

such concepts as

absolute speed for the

same reason. In a letter

to Pauli, dated 9 July

1925, Heiseberg wrote,

“My entire meager

efforts go toward killing

off and suitably replac-

ing the concept of the

orbital paths that one

cannot observe.”

—HANS CHRISTIAN VON

BAEYER, TAMING THE ATOM: THE

EMERGENCE OF THE VISIBLE

MICROWORLD

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conduction electrons in superconductors have totalspin zero, that is, their paired spins are opposite, eventhough their spatial separation can be enormous—cen-timeters to meters, for example—because they haveopposite momenta. These pairs can act like bosons ofspin zero, which obey Bose-Einstein statistics. Anynumber of bosons can be in the same quantum state,that is, have the same four-momentum (e.g., defined bythe energy and three-momentum) and spin. Therefore,all bosons in the same collective superconducting statehave exactly the same energy. Yet this boson collectivestate in a superconductor has a small energy width.Any thoughts about the cause of this energy width?

173. Superfluidity

He-4 below the lambda transition temperature 2.7 Kcan be analyzed as a two-fluid liquid composed of Heatoms in the normal state and He atoms in the macro-scopic superfluid state. Superfluidity is a property ofHe-4 in the liquid state because He-4 atoms obey Bose-Einstein statistics. Many He-4 atoms can be in the samemacroscopic quantum state—that is, the same momen-tum states for these atoms moving in the superfluid. Ifso, then why can He-3 at low temperatures alsobecome a superfluid?

174. Gap Jumping

Tiny detectors on a person’s head have been used tosense tiny fluctuations in the brain’s magnetic field.These SQUIDS, short for superconducting quantuminterference devices, are the most sensitive of any kindand rely on the Josephson Effect, in which the Cooperpairs of electrons in a superconductor can sometimesjump a physical spatial gap in the material to anotherpart of the superconductor. Manufactured SQUIDS

Creative thinkers seem

to possess the following

characteristics in

common:

• an acutely sensitive

awareness of their

environment;

• the ability to generate

a large number of

ideas in response to a

given problem;

• the ability to focus

their faculties in sus-

tained concentration;

• in most cases the cre-

ative individual’s work

place is likely to be a

cheerfully haphazard

conglomeration of

complete disorder;

• the majority of truly

creative persons are

introverts;

• they tend to be much

less concerned with

what others think of

them than most peo-

ple are; also, they are

often comparatively

indifferent to clothing

and appearance. Cre-

ative people do not

seem to have a need

to present themselves

in a favorable light to

others.

OTTO H. THEIMER, AGENTLEMAN’S GUIDE

TO MODERN PHYSICS

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have a thin film filling the gap. In fact, the direct cur-rent (DC) SQUID used in laboratories worldwide todayfor sensing small magnetic fields is a superconductingring with two gaps! The best DC SQUIDS have anenergy sensitivity capable of detecting a magnetic fluxchange corresponding to about 10–34 joule in one sec-ond, about the mechanical energy required to raise anelectron 10 centimeters in one second. Why do thepaired electrons jump the gap?

175. Nuclear Decay

In the nucleus of an atom, neutrons and protons areheld by nuclear forces. Their total energy (ignoring themc2 contributions) is less than the barrier height poten-tial energy. Yet some nuclear particles do escape. Anythoughts about the reason for an escape?

176. Total Internal Reflection

In total internal reflection of light—for example, at aglass-air interface or from a water-air surface—if theincident light is in the more dense medium, does thelight penetrate into the air beyond the interface?

86 Mad about Modern Physics

We all share a strange

mental time lag, a phe-

nomenon first brought

to light in the 1970s by

neurophysiologist Ben-

jamin Libet of the Uni-

versity of California at

San Francisco. In one

experiment, Libet docu-

mented a gap between

the time an individual

was conscious of the

decision to flex his fin-

ger (and recorded the

exact moment of that

consciousness) and the

time his brain waves

indicated that a flex was

imminent. The brain

activity occurred a third

of a second before the

person consciously

decided to move his

finger.

—ANTONIO R. DAMASIO,“REMEMBERING WHEN,”

SCIENTIFIC AMERICAN

(SEPTEMBER 2002)

Oxide layer

Superconductor

Current outCurrent in

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Chances Are 87

177. Annihilation

We know that particles and their antiparticles annihi-late each other. For example, the electron and thepositron in positronium can annihilate into two pho-tons or three photons in the final state, depending onthe total angular momentum of the positronium. Whywould they do such a violent action?

Hint: why does any event occur in nature? Weknow that the rate of any quantum mechanical event,by Fermi’s Golden Rule, is proportional to the proba-bility for the event times the density of final states. Isthis statement all we need to say?

178. A Bouncing Ball

We see a kid bouncing a ball. According to quantummechanics, which applies to everything that happens,why does the ball bounce?

179. The EPR Paradox

First of all, a short explanation. Although there areother examples of the Einstein-Podolsky-Rosen (EPR)paradox and violations of Bell’s inequalities, we choose this version because we can provide you with

The first wrist watches

were decorative orna-

ments made for women,

and men consequently

shunned them as being

too effeminate. Again

war had its effect—the

first World War. A watch

that has to be fished

out of a pocket in order

to be read is a lot less

convenient than one

strapped to the wrist,

particularly when you’re

trying to fire a machine

gun or charge over a hill

at a particular moment.

Realizing this, govern-

ments made wrist

watches part of the

standard equipment

issued to their soldiers.

After the war, men sud-

denly felt that wrist

watches were accept-

able for civilian wear, a

trend which clearly has

continued for both

sexes until the present

day.

—JO ELLEN BARNETT, TIME’SPENDULUM: FROM SUNDIALS TO

ATOMIC CLOCKS, THE FASCINATING

HISTORY OF TIMEKEEPING AND

HOW OUR DISCOVERIES

CHANGED THE WORLD

Light ray Air

Glass prism

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actual data to use in formulating your own solution tothe paradox.

A source of two correlated identical particles ofopposite spins sits on the straight line between twoidentical particle detectors. Each detector can measurethe polarization state of the entering particle, and eachdetector has three polarization switch positions (1, 2,and 3) and two display lamps (green and red). Eachtime the experimenter pushes the button, the two cor-related particles are shot out of the source in oppositedirections into the detectors. The data show two pat-terns: (1) For runs that have the same switch settings onthe two detectors, the same color lights flash on them.(2) For all runs, without regard for switch settings, thepattern of flashing is completely random.

This experiment gets to the heart of QM and theapplication of its three rules for events. We can useclassical mechanics to explain the first pattern: let thetwo particles carry the same instructions to be appliedat the detectors. For example, this instruction set mightwork: flash red at switch positions 1 and 3; flash greenat switch position 2. But this classical scheme with pre-determined instruction sets will not handle the secondpattern. Why not? What is the surprising conclusion?

Reproduced here is a small part of a data set for theexperiment (from the Mermin reference in the answer).Each entry shows the switch settings and the colors thelights flashed for each run. The switch settings are ran-domly changed from run to run.

31GR 13RG 31RR 33GG

21RR 31RG 33GG 11GG

22RR 12RG 31RG 13RG

33GG 13GR 31RR 31RG

88 Mad about Modern Physics

Einstein’s unease with

quantum mechanics

stemmed from a firm

belief in determinism. In

a 1931 essay “The World

as I See It,” reprinted in

he wrote, “I do not at all

believe in human free-

dom in the philosophical

sense. Everybody acts

not only under external

compulsion but also in

accordance with inner

necessity. Schopen-

hauer’s saying, ‘A man

can do what he wants,

but not want what he

wants,’ has been a very

real inspiration to me

since my youth; it has

been a continual conso-

lation in the face of

life’s hardships, my own

and others’, and an

unfailing well-spring of

tolerance.”

—ADAPTED FROM

ALBERT EINSTEIN,IDEAS AND OPINIONS

Ideas and Opinions,

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Chances Are 89

11GG 22GG 33RR 23GR

23RR 12RG 32RG 31GR

32GR 12GR 31RG 23RG

12GR 22GG 11RR 22RR

12RG 23GR 23GR 12GR

11GG 33RR 12GG 32GR

12GR 23GG 21GR 12GG

22RR 23GG 13GR 31GG

12GG 33RR 33GG 32RG

33RR 23GR 11GG 21GR

11RR 21GG 12RR 22GG

180. Information and a Black Hole

Classical information and quantum information arenot the same. Why? Because QM rule 2 tells us that inQM there can be a coherent superposition of quantumstates. No such state exists in classical physics. Soquantum information supersedes classical information.

The classical and the quantum information contentin a system, such as a chair, can be determined or esti-mated by standard techniques of classical and quantuminformation theory. Suppose the chair is tossed into ablack hole. The quantum information in the chairseems to have gone with the chair into never-neverland. Why should we worry about this informationloss?

HEISENBERG’S

UNCERTAINTY PRINCIPLE

SIMPLIFIED

“If you know where it is,

you don’t know where

it’s going,” or, “If you

know where it’s going,

you don’t know where it

is.”

The ordinary adult never

gives a thought to space-

time problems. . . . I, on

the contrary, developed

so slowly that I did not

begin to wonder about

space and time until I

was an adult. I then

delved more deeply into

the problem than any

other adult or child would

have done.

—ALBERT EINSTEIN (TO

NOBEL LAUREATE

JAMES FRANCK) IN ALICE

CALAPRICE, THE EXPANDED

QUOTABLE EINSTEIN

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Can ThisBe Real?

91

Can ThisBe Real?

AFTER QUANTUM MECHANICS EXPLAINED

the internal behavior of the atom in the 1920s

and the chemistry of atoms and molecules, physicists

turned toward understanding the atomic nucleus in the

1930s and 1940s. Rutherford in 1911 had determined that

practically all the atomic mass was in the nucleus, and of

course everyone knew that its positive protons balanced

the electron negative charges in the neutral atom. But what

held the nucleus of positive protons together? A nuclear

strong force was eventually identified in the 1970s as the

color interaction acting between quarks, and it is one of the

four known fundamental forces in nature. The second

nuclear force, the weak interaction, responsible for many

nuclear decays, was identified completely in the 1960s. By

the early 1980s three of the four fundamental interactions

had been unified into the Standard Model (SM) of Leptons

9

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and Quarks. Only gravitation needs to be incorporated

into the unified model of nature. The selected chal-

lenges in this chapter range through the whole gamut

of nuclear and particle physics.

92 Mad about Modern Physics

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Can This Be Real? 93

181. Carbon-14 Dating

Carbon-14 is produced when cosmic rays collide withatoms in the atmosphere to create an energetic neutronthat then collides with a nitrogen-14 atom (seven pro-tons, seven neutrons) to make a carbon-14 atom (sixprotons, eight neutrons) and a hydrogen atom (oneproton, zero neutrons). Carbon-14 is radioactive, witha half-life of 5,730 years.

These C-14 atoms combine with oxygen to formcarbon dioxide, which plants absorb into plant cellsthrough photosynthesis. Animals and people eat theplants and take in the C-14 as well as the normal non-radioactive isotope C-12. The ratio of C-14 to C-12 inthe air and in all living things at any given time isassumed constant; about 1 in 10 trillion carbon atomsare C-14. The C-14 atoms are always decaying, so afteran organism dies, no new carbon atoms are taken inand this ratio of C-14 to C-12 atoms decreases.

The carbon-14 radiocarbon dating of living andonce-living materials began with Willard Libby in the1940s. Antiquities dated by C-14 agree with other daterecords until they begin to disagree for dates more thanseveral thousand years ago. Why is there disagreementin the dates between C-14 dating and the writtenrecords?

182. Nuclear Energy Levels

In the 1930s and 1940s, physicists working on theenergy states of the nucleus of an atom concentrated onvarious models, including a shell model using theSchrödinger equation with an approximately constantelectrical potential inside the nucleus. Conceptually,each nucleon is in a well-defined orbit within thenucleus and moves in an averaged field produced by allthe other nucleons. However, even though quantum

Roughly once a second,

a subatomic particle

enters the earth’s

atmosphere carrying as

much energy as a well-

thrown rock. Somewhere

in the universe, that

fact implies, there are

forces that can impart

to a single proton 100

million times the energy

achievable by the most

powerful earthbound

accelerators.

—JAMES W. CRONIN, THOMAS

K. GAISSER, AND SIMON P.SWORDY, “COSMIC RAYS AT

THE ENERGY FRONTIER,” SCIENTIFIC AMERICAN

(JANUARY 1997)

[Pierre Curie] was

impressed by Marie’s

courage and her amaz-

ing love of work and

fascinated by her

lucidity, her challenging

questions, her reflective

answers.

—J. A. DEL REGATO,RADIOLOGICAL PHYSICISTS

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states such as n = 1, with l = 0, 1, 2, 3, etc., are possi-ble in the shell model, the predicted energy levels didnot fit the data. In fact, the actual energy levels were allscrambled compared to the shell-model theoreticalpredictions. Why?

183. Nuclear Synthesis

The championship of nuclear binding energy is oftenattributed to Fe-56, meaning that Fe-56 has the great-est binding energy per nucleon and therefore is themost stable nucleus. Most elements are synthesized instars. Supposedly, elements higher on the periodic chartthan Fe cannot be synthesized in normal star burningcycles. Why not? Actually, the sequence of nuclear syn-thesis does not stop at iron, because Ni also is synthe-sized. What happens to the Ni isotopes that aresynthesized?

94 Mad about Modern Physics

On August 6, 1945, an

atom bomb dubbed

“Little Boy” was dropped

from an American B-29

bomber called the

on the city of

Hiroshima. It detonated

at 8:16 A.M. at a height

of 1,900 feet. Of

Hiroshima’s 330,000

inhabitants, approxi-

mately 70,000 were

killed instantly. By the

end of 1945, the death

toll had risen to

140,000. “Little Boy”

used the gun assembly

design and uranium-

235 as the fissionable

material. Because the

gun design was an inef-

ficient means of caus-

ing the chain reaction,

about 50 kilograms of

89 percent U-235 and

14 kilograms of 50 per-

cent U-235 ended up

being used. Of this it is

estimated that only

about 2 percent actually

fissioned. Three days

later another atom

bomb, dubbed “Fat

Man,” was dropped on

the city of Nagasaki.

Approximately 40,000

were killed instantly.

Enola Gay

Energy

2p

1f

2s1d

1p

1s

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Can This Be Real? 95

184. Heavy Element Synthesis

If we are “truly the stuff of stars,” then where do all theheavier elements beyond iron come from if they are notmade in normal star burning cycles?

185. Neutron Decay

A free neutron will decay with a half-life of about 14.8minutes, but it is stable if combined into a nucleus.Why would the neutron be stable in the nucleus?

186. Finely Tuned Carbon?

Eventually a star exhausts its supply of hydrogen in itscore, gravitational contraction occurs, the temperaturereaches about 108 K, and helium burning can occur viathe reaction 3He-4 → C-12 + 2 photons. In fact, thenucleosynthesis of all the heavier elements essential forlife relies on this reaction. However, the chance thatthree helium nuclei get together fast enough to form thecarbon nucleus is negligible. So this critical reactionactually proceeds via an intermediate beryllium stepgiven by 2He-4 + (99 ± 6) keV → Be-8 followed by

By the time she enrolled

at the Sorbonne in 1891,

Maria Sklodowska was

twenty-four years old. In

1893 she passed the

in physics, com-

ing first in her class. In

that year Maria

Sklodowska met Pierre

Curie. The meeting

between the two was

arranged for scientific

purposes, with little hint

of matchmaking. At the

time they met, both

Maria and Pierre consid-

ered themselves des-

tined for single lives.

After graduation Maria

intended to return to

Warsaw to look after her

aging father and teach

science. Pierre, mean-

while, at age thirty-four

one of France’s leading

young physicists, was

convinced that he would

never find a wife who

would tolerate his com-

plete devotion to sci-

ence. He was the first to

fall. Almost from the

beginning he realized

that in this severe Polish

girl he had found the

woman of his dreams.

—ADAPTED FROM MARGARET

WERTHEIM, PYTHAGORAS’TROUSERS: GOD, PHYSICS, AND

THE GENDER WARS; SUSAN

QUINN, MARIE CURIE: A LIFE

license

7.70 MeV7.65 MeV

7.40 MeV

0.00 MeV

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Be-8 + He-4 → C-12 + 2 photons. Since the Be-8 lifetime of about 10–17 second is much longer than theHe-4 + He-4 collision time in a star, the beryllium willbe around long enough for the reaction to occur.

The total energy of the Be-8 nucleus and a He-4nucleus at rest is 7.4 MeV above the energy of the nor-mal state of the C-12 nucleus. The radioactive state ofthe C-12 is 7.65 MeV above the normal state. If theenergy of the radioactive state were more than 7.7 MeVabove the normal state, the formation of C-12 via Be-8plus He-4 would require the reactants to have at least0.3 MeV of total kinetic energy, which is extremelyunlikely at the temperatures found in most stars.

The importance of this process is emphasized byphysicists who inject the Anthropic Principle, that cer-tain constants of nature have values that seem to havebeen mysteriously fine-tuned to just the values thatallow for the possibility of life. Recently, others haveintroduced a further extension that claims that this car-bon nucleus coincidence can be explained only by theintervention of a designer with some special concernfor life. Both groups cite the closeness of the requiredenergy to the actual limit, 7.7 MeV – 7.65 MeV = 0.05MeV, a quantity less than 1% of 7.65 MeV, as their evi-dence for the fine-tuning. Why is their reasoning sus-pect with regard to this carbon formation process?

187. Proton-Proton Cycle

The thermonuclear reactions in the proton-protoncycle inside the Sun convert four protons into an alphaparticle, two positrons, two electron neutrinos, andtwo photons with the release of 26.7 MeV of energy.First, two protons collide to form a deuteron H-2, thenthis deuteron collides with a proton to form He-3, thenfinally two He-3 nuclei must find each other to collide

96 Mad about Modern Physics

An ingenious “explana-

tion” of the Michelson-

Morley null result was

found by George F.

FitzGerald of Dublin in

1889. He suggested

that the lengths of bod-

ies moving through the

ether at velocity con-

tract in the direction of

their motion by a factor

(1 – 2 / c2)1/2—which

would just compensate

for the ether drift in the

Michelson-Morley

apparatus. A few years

later [Dutch physicist

Hendrik A. Lorentz]—

apparently independ-

ently—made the same

hypothesis and incorpo-

rated it into his ever

more comprehensive

ether theory. This

“Lorentz-FitzGerald

contraction” then

quickly diffused into the

literature.

—WOLFGANG RINDLER,RELATIVITY: SPECIAL, GENERAL,

AND COSMOLOGICAL

v

v

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Can This Be Real? 97

and form an He-4. The overall representation of thisproton-proton cycle is:

4H → He-4 + 2e+ + 2ν + 2γ.

The six photons ultimately produced, including thefour 0.511 MeV photons from two positron-electronannihilations, take about a million years to reach theSun’s surface to be emitted eventually as visible pho-tons, which then take about another eight minutes toreach Earth. The two neutrinos carry away about 3percent of the energy to balance the energy conserva-tion equation and to conserve lepton family number.

Presented as the primary source of our Sun’s energy,this method of burning hydrogen is not the primarymethod for fusion energy in many stars. Why not?What reaction sequence is the primary candidate?

188. Oklo Nuclear Reactor

In the 1970s, uranium samples from the Oklo uraniummine in Gabon, Africa, were discovered to have abnor-mally high concentrations of the isotope U-235, as highas 3 percent, when only about 0.72 percent of the iso-tope was expected in a natural source. Supposedly thehigh concentration of U-235 is explained by realizingthat the uranium deposits at Oklo acted as a naturalnuclear reactor. Could this natural reactor have been abreeder reactor making its own Pu and U-235?

189. Human Radioactivity

Radiation doses are expressed in SI units as milliSievert(mSv) effective doses. This unit takes into account the type, the intensity and duration of radiation, theamount and type of body tissues irradiated, and thedifferent radiation sensitivity of the irradiated tissues.The average natural background dose rate in many

Bertrand Russell would

sometimes liken the

scientific method to the

following syllogism:

Bread is made of

rock;

Rock tastes good;

Therefore bread

tastes good.

In other words, you

can never be sure that

correct conclusions

don’t follow from incor-

rect premises.

The illumination pro-

vided at eye level in

artificially lighted rooms

is commonly from 50 to

100 footcandles, or less

than 10 percent of the

light normally available

outdoors in the shade of

a tree on a sunny day.

As a result, the total

amount of light to which

a resident of Boston,

say, is exposed in a

conventionally lighted

indoor environment for

16 hours a day is con-

siderably less than

would impinge on him if

he spent a single hour

each day outdoors.

—RICHARD J. WURTMAN, “THE

EFFECTS OF LIGHT ON THE

HUMAN BODY,” SCIENTIFIC

AMERICAN (JULY 1975)

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countries is 1–5 mSv a year. On average, medical expo-sures contribute about another 0.5–0.7 mSv a year. Thecurrent recommended limit for occupational exposurein many countries is about 20mSv effective dose peryear averaged over five consecutive years.

The typical human adult body has an inherentinternal radiation dose from its natural amounts ofradioactive elements, including its major contributionof about 40 milligrams of radioactive potassium as theisotope K-40, which has a half-life of about 1.3Gigayears. This isotope is not the result of artificialradioactivity but remains from the formation of potas-sium in the supernova that gave birth to our Solar Sys-tem about 5 billion years ago. There has not beenenough time for all of the radioactive potassium todecay, so that is why there is so much in our bodies.Eileen wonders whether this inherent K-40 radioactivesource is exposing our bodies to more than the recom-mended limit? Is the limit exceeded when several peo-ple gather together in a small circle?

190. Nuclear Surprises?

Which of the following statements is true?

1. A typical coal burning power plant releases moreradioactive materials into the air than a typicalnuclear reactor plant.

2. Spreading all the nuclear waste equally around thesurface of the planet will hardly change the back-ground radiation level at all.

191. Cold Fusion

Is cold fusion—that is, the fusion of two deuteriumnuclei at about room temperature—a possibility, or can this process be eliminated by theoretical argumentsalone?

In early 1940, Paul

Harteck, a German physi-

cal chemist, felt he’d need

up to 300 kilograms of

uranium to test his idea of

using carbon dioxide as a

moderator. He arranged to

get the frozen carbon

dioxide (dry ice) from I. G.

Farben, and the necessary

uranium from Heisenberg.

But at the last moment,

Farben declared they

could only supply the dry

ice until early June; they’d

need it after that for

keeping food fresh during

the hot summer months.

Harteck scraped together

about 200 kilos of ura-

nium, but with that low

amount his results were

inconclusive; Germany did

not go ahead with the

easy, dry ice reactor that

would almost certainly

have given them plenty of

radioactive metal early on

in the war. Thus was the

clear hot weather of that

summer—so often cursed

by the Allies for letting

Panzer armies advance

into France—central to

forestalling this greater

evil.

—MARK WALKER, GERMAN

NATIONAL SOCIALISM AND THE

QUEST FOR NUCLEAR POWER

1939–1949

98 Mad about Modern Physics

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Can This Be Real? 99

192. Fission of U-235

During World War II the Germans and the Allies wereboth working on projects related to nuclear weaponsdevelopment. One can calculate the minimum mass ofU-235 required for a fission weapon from present-daynuclear physics data sheets. That value is the amountrequired if the neutrons produced by the fission of U-235 encounter stationary target nuclei. The problemis much more difficult for two important reasons. Canyou identify them?

193. Minimal Nuclear Device

What is the minimum mass of pure U-235 or Pu-239required in a device for a nuclear event? How wouldyou estimate this value?

194. Large Nuclei

Small nuclei that become excited and deformed losetheir energy by breaking up into smaller fragments. Alarger nucleus, with 150 or more nucleons, stores most of its excitation energy as rotational energy. Asthey slow down and de-excite, these nuclei lose energy

Although we are quite

unaware of their pres-

ence, there are, on the

average, some 400

microwave photons in

any cubic centimeter in

the universe left over

from the big bang.

In A.D. 499 the Indian

astronomer Aryabhata

presented a treatise on

mathematics and astron-

omy, the

The is a

summary of Hindu math-

ematics up to his time,

including astronomy,

spherical trigonometry,

arithmetic, algebra, and

plane trigonometry. The

presented a

new treatment of the

position of the planets in

space. It proposed that

the apparent rotation of

the heavens was due to

the axial rotation of the

Earth. Moreover, Aryab-

hata conceptualized the

orbits of the planets as

ellipses, a thousand

years before Kepler.

—DICK TERESI, LOST

DISCOVERIES: THE ANCIENT ROOTS

OF MODERN SCIENCE—FROM THE

BABYLONIANS TO THE MAYA

Aryabhatiya

Aryabhatiya

Aryabhatiya.

U-235

Fragments

Neutrons

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and return to their unexcited shape. What do thesenuclei emit, and how would you characterize theenergy spectrum?

195. Human Hearing

The human eardrum is sensitive to displacements ofless than the diameter of an atomic nucleus. How havesuch minute displacements been measured via nuclearphysics techniques?

196. 1908 Siberia Meteorite

In an article by Andrew Chakin in Sky & Telescope inJanuary 1984, pages 18–24, the author states:

A grande dame of scientific mysteries—the Tun-guska event—turned 75 last summer, her charmvery much intact. She continues to seduce bothscientist and charlatan alike, both hoping toexplain what happened over a remote stretch ofSiberian taiga on June 30, 1908. All that can besaid from direct eyewitnesses is that a fireballnearly as bright as the Sun streaked to Earth out ofa cloudless morning sky. The bolide’s plunge wasabruptly terminated by an explosion so great thatit registered on seismic stations across Eurasia.The resulting shock wave circled the Earth twice.

The article relates that in 1908 in Siberia a hugemeteorite is supposed to have crashed in the forest,causing huge fires and a crater many kilometers long,but no rocky debris was ever found.

By radiocarbon dating tree rings from old trees thathave been living since 1908, Willard Libby and EdwardTeller in 1963 may have learned something veryimportant about the constitution of the meteorite.What could the radiocarbon data have suggested?

100 Mad about Modern Physics

In the end we—that is,

Bohr, Pauli and I—knew

that we could now be

sure of our ground, and

Einstein understood

that the new interpreta-

tion of quantum

mechanics cannot be

refuted so simply. But

he still stood by his

watchword, which he

clothed in the words:

“God does not play at

dice.” To which Bohr

could only answer: “But

still, it cannot be for us

to tell God, how He is to

run the world.”

—WERNER HEISENBERG,ENCOUNTERS WITH EINSTEIN

When Rutherford was

offered a position at

Yale that required some

teaching, he turned it

down, commenting,

“They act as if the uni-

versity was made for

students.”

—ADAPTED FROM E. SEGRÈ,FROM X-RAYS TO QUARKS

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Can This Be Real? 101

197. The Standard Model

The Standard Model (SM) of Leptons and Quarks isthe most successful physics model of all in terms of test-ing its concepts. The model has six leptons in pairs inthree lepton families and six quarks in pairs in threequark families, with the quarks in three different col-ors. Aesthetically, the matching of three to three ispleasing. Mathematically, this matching of numbers oflepton and quark families cancels out infinities in quan-tum field theory calculations, such as the infinities thatwould arise from the famous triangle anomaly. How-ever important this family matching may be, can youprovide a fundamental physics argument for the spe-cific matching of the first lepton family to the firstquark family, of the second lepton family to the secondquark family, and so on?

198. Spontaneous Symmetry

Breaking

Spontaneous symmetry breaking is a concept first intro-duced by W. Heisenberg in describing ferromagnetic

The Schrödinger equa-

tion, published in March

of 1926, was designed to

explain almost all

aspects of the behavior

of electrons in terms of

de Broglie waves, rather

than of matrices. Physi-

cists now could visualize

the atom in terms of

continuous processes—

the ripple and flow of

standing waves—

whereas with matrices

they had to deal with

Heisenberg’s assertion

that the nature of the

microworld was discon-

tinuous and impossible

to picture. Little wonder

that many physicists

threw away their matri-

ces and started working

with Schrödinger’s

methods. Even today,

most physicists would

say that the Schrödinger

equation, being nonrela-

tivistic, has no right to

be this good.

—ADAPTED FROM

ROBERT P. CREASE AND

CHARLES C. MANN,THE SECOND CREATION:

MAKERS OF THE REVOLUTION IN

20TH-CENTURY PHYSICS

Electron neutrinoElectron

Muon neutrinoMuon

Up quarkDown quark

Charm quarkStrange quark

Top quarkBottom quark

Tau neutrinoTau

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materials. A ferromagnet has a perfect geometric sym-metry until the Curie transition temperature is reached;then the material becomes magnetized and one particu-lar direction of magnetization is chosen. The theory issymmetrical still, but the actual material is not. One cansummarize the process by stating that microscopicevents can have macroscopic consequences. Near thecritical point of a phase transition, small, random fluc-tuations can grow to make their presence felt through-out the material. A few aligned spins can propagatetheir influence throughout the whole crystal, and thesymmetry is broken.

Other examples are the Schrödinger equation andMaxwell’s equations. As successful in helping todescribe nature as they have been, these equations have more symmetry than the underlying phenomenathey describe. Interest in their symmetry-breakingapplications has led to significant new insights into newconnections between macroscopic and microscopicphenomena.

In particle physics, the spontaneous symmetrybreaking is achieved by the Higgs mechanism. TheStandard Model of Leptons and Quarks relies uponthe Higgs particle to spontaneously break symmetry toprovide three of the electroweak bosons with masswhile leaving the photon massless. Simultaneously, allthe leptons and quarks get their mass values. Moreover,the effect of the Higgs field is to provide a frame of ref-erence in the vacuum for the isotopic spin directionsthat distinguish the particles of each grouping—forexample, neutrons from protons.

Is spontaneous symmetry breaking by the Higgsmechanism the only way to go? Are there other ways tospontaneously break symmetry to achieve the StandardModel of Leptons and Quarks?

102 Mad about Modern Physics

Maria Sklodowska was a

daughter of a Polish

freethinker but reared

by a Catholic mother.

She abandoned the

church before she was

twenty and her marriage

to Pierre Curie was a

purely civil ceremony

because she says in her

memoir of him, “Pierre

belonged to no religion

and I did not practice

any.”

One must make of life a

dream, and of that

dream a reality.

—PIERRE CURIE

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Can This Be Real? 103

199. Proton Mass

Kate sees that the chart of the fundamental leptons and quarks shows that the up and down quark massesare ~ 5 MeV/c2 each. Yet the proton, which is com-posed of two up quarks and one down quark as thecombination uud, has an enormous mass of 938MeV/c2. She asks why there is such a large mass differ-ence between constituents and the final product.

200. Right- and Left-Handed

Neutrinos?

Neutrinos are lepton family partners to the electron,muon, and tau particles of the Standard Model of Lep-tons and Quarks. Each neutrino is thought to be dis-tinct, the electron neutrino being different from themuon neutrino, for example. We now know, however,that each lepton family neutrino type has a very smallmass and is actually a linear combination of three fun-damental neutrino states: ν1, ν2, and ν3.

For the weak interaction, there is the left-handeddoublet state | νL, eL> and the two right-handed singletstates | νR > and | eR >, with the consequence that theright-handed states interact with the Z0 boson but donot participate in the weak interaction mediated by theW+ and W – bosons. The left-handed doublet interactswith all three weak bosons. Must one resort solely tothe explanation “that is how Nature behaves,” or isthere another fundamental reason for left-handed dou-blet and right-handed singlet states?

201. Physics without Equations

John von Neumann and Stanislaw Ulam in the 1940s were among the first to consider attempting to

Contrary to the claim

found in some dictionar-

ies, the word

does not derive from an

Arabic expression for

but rather

it means

as in compelling the

unknown to assume a

numerical value.

The amount of ultra-

violet radiation that

penetrates the atmos-

phere varies markedly

with the season: in the

northern third of the

U.S. the total amount

of erythemal (skin-

inflaming) radiation that

reaches the ground in

December is only about

a fifteenth of the

amount present in June.

—RICHARD J. WURTMAN, “THE

EFFECTS OF LIGHT ON THE

HUMAN BODY,” SCIENTIFIC

AMERICAN (JULY 1975)

x

compulsion,

bone setting

algebra

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understand natural phenomena via cellular automataand computers. Cellular automata (CA) involve adja-cent cells in a 1-D, 2-D, 3-D, and so on grid of cells (ornodes) that take on new numerical values at each tickof the clock according to given rules. The future stateof each cell is determined only by the present state of itslocal neighborhood. One can even remove the externalclock and still maintain a progression of states withinthe CA grid to simulate the passage of time.

Some people claim that all of nature will be simu-lated eventually on computers using cellular automata.Certainly, fluid flows and other large-scale systems innature can be simulated to a reasonable degree by CA.But concerning the motion of electrons and other fun-damental particles, which involves quantum mechanicsand the fundamental interactions, how will these parti-cles show their behavior with this CA technique?

104 Mad about Modern Physics

In 1911 Marie

Sklodowska-Curie, by

then a double Nobel

Prize winner, was asked

to write a letter of rec-

ommendation for Albert

Einstein who was being

considered for a posi-

tion at the ETH, the

Zurich Polytechnic. She

wrote, “In Brussels,

where I attended a sci-

entific conference in

which M. Einstein also

participated, I was able

to admire the clarity of

his intellect, the breadth

of his information, and

the profundity of his

knowledge. Considering

that M. Einstein is still

very young, one is justi-

fied in placing great

hopes in him and in

regarding him as one of

the leading theoreti-

cians of the future.”

—ALBRECHT FÖLSING, ALBERT

EINSTEIN: A BIOGRAPHY

Cellular Automaton Rule 30 for 50 steps

Time steps

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Over MyHead

105

Over MyHead

U NTIL THE 1920S NO ONE WAS SURE THAT WE

were seeing stars outside our own Milky Way galaxy.

Then, after Edwin Hubble established in 1927 that extra-

galactic galaxies existed and had recession velocities pro-

portional to their distance from us, the cosmology game was

afoot. The rules of the game had been established already by

Einstein in 1916 with his general theory of relativity (GTR).

The verification of one of its major predictions by analyzing

the deflection of starlight passing near the Sun during the

1919 total solar eclipse told everyone that solid theoretical

foundations were in place. But only in the 1990s did the vast

accumulation of data on distant objects, by orbiting satellites

such as the Hubble Space Telescope and the COBE micro-

wave detector, and by a new generation of ground-based

telescopes, transform a conjectural science into real testing

of models of the universe. We present a sample of challenges

from a vast range of possibilities.

10

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202. Olbers’ Paradox

While walking through the fields one night with hisdog, Jan looked up to see a remarkably clear night sky.In an instant, a famous question flashed in his mind:“Why is the sky dark at night?” With his engineeringbackground, he determined that if the universe is uni-formly filled with stars, then their successive sphericalshells would contribute equal amounts, and the skyshould be ablaze with light from all directions. Yet thenight sky remains dark. What is the resolution of thisparadox?

203. Headlight Effect

We live in a universe in which very distant stars haveenormous cosmological redshifts of their light. Thisfact is interpreted as a cosmological recession velocityat nearly the speed of light. Unusual relativistic effectscan be observed when looking at such fast-moving lightsources. We consider a more local version here.

Suppose you are standing next to a straight testtrack that carries a vehicle with a light that shines in acone with an apex angle 45 degrees about the forwarddirection. In the past, you have always seen the light asthe vehicle approached. One day the vehicle for the firsttime is able to reach its highest speed ever: v = 0.9999times the speed of light. But this time you do not see thelight as it approaches. Why not? Do you see the vehicleas it passes and then recedes into the distance? Supposea distant star or galaxy is approaching you at this speed.What would you see? And if receding?

204. Incommunicado?

In later problems we will encounter the behavior of lightnear a black hole, particularly its inability to escape

106 Mad about Modern Physics

The possibility that a

massive object could bend

light rays was discussed

by Newton as early as

1704, and later by Henry

Cavendish. However, the

first actual calculation

of the deflection angle

was published by a

Bavarian astronomer

named Johann Georg

von Soldner in 1803.

Assuming that light was a

corpuscle undergoing the

same gravitational attrac-

tion as a material particle,

Soldner determined how

much bending would occur

for a path that skimmed

the surface of the Sun.

The deviation, while small,

is calculable, and

Soldner’s value was

0.875 seconds of arc. In

1911 Einstein, using the

principle of equivalence,

obtained the same result.

Then in 1915, using the

equations of the general

theory of relativity,

Einstein found that the

deflection had to be 1.75

seconds of arc, twice the

previous value. In recent

decades the result as

been confirmed to a

precision better than 0.1

per cent.

—CLIFFORD M. WILL,WAS EINSTEIN RIGHT?

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Over My Head 107

from a black hole. That is, if you are trapped inside ablack hole and are still alive, you cannot communicatewith your friends outside because nothing escapes.

Meanwhile, consider a related problem in a normalspace environment. Suppose you and your friend are inseparate rocketships that begin next to each other andaccelerate with respect to the stars in opposite direc-tions. You both maintain steady pulsed light communi-cation with each other via intense, nondiverging laserbeams. But your relative speed is increasing each sec-ond as the separation distance grows ever faster. Willthere come a time when neither of you will receive theother’s light beam?

205. Local Accelerations

Einstein formulated the general theory of relativity(GTR) in 1915 based on his Equivalence Principle. Inprerelativistic terms, a uniform gravitational field ofstrength g may be exactly simulated inside a rigid lab-oratory in a completely gravity-free region of space byaccelerating this laboratory with a constant accelera-tion g m/s2 relative to an inertial frame. By releasingtwo small test masses, their behavior reveals the physi-cal environment.

Suppose an unseen massive body is near the rigidlaboratory. What behaviors of the two small test masseswill reveal its presence?

At age twelve Einstein

suddenly became com-

pletely irreligious. Ironi-

cally, this conversion was

the consequence of the

only religious custom his

parents observed, namely

to host a poor Jewish stu-

dent for a weekly meal.

The beneficiary was Max

Talmud (later “Talmey”),

a medical student from

Poland, ten years older

than Albert. Talmud

directed his attention to

popular science books as

well as to various books in

mathematics. Einstein

summed up the results of

Talmey’s influence:

“Through the reading of

popular scientific books I

soon reached the convic-

tion that much in the sto-

ries of the Bible could not

be true. The consequence

was a positively fanatic

[orgy of] freethinking

coupled with the impres-

sion that youth is inten-

tionally being deceived by

the state through lies.

Suspicion against every

kind of authority grew out

of this experience.”

—ADAPTED FROM MAX

JAMMER, EINSTEIN AND

RELIGION—PHYSICS AND THEOLOGY

Massive body

Test masses

Laboratory

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206. Twin Paradox

The twins are five years old when one of then is sent offin a spaceship that travels nearly the speed of light andthe other remains on spaceship Earth. After 50 yearsEarth time the spaceship returns. The twins greet eachother and compare their experiences. We know thatthe twin who experiences accelerations will age slowerand return to Earth much younger than 55 years old.Precisely how does the general theory of relativityexplain the aging of the twin during accelerations?

207. Twin Watches

This problem came from Richard P. Feynman in the1960s while one of us (F. P.), an undergraduate, was withhim in his car on the way to Malibu, California, wherehe gave weekly physics lectures. The third person in thecar, B. Winstein, then a graduate student in physics, con-tributed to a discussion that became quite involved!

Charlotte holds two identical ideal watches at thesame height, one in each hand. She holds one steady inher left hand and tosses the other into the air straightup. At the instant the upward-moving watch is along-side the other at the same height above the ground, shesees that the two watches are synchronized, with theexact same readout value. Later, on its downward free-fall path, she reads the time on both watches when theyare again alongside each other and at the same heightabove the ground. Assuming that the moving watch isalways in free fall, what would you predict for the twowatch readings?

208. Global Positioning Satellites

The global positioning system (GPS) is a modern marvel,with a constellation of at least 24 satellites, each in a 12-hour orbit at an altitude of about 20,200 kilometers,

108 Mad about Modern Physics

Are the planets “arranged”

so that the gravitational

perturbations between

them are smaller than

would be expected from a

random configuration, thus

resulting in long-term sta-

bility of the system? Some

argue that an example of

this favorable arrangement

is given by Neptune and

Pluto, whose orbits appear

to cross if we neglect their

inclinations. Since their

orbital periods are in the

ratio 3:2 Pluto never gets

near to Neptune and actu-

ally approaches more

closely to Uranus. It is

worth noting that the

orbital planes of the plan-

ets are not coincident with

each other or with the

equatorial plane of the sun,

which is inclined at about

7° to the ecliptic.

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Over My Head 109

whizzing around Earth at enormous speeds with respectto the ground beneath them. Each satellite knows itsown position and sends out signals with this informa-tion. The GPS handheld receiver uses the signals from atleast four different satellites to calculate its own positionto within a few meters or better when a local referencesignal is present. Yet within minutes the accuracy wouldreduce to many kilometers of error if one of Einstein’sdiscoveries were not an essential part of the calculationsin the GPS system. What are we referring to?

209. Solar Redshift

The light emitted from the Sun shows a redshift of thespectral lines even though our distance to the Sun isfixed during the measurement process. Why so?

210. Orbiting Bodies

When a body such as a planet orbits around a moremassive body such as the Sun, the orbit does not closeon itself, as expected from Newton’s universal law ofgravitation and Kepler’s laws. The general theory ofrelativity (GTR) calculates the correct value for thisprecession of the orbital ellipse, determining that itscomplicated equations reduce to an equation similar inform to that of the classical Kepler problem, with anadditional quadratic term that causes the precession.Can you provide a conceptual argument in GTR for theprecession of the ellipse?

211. Gravitational Lensing

In examining the universe, astronomers utilize a tech-nique called gravitational lensing of the light from dis-tance stars. Supposedly, space itself can act as a lens forlight rays. How can the emptiness of space—the vac-uum itself—around stars and galaxies focus light?

The Russian astrophysi-

cist George Gamow

decided early in life

that traditional religion

could not be trusted.

After watching Commu-

nion in the Russian

Orthodox Church, he

decided to see for him-

self whether red wine

and bread could trans-

form into the blood and

flesh of Jesus. He held

a bit of the blessed

bread and wine in his

mouth, ran home from

church, and placed the

specimen under the

lens of his new toy

microscope. It looked

identical to an ordinary

bread crumb that he

had prepared at home

earlier for comparison.

“I think this was the

experiment which made

me a scientist,” he

recalled. In the 1940s,

Gamow and others pre-

dicted the existence of

a cosmic background

radiation.

—COREY S. POWELL, GOD IN

THE EQUATION: HOW EINSTEIN

BECAME THE PROPHET OF THE

NEW RELIGIOUS ERA

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212. Cosmological Redshifts

The light from a distant galaxy can exhibit a significantcosmological redshift. If the cosmological redshift isnot a velocity redshift, what is its origin? Can the twoeffects be distinguished from each other by observingthe spectrum of the galaxy or other light source?

213. Tired-Light Hypothesis

Since the 1920s there has been a popular hypothesistrying to explain the cosmological redshift as a so-called tired-light effect—that is, the light loses energy asits photons race through space, getting more tired withdistance, like a long-distance runner completing a race.What two specific pieces of evidence rule out this expla-nation for the cosmological redshift?

214. Black Hole Entropy

A black hole has an entropy proportional to its surfacearea, so it must have a temperature above absolutezero. What would be evidence for this temperature?

215. Black Hole Collision

Two black holes collide head-on. Will they coalesceinto one black hole?

110 Mad about Modern Physics

In various writings, not

all published, Isaac New-

ton intimated that

comets were divine

agents destined to

reconstitute the entire

solar system, to prepare

sites for new creations,

and to usher in the

Millennium.

—SARA SCHECHNER GENUTH,COMETS, POPULAR CULTURE,AND THE BIRTH OF MODERN

COSMOLOGY

In a letter to E. Büsching,

the author of

dated 25

October 1929, Einstein

declared that a belief in

a personal God seems

“preferable to the lack of

any transcendental out-

look of life.”

—MAX JAMMER, EINSTEIN

AND RELIGION—PHYSICS

AND THEOLOGY

My comprehension of

God comes from the

deeply felt conviction of

a superior intelligence

that reveals itself in the

knowable world. In

common terms, one

can describe it as

“pantheistic” (Spinoza).

—ALBERT EINSTEIN, 1923 IN

ALICE CALAPRICE, THE

EXPANDED QUOTABLE EINSTEIN

No God,

There Is

Earth Quasar

Galaxy lens

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Over My Head 111

216. Centrifugal Force Paradox

The general theory of relativity (GTR) predicts that incertain circumstances the centrifugal force may bedirected toward, not away from, the center of circularmotion. In fact, if an astronaut could steer a spacecraftsufficiently near to a black hole, the astronaut wouldfeel a centrifugal force pushing inward, not outward,contrary to everyday experience! What is the concep-tual explanation of the unusual result?

217. Geodesics and Light Rays

In conventional geometry, the geodesic is the shortestcurve between two points measured by counting howmany rulers fit along the curve. When learning relativ-ity theory, one often reads statements that conflict withintuition, such as the following: “In any space-time,with or without a gravitational field, light alwaysmoves along geodesics and traces out the geometry ofspace-time.” “In a space warped by a gravitationalfield, the light rays are curved and in general do notcoincide with geodesics.” Why are these phrases, takenfrom the general theory of relativity (GTR), not reallyin conflict with each other?

218. Galaxy Rotation

RIEMANN ANTICIPATES

EINSTEIN

Ever since Newton, scien-

tists had considered a

force to be an instanta-

neous interaction between

two distant bodies. How-

ever, over the centuries,

critics argued that this

action-at-a-distance was

unnatural, because it meant

that one body could change

the direction of another

without even touching it.

Georg Bernhard

Riemann (1826–1866)

developed a radically new

physical picture, banishing

the action-at-a-distance

principle. To Riemann,

He con-

cluded that electricity,

magnetism, and gravity are

caused by the crumpling

of our three-dimensional

universe in the unseen

fourth dimension. Thus a

“force” has no independent

life of its own; it is only

the apparent effect

caused by the distortion

of geometry.

—MICHIO KAKU, HYPERSPACE:A SCIENTIFIC ODYSSEY THROUGH

PARALLEL UNIVERSES,TIME WARPS, AND THE TENTH

DIMENSION

of geometry.

“force” was a consequence

Velocity

Radius

Visible edge

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One of the great surprises in astronomy is the rota-tional behavior of galaxies—that is, all the stars in thegalactic disk revolve at roughly the same tangentialspeed! Two immediate conflicts with conventionalphysics arise: (1) If Newtonian gravitation and Kepler’slaws apply, they would dictate a decrease in star veloc-ity with increasing radius from the galactic nucleus,like the planets of the Solar System do. (2) If the spiralarms of a spiral galaxy are to retain their integrity andpersist for at least ten complete revolutions, as theyhave for the Milky Way, there must be something pre-venting them from wrapping numerous times. Byassuming that Newton’s universal law of gravitationapplies to these galactic problems, what general type ofmass/energy distribution must be proposed to explainthe rotational velocity curve? What further hypothesesmight you propose?

219. Cosmic Background Radiation

Cosmic background radiation was first detected in themicrowave region in the 1960s and exhibits a perfectblackbody spectrum equivalent to the radiation from asource at a temperature of 2.72 K. One would expect

112 Mad about Modern Physics

Venus always presents

the same face to Earth

when the two planets

are at their closest

approach, suggesting

that its peculiar rotation

may be due in part to

terrestrial action. A

simple calculation will

show, however, that

solar tidal action will

dominate that of Earth

on Venus, and so it is

difficult to see how

such a situation has

evolved.

God is what mind

becomes when it has

passed beyond the

scale of our comprehen-

sion. God may be either

a world-soul or a collec-

tion of world souls. So I

am thinking that atoms

and humans and God

may have minds that

differ in degree but not

in kind.

—FREEMAN DYSONIntensity (W/m3)

0.005 0.001 0.0015 0.002

Wavelength (m)

2.72 K

3

2

1

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Over My Head 113

lots of remnant starlight all over the universe in allparts of the electromagnetic spectrum emitted for thepast 10 billion years or so. Yet this microwave back-ground radiation is not this light emitted from stars.How do we exclude this starlight?

220. Planetary Spacings

For some people the orbital radii for the planets in theSolar System seem to follow a regular pattern. The pat-tern was originally called the Titius-Bode law beforePluto was discovered. According to this numericalscheme, the semimajor axis of a planet’s orbit a = 0.4 +(0.3) 2n, where n is taken as negative infinity for Mer-cury, zero for Venus, and has increasing integer valueby one unit for each successive planet. Neptune doesnot fit in this scheme, and the scheme may not repre-sent an underlying physics.

In the 1990s, applying chaos theory to gravitation-ally bound systems, L. Nottale found that statistical fitsindicate that the planet orbital distances, including thatof Pluto, and the major satellites of the Jovian planets,follow a numerical scheme with their orbital radii pro-portional to the squares of integers n2 extremely well!The planets were divided into two groups, the inner

NOT ALL STARS ARE

ROUND!

Many red giant stars,

including the bright star

Betelgeuse and the well-

known variable star Mira,

exhibit peculiar egglike

shapes, presumably

because of the huge

convection currents roil-

ing their filmy outer lay-

ers. Astronomers also

found huge cocoons of

hydrogen gas surround-

ing hot blue stars and

clouds of titanium oxide

billowing off red giants’

surfaces.

—“MOONBALL: ASTRONOMERS

BEAT A PATH TO HIGH

RESOLUTION,” SCIENTIFIC

AMERICAN (JULY 1993)

Many of the oldest

stars and star clusters

in the galactic halo of

the Milky Way move in

retrograde orbits—that

is, they revolve around

the galactic center in a

direction opposite to

that of most other stars.

—SIDNEY VAN DEN BERGH

AND JAMES E. HESSER, “HOW

THE MILKY WAY FORMED,” SCIENTIFIC AMERICAN

(JANUARY 1993)

Jupiter

Planet distances

Actual

Bode

Nottale

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planets Mercury, Venus, Earth, and Mars being at n = 3, 4, 5, and 6, respectively, and the outer planets,starting with Jupiter at n = 1. The two sets can becombined into one set with Mercury at n = 3, Jupiter atn = 10, and so on. The lack of planets at some integerscan be attributed to the history of the Solar System anddoes not indicate a failure of the prediction.

Other researchers claim that the Nottale sets ofintegers are not unique and that several alternative setsof integers exhibit excellent fits, raising the question ofwhether there is actually a unique pattern in the orbitalspacings. In addition, there are known orbital reso-nances for the satellites of the Jovian planets that causesome of the apparent patterns in the satellite spacings.

How would you determine whether any of theclaimed patterns are physically significant or simplynumerology?

221. Entropy in the Big Bang

“The primordial fireball was a thermal state—a hot gasin expanding thermal equilibrium. But the term ‘ther-mal equilibrium’ refers to a state of maximum entropy.However, the second law demands that in its initialstate, the entropy of our universe was at some sort ofminimum, not a maximum!” How would you resolvethis paradox raised by R. Penrose?

222. Gravitational Wave Detectors

Radio wave detectors are calibrated by sending outradio waves from a transmitter several wavelengthsaway and more. Why can’t builders of gravitationalwave detectors do the same thing? After all, one couldput two large masses at opposite sides of a rotatingplatform and spin them around to have a gravitationalwave source for detector calibration.

114 Mad about Modern Physics

Experience may suggest

the appropriate mathe-

matical concepts, but

they most certainly can-

not be deduced from it.

Experience remains, of

course, the sole criterion

of physical utility of a

mathematical construc-

tion. But the creative

principle resides in math-

ematics. In a certain

sense, therefore, I hold it

true that pure thought can

grasp reality, as the

ancients dreamed.

—ALBERT EINSTEIN, 1933, IN

GERALD HOLTON, THEMATIC ORI-GINS OF SCIENTIFIC THOUGHT

Popular books claim that

Fred Hoyle coined the

term to ridicule

the theory; but Hoyle dis-

puted that. “The BBC was

all radio in those days,

and on radio, you have no

visual aids, so it’s essen-

tial to arrest the attention

of the listener and to hold

his comprehension by

choosing striking words.

There was no way in which

I coined the phrase to be

derogatory; I coined it to

be striking, so that people

would know the difference

between the steady state

model and the big bang

model.”

—KEN CROSWELL,THE UNIVERSE AT MIDNIGH

big bang

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Over My Head 115

223. Space Curvature

The general theory of relativity (GTR) has beenchecked and verified at local distance scales. We knowthat the GTR may not explain the rotation curves ofgalaxies without the introduction of “dark matter.” Wedo not expect the GTR to work for extremely small dis-tances, for extremely short time intervals, or for cos-mological distances—that is, whether the GTRcorrectly explains the universe on a global scale. TheGTR, for example, allows for the overall curvature ofspace but does not predict its global value. In betterwords, the GTR does not fully predict the geometry ofspace, neither determining the global shape nor theconnectedness of space.

Suppose you were given the task of measuring theoverall curvature of space. One way might be to countthe number of stars at each radial distance, say, and plotthe number found versus distance. How does thismethod determine the curvature of space? Does thistechnique work for both continuous and discrete spaces?

224. The Total Energy

The total energy in the observable universe can beshown to be zero by adding the total mass energy inmatter and radiation to the total gravitational potentialenergy. That is: energy total = mass energy + gravita-tional energy. Does this result mean that the creation ofmatter out of nothing contradicts no physical conser-vation law?

225. Different Universes?

The present limited understanding of our universeallows for much speculation with regard to whether we live in but one of many universes, possibly with

WHAT’S IN A NAME?

Planck happened upon

when

in 1906 he was groping

for a name to distin-

guish the Lorentz-

Einstein theory (of the

deformable electron)

from Abraham’s (theory

of the rigid, spherical

electron). It was in the

discussion following

Planck’s 1906 lecture

on Kaufmann’s experi-

mental electron data

and their theoretical

interpretation that

Planck’s term

was

embellished by the

experimentalist

A. H. Bucherer to

Although Einstein

had in his published

work from the very

first referred to the

he

did not use the designa-

tion

in print until 1907, as

a variant of Planck’s

expression. Felix

Klein’s suggestion of

did

not catch on.

—ERIC SHELDON, “RELATIVITY

OR INVARIANCE?,” AMERICAN

JOURNAL OF PHYSICS

(SEPTEMBER 1986)

Invariantentheorie

Relativitätstheorie

Relativitätsprinzip,

Relativitätstheorie.

Relativtheorie

“Relativtheorie”

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connections among them. These wild conjectures arepermitted simply because we do not know enoughabout the origins of the fundamental constants such asPlanck’s constant, the gravitational constant, or the lep-ton and quark masses, for example. Indeed, the otherpossible universes could have different values for theseconstants. Suppose that the lepton and quark mass values are discovered to be determined by some funda-mental properties in mathematics such as the invariantsof elliptic functions. How might this discovery endspeculation about many universes?

116 Mad about Modern Physics

This world is indeed a

living being endowed

with a soul and intelli-

gence . . . a single living

entity containing all

other living entities,

which by their nature

are related.

—PLATO

Cosmic bombardment

of Mars by planetary

debris has sent hun-

dreds of tons of Martian

rocks falling to Earth.

Experts estimate that a

Mars rock lands as

often as every three

days, and that billions

have done so over

time—even though only

13 of them have been

identified so far. The

rain of debris from

Mars was hardest,

experts agree, in the

Earth’s early days. And

the reverse trip was far

less likely because the

Sun pulls Earth debris

away from Mars and

toward itself.

—WILLIAM J. BROAD, “WANNA

SEE A REAL LIVE MARTIAN? TRY

THE MIRROR,” THE NEW YORK

TIMES (MARCH 14, 1999)

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Crystal BluePersuasion

117

Crystal BluePersuasion

S OME PROBLEMS THAT COULD HAVE BEEN PUT

in the previous chapters are presented in this chap-

ter. We have collected these special problems for this grand

finale. Some of the previous problems will provide significant

clues toward answering challenges in this hodgepodge of a

collection, while many challenges here are new to most

readers. We hope that you enjoy them as much as we have.

11

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226. Iodine Prophylaxis

Supposedly, in the event of a nuclear emergency, iodinetablets offer protection from radioactive iodine. Howcan this preventive measure work? Isn’t all the iodine,tablets or not, exposed to the ambient radiation?

227. Bicycle Tracks

If you came upon this set ofbicycle tracks meanderingaround in the mud,could you determinewhich way the bicyclewas going simply byexamining the tracks?Remember that thefront wheel and the rear wheel make separate tracks.

228. Earth Warming

Over the past several decades there has been consider-able concern over the possible slow rise of a few tenthsof a degree Celsius in the average temperature ofEarth’s atmosphere and its surface. Most of the concernseems to be associated with the greenhouse effect onthe radiation from the Sun. However, assuming that theaverage temperature of Earth can be defined unam-biguously so that the small rise in average temperatureis true, can there have been additional thermal energycoming from within Earth to cause this rise?

229. Frequency Jamming

Suppose one desired a noisy emitter of electromagneticwaves at all frequencies simultaneously. Such a devicemight be useful to jam undesired cell phone signals, forexample. How could you do this simply?

118 Mad about Modern Physics

HIMMEL’S THEORY OF

ACADEMIC TYPES

The greater the cer-tainty of one’s results,the less the concernwith others’ opinions ofoneself.

Thus at the end ofthe spectrum occupiedby sociologists and pro-fessors of literature,where there is uncer-tainty as to how to dis-cover the facts, thenature of the facts tobe discovered, andwhether indeed thereare any facts at all, allattention is focused onone’s peers, whoseregard is the sole crite-rion for professionalsuccess. Great painsare taken in the devel-opment of the impres-sive persona, withexcessive attentiongiven to distinguishedappearance and fault-less sentence structure.

At the other end,where, as the mathe-maticians themselvesare fond of pointing out,“a proof is a proof,” noconcern need be givento making oneselfacceptable to others;and as a rule nonewhatsoever is given.

—REBECCA GOLDSTEIN, THE

MIND-BODY PROBLEM: A NOVEL

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Crystal Blue Persuasion 119

230. Light Energy

We know that the speed of light is the same for allobservers in inertial frames. If so, are the momentumand energy of light as measured by all observers thesame even when the light source is moving toward theobserver?

231. Acid Rain

Chemists define the quanity pH = -Log [H+], where[H+] is the aqueous concentration of H+ ions. Purerainwater is a neutral solution that has a pH of 7 whenthe droplets form. Will these droplets falling throughclean, unpolluted air have a pH of 7 by the time theyhit the ground?

232. Electrical Current

Upon turning on a lamp, Raymond wonders, “Abouthow fast do the electrons in house wiring move as theyprovide electrical energy to the lamps and other elec-tronic devices?”

233. Earth’s Orbit

Earth’s elliptical orbit is not fixed in orientation withrespect to the stars. Why not?

If the universe of sci-

ence is not evident to

our ordinary senses but

is elaborated from cer-

tain key perceptions, it

is equally the case that

these perceptions

require their appropriate

instruments: micro-

scopes, Palomar tele-

scopes, cloud chambers,

and the like. Again, is

there any reason why the

same should not hold for

religion? A few words by

that late, shrewd lay the-

ologian, Aldous Huxley,

make the point well. “It is

a fact, confirmed and

reconfirmed by two or

three thousand years of

religious history,” he

wrote, “that Ultimate

Reality is not clearly and

immediately appre-

hended except by those

who have made them-

selves loving, pure in

heart, and poor in spirit.”

—HUSTON SMITH, BEYOND THE

POST-MODERN MIND

I swear to you that to

think too much is a

disease, a real actual

disease.

—FEODOR DOSTOEVSKY, NOTES

FROM UNDERGROUND

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234. Crystal Growth

Many children grow crystals in solution as a scienceproject. How does a crystal grow from a small “seed”to its final size? That is, exactly how do the atomsknow where to adhere to the growing structure withoutfouling up the precise cubic crystal structure develop-ment, for example? Do you see the dilemma? And thesurprise?

235. Ruby, Sapphire, and Emerald

How are ruby, sapphire, and emerald crystals related?How do they produce their colors?

236. Kordylewski Clouds

By Kepler’s laws any object orbiting the Sun in an orbitsmaller than Earth’s has a faster speed. So how can dustparticles placed in solar orbit along the Earth-Sunradial line but closer to the Sun have the same speed asEarth?

237. Twist

Scooter

There is a type of three-wheeled scooter with ahandlebar extending ver-tically upward from theapex of a V-shaped hori-zontal metal frame thathas three wheels. Thefront wheel can rotateabout a vertical axis atthe foot of the verticalhandlebar at the apex

120 Mad about Modern Physics

Even when all the possi-

ble scientific questions

have been answered, the

problems of life remain

completely untouched.

—LUDWIG WITTGENSTEIN

The shooting stars are

meteors usually the size

of a grain of sand.

General relativity is

actually less relativistic

than special relativity.

The complete feature-

lessness of flat space-

time, its homogeneity

and isotropy, are what

ensure that positions

and velocities are

strictly relative. As soon

as spacetime acquires

“bumps,” or local regions

of curvature, it becomes

absolute because posi-

tion and velocity can be

specified with respect

to the bumps. Space-

time, instead of being

merely a featureless

arena for physics, is

itself endowed with

physical properties.

—BRYCE S. DEWITT,“QUANTUM GRAVITY,”

SCIENTIFIC AMERICAN

(DECEMBER 1983)

Feet go here

Flexible hinge

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Crystal Blue Persuasion 121

intersection, and the two rear wheels are at the ends ofthe arms of the V. The two arms of the V are hinged attheir intersection at the front of the scooter so they canrotate about the vertical axis of the hinge—that is, theycan form a wider or a narrower V angle within limits.The rider places one foot on each arm of the V andswivels his or her body from side to side to tilt the verti-cal handlebar from side to side to make the vehicle goforward or backward. Can you explain the physics of itsforward motion?

238. Unruh Radiation

Physicist J. Bekenstein determined that a particle accel-erating in a vacuum experiences a blackbody radiationbath around itself at a temperature directly propor-tional to its acceleration. By the equivalence principle,would a particle at rest in a gravitational field alsoexperience this blackbody radiation bath?

239. Star Diameters

One can determine the diameter of a distant star eventhough the diameter cannot be measured by parallax.The process uses the intensity interference, not the

A 1972 study of the polit-

ical views of academic

scientists shows that the

high achievers in acade-

mia, particularly the

physicists and mathe-

maticians, are signifi-

cantly more liberal than

the rank and file of the

academic scientists and

engineers. The study

ranks the percentage of

scientists ranked liberal,

from the highest to the

lowest, as follows: math-

ematics, physics, biologi-

cal sciences, chemistry,

geology, medicine, and

engineering. The authors

suggest that if the scien-

tist’s work is character-

ized by a high degree of

intellectuality, creativity,

and orientation toward

basic research, the sci-

entist will very likely be

near the liberal end of

the scale, and will tend to

be critical of existing

social institutions and

practices. Conversely,

if his work is character-

ized by practicality,

routine, and know-how,

the scientist will tend to

be conservative.

—E. C. LADD JR. AND S. M.LIPSET, “POLITICS OF ACADEMIC

NATURAL SCIENTISTS AND ENGI-NEERS,” SCIENCE (JUNE 9, 1972)

Star light Star light

Multiplier

Correlator

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amplitude interference, between the light entering twoidentical photodetectors (telescopes) from the left sideof the star surface, and from the right side of the starsurface. This process is valid even though the star issimply a point in the light-gathering optics of eitherphotodetector. Can you explain the physics?

240. Glauber Effect

Does a standard incandescent lightbulb emit singlephotons? Paired photons? Triplets?

241. Bird Sounds

Practically all birds and other animals emit sounds thathave a fundamental frequency and several harmonics.Some birds, however, can emit just the fundamentalwith no harmonics! Measurements inside the birdreveal that the original sound has harmonics. So howdoes the bird eliminate them before they escape into thegreat outdoors?

242. Spouting Alligator

Some alligators can submerge themselves slightlyunderwater and vibrate their bodies so that numerous

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Water surface

Talented alligator

The surest sign that

intelligent life exists

elsewhere in the uni-

verse is that it has never

tried to contact us.

—BILL WATTERSON, CALVIN

AND HOBBES COMIC STRIP

Among those who

become famous, one

specific category is

greatly overrepresented:

firstborn sons who lost

their fathers at a young

age and were emotion-

ally rejected by their

mothers. The foremost

example in physics:

Isaac Newton.

Q: Why do chemists call

helium, curium, and

barium the medical

elements?

A: Because if you can’t

helium or curium, you

barium!

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Crystal Blue Persuasion 123

individual water droplets are projected upward simulta-neously a foot or more directly above the backs of theirheads. What is the physics here?

243. Hair-Raiser Function

One of us (F. P.) first heard about the hair-raiser func-tion (HRF) from physicist Richard Feynman. We intro-duce this function here as a curiosity to stimulate themind. And if one pulls a hair on top of the headupright, this function is probably a good representationof its fast vertical increase with horizontal distance.

Most mathematical functions are easy to define,and so are their inverses. Physicists utilize a tremendousvariety of mathematical functions, the two most com-mon being the power of a quantity and the exponentialof a quantity. Physicists also use mathematical opera-tions that may not seem to qualify as a function, suchas the Dirac delta function δ(r – r0). Powers are preva-lent in fundamental laws dictated by geometrical sym-metries such as the universal law of gravitation and theCoulomb law, both having potentials proportional to1/r and forces proportional to 1/r2 for ideal pointsources. The exponential function increases more

When Glenn T. Seaborg

and his colleagues at the

Lawrence Berkeley

National Laboratory in

California were able to

make a new element in

1940 with 94 protons in

its huge nucleus, they

could not at first imag-

ine that anything more

massive would ever be

obtained, and so they

called their new element

(later it would

be renamed plutonium).

A typical person age fif-

teen to forty-five grows

about an inch overnight.

When we lie down to go

to sleep at night, the

spaces in our spine

expand and we get taller.

In the morning when we

get up, gravity reasserts

its downward pull and

very quickly we go down

a bit. Astronauts typi-

cally stretch more than

two inches in weightless-

ness, but as soon as

they return to Earth

gravity, over a matter of

hours they shrink to

normal height.

ultimium

Hair-raiser function

1200

0 1 2 3

0 0

0 1 2 3 4

2 × 1024

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rapidly than practically any other known function andis used whenever the change in a quantity is propor-tional to the quantity itself, such as in growths anddecays.

The hair-raiser function HRF(x) can be defined byexample on how it maps integers to integers. TheHRF(1) is 1. The HRF(2) = 22. The HRF(3) = (33)3, andso on. Notice the grouping with parentheses. Oneneeds a calculator for most of the higher-integer HRFs.Certainly the HRF is a one-to-one mapping.

How does one calculate the HRF of a non-integer?Of a complex number? How does one determine theinverse of the HRF? That is, given a number such as 42,how does one determine what number is mapped bythe HRF into 42? And finally, of what potential use isthe hair-raiser function?

244. Space Crawler

In 1999, the U.S. Patent Office awarded a patent to apropulsion device that is a base frame with a slidingcarriage on the frame and two counterrotating massesthat together couple and decouple to the base frame tomove the carriage forward and backward in a compli-cated motion. An onboard battery provides the energyfor all internal movement. When the rotating massesare not coupled to the frame, and when placed on anearly frictionless air table, the whole device simplyoscillates forward and back repeatedly, as expected.When the coupling is allowed to occur at specificphases in their rotational cycle, the whole device movesonly forward in a continual sequence of spurts that arelonger with less air table friction! Will this device oper-ate likewise in space?

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Cosmic radiation comes

from outer space. The

radiation dose from

cosmic radiation

increases with altitude,

roughly doubling every

6,000 feet. Therefore,

a resident of Florida (at

sea level) on average

receives about 26 mil-

lirem, one-half the dose

from cosmic radiation

as that received by a

resident of Denver,

Colorado, and about

one-fifth of that by a

resident of Leadville,

Colorado (about two

miles above sea level). A

passenger in a jetliner

traveling at 37,000 feet

would receive about 60

times as much dose

from cosmic radiation

as would a person

standing at sea level for

the same length of time.

I don’t think there is one

unique real universe. . . .

Even the laws of physics

themselves may be

somewhat observer-

dependent.

—STEPHEN HAWKING

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Answers

Chapter 1The Heat Is On

1. Egg into a Bottle

Newton’s second law explains theresult. When the egg is resting on thebottle, the ambient air in the roomplus the gravitational force on the eggby the Earth together exert a totaldownward force on the egg that isequal to the total upward force pro-vided by the contact force of the bottleplus the force of the air inside. ByNewton’s second law, a downwardacceleration begins when there is a netforce downward. To get the egg toaccelerate downward, you mustreduce the upward force of the airinside the bottle. This action reducesthe total upward force, allowing a netforce downward, with the resultingacceleration downward dictated by netforce = mass times acceleration.

The correct timing requires one towait until the paper dropped inside thebottle has stopped burning, thenimmediately and carefully place theegg on the opening. The warmed air

inside the bottle will begin to cool justas the burning finishes. The egg sealsthe opening, so the air pressure insidedecreases as the cooling progresses.The total downward force will eventu-ally be greater than the total of theforces resisting the egg at the entrance,and the net force downward will accel-erate the egg into the bottle. Themovement continues until the eggdrops to the bottom. Kerplop!

2. Egg out of a Bottle

The hard-boiled egg is in the bottleand the goal is to remove this eggwithout damage. When the total out-ward force acting on the egg exceedsthe total opposing force resisting itsexit, then the egg will accelerate out ofthe bottle. Assume that the bottle isheld vertically upside down. Earth’sgravitational force acting on the egg—that is, its weight—and the force of theair inside the bottle both contribute tothe total downward force acting onthe egg. The upward force is the con-tact force of the bottle (which includesstatic friction) plus the force of theambient air in the room.

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To remove the egg, you need tocreate a pressure difference betweenthe air inside and the air outside thebottle, with the greater pressure inside.Hold the bottle with its mouth nearyour mouth and its bottom somewhathigher so that the egg is positioned farforward, to the neck of the bottle, butnot completely blocking the neck.Exhale a tremendous burst of breathinto the bottle to suddenly increase theinside air pressure. The Bernoullieffect, the reduction in air pressureperpendicular to the air stream flowdirection, caused by the rush of airflowing around the egg plus theincreased inside air pressure, will aidin pushing the egg through the neckand mouth of the bottle. When the netforce outward occurs, the egg acceler-ates outward. Sometimes the egg justpops out and must be caught in themouth, and at other times the eggmust be gently removed at the open-ing. The more vertical the bottle is, themore help one has from the gravita-tional force of the Earth.

3. Sugar

One can usually dissolve about fivecups of sugar in one cup of water!Very simply, sugar molecules cansqueeze into the empty spaces amongthe water molecules, so they are notreally occupying more space. Thewater forms somewhat of an open

latticework with water moleculesloosely connected, so the “holes” inthis water lattice can accommodate alarge number of other molecules. Thesugar molecules form temporaryhydrogen bonds to the water mole-cules, these bonds breaking and re-forming continually. Essentially therule is “like dissolves like.” Of course,the sugar molecules are quite large,and one cup of sugar molecules con-tains only about one twenty-fifth thenumber of water molecules in a cup.So there are many water molecules toevery sugar molecule in the solution.

Wolke, R. L. What Einstein Told His Cook:Kitchen Science Explained. New York: W. W. Norton, 2002, pp. 21–22.

4. Kneading Bread

Each successive kneading of the breaddough distributes the CO2 gas releasedby the action of the yeast to make afiner texture—that is, smaller holesmore evenly distributed throughoutthe bread volume.

Initially, the concentration of yeastis not uniform but has some non-uniform volume distribution in thebread dough. Where there is moreyeast, there will be more CO2 gas pro-duced by the yeast chemistry and usu-ally bigger bubbles in the region. Atthe molecular level, CO2 moleculesreleased by the yeast will diffuse intothe surrounding dough somewhat,

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probably not moving very far in theavailable time. Some of the gas bub-bles may even coalesce to form biggerbubbles. Without further kneading,some places in the bread will possessmany bubbles or large bubbles, andother places may have very tiny bub-bles or none. We have all seen breadwith a non-uniform distribution ofbubbles, or even with one large bubblesomewhere. A more thorough knead-ing would eliminate these oddities,unless they are planned.

5. Measuring Out

Butter

Butter floats on water because its den-sity is less than the density of water.The recommended procedure in manycookbooks for measuring one-half cupof butter is: put one-half cup of waterinto the measuring cup and then addpieces of butter until the water level ispushed up to the one-cup level. Oftena reference to Archimedes’ principle isgiven. (According to Archimedes’ prin-ciple, a body wholly or partiallyimmersed in a fluid will be buoyed upby a force equal to the weight of thefluid that it displaces.)

However, this recommended meas-uring procedure does not useArchimedes’ principle! And the meas-ured amount of butter is not exactlyone-half cup!

If ice chunks were used instead ofbutter, then the procedure is correct.As a check, when floating ice melts,the water level does not change.Therefore, the recommended proce-dure is correct for measuring one-halfcup of ice. But the density of butter isnot the same as that of the ice, and thedensity of melted butter is not thesame as that of the water. Therefore,the immersed volumes of ice and but-ter would be different.

However, if the butter is heldunder the water surface when water isadded, then the butter measurement iscorrect when the water reaches theone-cup level. One is simply measur-ing the volume of the butter, and onedoes not use Archimedes’ principle forthis measurement.

6. Milk and Cream

The milk is “heavier,” meaning moredense. Cream floats to the top in a milk-cream mixture, so the cream is lessdense. Quite often people confuse massdensity with liquid flow sluggishness.The two properties are unrelated. Manyof these people will think that the creamis more dense. Anyone who has milkeda cow or a goat has seen the cream atthe top. A pint of light cream weighsabout 1.5 grams more than a pint ofheavy cream on average. Why? Becauseheavy cream has more fat per volume,and fat is less dense than water.

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Separation of materials by densityhas been an important method for mil-lennia. Gold and platinum have beenseparated from other elements bydumping the ores into a hot bath oflead. The specific density of lead is11.36, of gold 19.32, and of platinum21.45, so gold and platinum atomssink and practically all other elementsand compounds float. Of course, theworkers must not breathe in the leadatoms in the vapors.

7. Straw and Potato

Simply placing the straw in one handand trying to push the straw throughthe hard potato leads to a frustratingfailure. The straw material—paper orplastic—cannot take much compres-sion before bending sideways. So thissideways bending must be prohibitedif success is possible. We can use air pressure to help make the strawmore rigid.

Pinch the straw between yourthumb and forefinger about twoinches from the end that is farthestfrom the potato and squeeze tightly.Hold the potato carefully but securelyin the other hand placed in a horizon-tal plane with thumb on one side andfingers on the other side of the potato.Make sure that no part of the handwill be in the path of the straw—thatis, avoid having any part of your hand

on the top or the bottom of the potato.With a sudden thrust, drive thepinched straw into the potato held inthe other hand. The straw goes rightthrough. Why? The trapped air uponcontact is compressed inside the strawand helps the straw remain rigid. Thepaper or plastic is stretched taut and ismore difficult to bend significantly.Alternately, one could pinch the strawin a vertical position in a vise or aclamp and drop the potato onto thestraw (or a collection of straws).

8. Blueberry Muffins

The downward drift of the blueberriesin the warm batter is caused by thegravitational force of the Earth, so onemust increase the friction between thebatter and the surface of the berry tohinder this settling. One could makethe batter thicker, but this solutionmay not produce the desired muffintexture. Instead, before mixing theblueberries in the batter, dampen themslightly and shake them in a bag offlour. The flour attaches to the berrysurface and increases the static frictionwith the batter, keeping the berriesuniformly distributed in the muffin.

The physics involves Newton’ssecond law, found in every high school textbook, that is, net force =mass × acceleration. The downwardgravitational force on the blueberry

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must be balanced by an upward forceapplied by the batter through frictionto produce zero net force in the verti-cal direction and therefore zero accel-eration downward from rest.

In this case, the maximum value ofthe static frictional force (equal to thecoefficient of friction times the force ofthe batter perpendicular to the blue-berry’s attempted movement) has notbeen exceeded. Without the flour coat-ing, the static friction coefficient is toosmall. The maximum static frictionalforce upward provided by the batteralone is too small. The berry acceler-ates downward until reaching the crit-ical velocity, for which the net force isagain zero, so the blueberry driftsdownward with no acceleration. Withthe flour coating, the coefficient ofstatic friction is large enough to not beexceeded and the downward gravita-tional force is always balanced by thestatic frictional force upward.

At the atomic level, friction involveselectrical forces acting between atomsand between molecules. Still an activeresearch area, the influence of quantummechanical effects is significant. Evensound waves contribute good vibrationsin this field, called tribology!

Krim, J. “Friction at the Atomic Scale.”Scientific American 275, no. 4 (1996):74–80.

Miller, J. S. The Kitchen Professor. Sydney:Australian Broadcasting Commission,1972, pp. 22–24.

9. Can of Soup

Turn the can of soup upside down andopen the bottom. Then turn the canover and watch the concentrate beingpushed out by the weight of the more-liquid stuff, assuming that theupward-acting static frictional force ofthe wall with the concentrate balancesthe weight of the concentrate. Theweight of the liquid thus provides thenonzero net force downward to accel-erate the liquid downward by New-ton’s second law.

If there is delay in this evacuationprocess, allowing some air into the liq-uid region behind the solid concentratemight expedite the motion. Sometimesthe air seal at the wall is very good, sothat as the concentrate slips outward, asignificant inward pressure differencecan build up to slow down the extrac-tion process. In addition, the moleculesin the soup may interact more vigor-ously than expected via the electricalforce between the concentrate and thewall of the can (i.e., the viscous force,the surface tension, etc., may be largeenough to make the extraction evenmore challenging).

10. Salt and Sugar

Salt and sugar do their work on bacte-ria by osmosis, dehydrating them sothat they die or are deactivated. Abacterium in very salty water has a

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saltier environment outside its cellmembrane than inside. Water mole-cules will move from its inside throughits water-permeable membrane to theoutside to balance the salt concentra-tions, a process called osmosis. Thebacterium shrivels up and dies. Sugarworks by the same process to preservefruits and berries. In the markets todayone can buy “cured” hams and otherpork products that use both salt andsugar to enhance the flavor.

Wolke, R. L. What Einstein Told His Cook:Kitchen Science Explained. New York: W. W. Norton, 2002, pp. 137–138.

11. Defrosting Tray

The “miracle” defrosting tray is simplyaluminum metal, and one could use athick aluminum frying pan or otherpiece of aluminum or copper metaltray to do the thawing as quickly, aslong as there is no coating on themetal. Nonstick pans have coatingsthat are poor heat conductors. Metalsare the best heat conductors becausethey have about 1023 conduction elec-trons per cubic centimeter available totransfer thermal energy from the hottersource to cooler regions. For thawingpurposes, the metal tray will conductthermal energy from the room air intothe frozen food very efficiently.

Wolke, R. L. What Einstein Told His Cook:Kitchen Science Explained. New York: W. W. Norton, 2002, pp. 202–203.

12. Ice Cream Delight

Good ice cream contains abundant airbubbles to keep it light with very smallice crystals so that the texture issmooth. There are several good smallappliances for making ice cream andsorbet. Some ice cream makers rely onhuman musclepower to turn a largespatula to control the ice cream tex-ture; others are electric. But with anabundant supply of liquid nitrogen at–196°C and about an equal volume ofice cream mixture, the freezing of theice cream occurs so fast that only smallcrystals have time to grow. As thenitrogen furiously boils, plenty of smallgas bubbles form in the mixture. Allthese effects make for a delicious treat.

Kurti, N., and H. This-Benckhard. “Chem-istry and Physics in the Kitchen.” ScientificAmerican 270, no. 4 (1994): 66–71.

Walker, J. “The Physics of Grandmother’sPeerless Homemade Ice Cream.” ScientificAmerican 250, no. 4 (1984): 150–153.

13. Cooking a Roast

The bone-in roast cooks faster becausethe bone, even though porous, rapidlyconducts thermal energy to the insidefaster than the meat itself does. So thebone-in roast cooks from both direc-tions—outside in and inside out. Therewill be some minor speedup effectfrom the difference in specific heats ofbone and meat, and there will be

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slightly less meat if both roasts weighthe same, but in the simplified, idealcase we ignore these differences. Onecould use computer modeling with theappropriate physics equations todetermine the temperature distributionin various parts of the meat and bonein the two cases, showing that someparts of the meat are cooked morethan others in both types of pieces.

Any general physics text discussingthermal conductivity and specific heatcontains the pertinent information foranalyzing this problem, but the actualtemperature profile as a function of the elapsed time is difficult withoutsome idealizations about the shape, theuniformity, and other things. We haveconsidered the idealized case abovethat ignored the change in bone prop-erties with temperature change, such as the specific heat of the bone and its thermal conductivity. SomehowNature has figured out all these thingswithout special computer modeling!

14. Cooking Chinese

Style

There are at least two good reasons forcutting up meats into small volumes: (1)marinades and spices penetrate morethoroughly into the meat in a shortertime because the inner-volume elementsare closer to the surface; (2) smaller

chunks cook faster and thereforerequire less fuel. The faster cookingoccurs because (a) the inside of thesmall cube is closer to the heat sourcethan for a thicker piece and (b) the meatis tumbled during stir-frying, exposingdifferent small surfaces to the higher-temperature direction. The temperaturesensed by the meat tends to decreasewith distance from the thermal energysource, in this case the pan bottom.

The amount of cooking experi-enced by a small volume of interiormeat is proportional to the tempera-ture experienced and the duration ofcooking at this temperature. Both ofthese quantities are changing duringthe cooking process. In addition, thethermal conductivity and the thermalheat capacity of the meat are changingbecause the meat material itself ischanging. For example, if the outsidebecomes charred, its thermal conduc-tivity is significantly reduced, so thatthe transmission of thermal energydecreases compared to its prior rate.Therefore hamburgers, which must becooked thoroughly inside to kill thebacteria on the surfaces of the ground-up meat, must never be charred on theoutside because the inside will tend toremain uncooked or partially cooked,creating a dangerous eating condition.

The physics here is found in anyhigh school physics text, but theapplication to cooking was devised

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millennia ago by ancient chefs whodesired a particular result with per-haps a minimum of fuel expenditure.Certainly one can go to the otherextreme by slow-roasting a whole car-cass in a revolving spit or in a hotember-lined pit in the ground.

15. Baked Beans

The hot beans have taken up waterand swelled so that the skins are undergreat tension. Blowing cool fast-mov-ing air across the beans from pursedlips (instead of an open mouth) lowerstheir surface temperature and reducesthe ambient air pressure. The inside isstill hot, so the larger pressure differ-ence results in the hot, high-pressurevapor under the skin pushing outwardjust a bit more. If the pressure gradientis great enough, the skin will rupture.The slight cooling of the skin materialincreases the skin tension to reduce thetime to rupture.

We can simplify the physics toapplying Newton’s second law perpen-dicular to the skin surface. Threeforces acting on the skin are impor-tant: (1) the inward force of the ambi-ent air pressure, (2) the inward forceof the tension in the skin, and (3) theoutward force produced by the hot,high-pressure gas within. Blowing theair reduces the ambient air pressureenough to create a net force outward,and skin rupture occurs when the skin

tension has exceeded its elastic limit.During the rupture process, bean skinmolecules have been separated becausethe electrical force holding one mole-cule to the next has been exceeded.

Miller, J. S. The Kitchen Professor. Sydney:Australian Broadcasting Commission,1972, pp. 81–82.

16. Ice Water

The ice at the top brings about fastercooling of the water in the pitcher. Assome ice melts, this cold water is moredense than the surrounding water andsinks, cooling the water it passesthrough. The warmer, less dense waterat the bottom is buoyed upward into acooler region. This mixing helps thewater cool faster than when the ice isheld at the bottom, because the coldwater produced by the ice wouldremain at the bottom. Thermal con-ductivity throughout the water wouldeventually cool the water above, butthe convection currents work faster.Of course, vigorous stirring of the icewater eliminates any need for the pre-vious discussion!

One could say that the discussionabove is incomplete because in our ide-alization we have ignored the ice ther-mal interaction with the ambient air.This interaction can be important,especially on hotter days. The ice doesits job when thermally interacting withthe water, not with the air! The ice held

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in the water would be a more efficientdirect interaction procedure. So whenthe ambient air temperature is greatenough, they could be competitive.

By the way, this cooling process isexactly the same as the sequence ofevents that occurs when a pond freezesover in winter. However, in that case,the pond water is prevented from cool-ing further and from freezing throughuntil all the water reaches 4°C first.This delay in freezing throughoutsaves the lives of pond organismsthrough the winter if spring comessoon enough. Evaluated in a differentway, there would be no life survivingthe worst ice ages on Earth if waterdid not reach its maximum density atabout 4°C!

17. Peeling Vegetables

When a tomato is held carefully over aflame and rotated, some of the thermalenergy gained by the tomato vaporizesthe water just under the skin to locallyrupture it. A paring knife can removethe skin easily after the tomato hascooled. Often, just pulling on the rup-tured skin is enough. Very hot watercan be used instead of a flame, but theeffects are not as dramatic, and thepeeling is a bit more difficult.

The boiling raises the temperatureof the beets to cook them and, simul-taneously, a small amount of hot water enters them, so they are slightly

swollen. Cold water on the outer sur-face of the hot beet causes the skin toshrink, but the innards remain hot andswollen. So the stretching skin burstsin several places and becomes easier toremove with a paring knife withoutbeing so messy.

The procedure is exactly oppositeto placing an ice cube in hot water.Now the outside of the cube tries toexpand, but the inside is still cold. Onecan hear the thermal stresses crack theice cube.

18. Igniting a Sugar

Cube

Very small particles tend to ignitemore easily. The large surface-area-to-volume ratio for a collection of smallparticles aids ignition, providing alarge combustion area for the chemicalinteraction of their surface moleculeswith oxygen and also providing anearby heat source for sustenance.Therefore, rub the far corner of thesugar cube in some cigarette ash ortiny ash particles from burned paper,then light the ashen cube with theburning match. Ignition is now easy.Oxygen molecules react with mole-cules in the ash to produce thermalenergy and product molecules, includ-ing water.

Historically, there have been manyexamples of the spontaneous ignition

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of dust particles in the air, such asexplosions in granaries where grainscrops are stored and in mills that grindgrain into smaller particles. A smallwarm spot in the dusty air, perhapsproduced by sunlight, by a match, orby friction, can rapidly spread into afull-scale explosion.

On a less violent scale, simplylighting a campfire outdoors beginswith kindling, very small sticks andshavings of wood, which have a verylarge surface-area-to-volume ratio.The slightly larger sticks can be addedonce the flame sustains itself. Finally, awhole faggot of sticks can be placed inthe firepit to generate a lasting fire.

19. Water Boiling

The major thermal effect is the raisingof the boiling point (i.e., boiling tem-perature) by the added salt in solutionfrom 100°C (standard conditions at 1atmosphere) to about 104°C (if thesalt is pure NaCl), a significant changethat results in a small time lag beforeboiling begins again if thermal energyinput continues.

In contrast, the actual amount ofthermal energy needed to raise thetemperature of the sprinkled salt itselfis minuscule because the specific heatof NaCl is much lower than for water,and the amount of water by weight inthe pot is enormous compared to theamount of salt. The actual boiling

process is more complicated than thesimple, ideal version we have consid-ered here, involving nucleate boiling,transition boiling, etc., in the real situ-ation. However, the complete analysisproduces the same general argument.

One should ask whether differentresults would occur with sea salt, amixture of KCl and NaCl and deadorganic matter in larger grain sizesthan normal table salt. The slow rateof dissolving the larger sea salt grainsmay delay the reboiling longer thanexperienced for NaCl.

The relevant physical data are inmost chemistry and physics hand-books as well as in some textbooks.The physics pertinent to the idealizedcase discussed above is part of tradi-tional physics and chemistry courses,but the more detailed complete analy-sis can be found only in the technicalliterature.

20. Put the Kettle On

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Steam

GapSpout

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No, you cannot see water vapor, thatis, water molecules in their gaseousstate. If you look closely at the orificeof the spout, there is a clear region per-haps up to one inch long. That’s wherethe water vapor is before it condensesinto the steam you can see. The tem-perature of the vapor in the clearregion is still too high for droplets ofsteam to form—that is, collisions ofwater molecules are too violent toallow them to bind together to formdroplets.

In the clear region at the orifice ofthe kettle, the water molecules aremoving so rapidly that when they docollide, the van der Waals attractiveforce—an induced dipole-dipole elec-tromagnetic interaction—at closerange cannot keep them together. Asthe water vapor cools farther awayfrom the end of the spout, these samecollisions produce droplets that growin size.

21. The Watched Pot

Put a pot of water on a flame atop astove. The thermal energy from theflame raises the water temperature. Ifthe pot has no cover, soon the watervapor pressure above the water sur-face equals the pressure of the ambientatmosphere. The water is now boiling.If we are high in the mountains, theboiling has occurred at a lower tem-perature than when we are near sea

level. So at higher elevations potatoesmay not cook as quickly in the openpot, and the lukewarm water will notmake good tea or instant coffee. In themountains we would be wise to use apot with a lid so that the total pressureacting downward on the water surfacecan be higher—vapor pressure plusatmosphere—and so that the waterboils at a higher temperature thanwithout the lid, hopefully almost at100°C.

Suppose the cooking at sea level isdone in a pot with a lid. Now theaction of lifting the lid to “watch thepot” reduces the thermal energy in theair above the liquid surface as somemolecules escape. These escaping water molecules are among the most ener-getic in the vapor, so they can carryaway much thermal energy. The pres-sure above the liquid is now lowerthan before, the boiling occurs at alower temperature, and the cookingtakes significantly longer. One shouldreplace the lid and let the food cookundisturbed! Hence the expression “Awatched pot never boils.” This state-ment actually refers to the extendedcooking time and involves some goodphysics.

22. Ice in a Microwave

Yes! The water molecules in the liquidstate rotate a bit in the microwavesand transfer energy to the surrounding

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molecules to make them jiggle ran-domly. The water molecules in ice arelocked into crystals and are unable torotate. (Note: The actual details ofmolecular bonding in the ice are morecomplicated and show that a minus-cule amount of rotation is possible,but an insignificant amount to changethe ice to water.) Using microwaves,therefore, one can boil water inside anice block!

Boiling the water inside the iceblock is an example of selective energyabsorption. Numerous examples ofselective absorption occur in the natu-ral world. For example, the greenleaves of plants have chlorophyll Aand B molecules that selectivelyabsorb bluish and greenish light forphotosynthesis. At an even smallerscale, nuclei are very selective inabsorbing gamma rays of specific ener-gies. At the macroscale of meters, weknow that rooms can absorb andamplify sound energy at selected reso-nance frequencies. Some materials areeven useful for just the oppositebehavior, such as window glass, whichhas no selective absorption in the visi-ble part of the electromagnetic spec-trum. You can take your pick, but thegame is played by the rules of nature.

The selective absorption by watermolecules (and some other molecules)in a microwave environment is a littledifferent from the other examples

given above. At the water molecule’sresonant frequencies in the microwaveregion of the electromagnetic spec-trum, the applied field changes so rapidly that very little energy is trans-ferred to the nearby molecules.Microwave ovens actually operate at afrequency that is lower than thefrequency at which the absorption isgreatest. The food needs to be heatedthroughout, and by lowering theapplied frequency a bit, more micro-waves penetrate farther inside, past theouter layer.

Kurti N., and H. This-Benckhard. “Chem-istry and Physics in the Kitchen.” ScientificAmerican 270, no. 4 (1994): 66–71,120–123.

Walker, J. “The Secret of a MicrowaveOven’s Rapid Cooking Is Disclosed.”Scientific American 256, no. 2 (1987):134–138.

23. The Glycemic Index

The rate of conversion from one typeof molecule to another is a chemicalprocess, with the ratio of surface areato volume for particles in the foodbeing converted being an importantfactor. Smaller spherelike particleshave a higher ratio of surface area (SA)to volume than larger ones. Conse-quently, since the reactions occur onthe surface, material consisting ofsmaller-diameter spheres convert tosucrose faster than the same material

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consisting of larger spheres. In fact, forthe limiting case of a sphere, the ratioSA/Vol. = 3/R, where R is the radius ofthe sphere. Physical and chemicalprocesses initiated by the environmentoccur first at the surface of the parti-cle. In addition, for biological systems,larger particles must be reduced tosmaller ones before passing throughmembranes. So a collection of smallparticles equal in total mass to onelarge particle will be reduced toacceptable size faster than the largeone because the same amount ofchemical solution acts upon a muchlarger total surface area.

The smaller the particles ofingested food, the faster can be thedigestion of the molecules in the intes-tines because the surface-area-to-volume ratio is higher, and the sooneris the uptake into the blood. Thehigher temperature used for bakingthe potato makes its particles smallerthan for the same potato when boiledat 100°C, so its glycemic index isgreater, reflecting its faster uptake intothe blood.

Dates contain some maltose, asugar that is even faster than glucosein its basic conversion to sucrose in the blood, so their glycemic index isabove 100.

A popular book The New GlucoseRevolution by J. Brand-Miller is avail-able in many libraries and has tables of

the glycemic index for numerous foodsand some of the recent results fromnutrition and diabetes research world-wide. Excerpts from the book andmany other resources on the glycemicindex and its comparison to the insulinindex can be found on the Internet.

24. Electric Pickle

Even though the electrical energysource is provided through an AC cur-rent, the pickle glows predominantlyat one end with a yellowish color thatis determined by the pickling solutionand the pickle type. Reliably predict-ing which end will glow has not beenachieved. There is no actual symmetryhere in the shape or chemical compo-sition of the pickle, so alternate glow-ing is less likely and never seen. Theconjecture is that the pickle is now act-ing like an electrical diode, passingcurrent in one direction only!

The authors listed below performedan experiment by taking a visible lightspectrum of the glowing pickle, using aspectrometer with a diode array detec-tor. A fiber-optic probe was used tochannel the yellow glow to the spectro-graph, and a calibration spectrum wastaken of a sodium chloride flame test.The emission spectra of the two arenearly identical

This pair of emission lines, at589.00 nanometers (nm) and 589.59

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nanometers, indicate a characteristicof sodium emission, called the sodiumD line doublet. Josef Fraunhoferobserved these lines in the emissionspectrum of the Sun, in about 1817.We know now that these lines are duespecifically to an electronic transitionof sodium atoms in the gas phase.

The pickle conducts electricity dueto the vinegar (acetic acid) and sodiumchloride salt used to make it. Sodiumions in the pickle liquid attach elec-trons from the flowing current. Theseions are neutralized electrically, form-ing excited sodium atoms in two dif-ferent excited electronic states (hencethe emission doublet). Because of theheat and sparks and general pandemo-nium around the electrodes stuck inthe pickle, these sodium atoms are inthe gas phase. They emit yellow lightas they relax to the ground state.

Appling, J. R., F. J. Yonke, R. A. Edgington,and S. L. Jacobs. “Sodium D Line Emissionfrom Pickles.” The Journal of ChemicalEducation 70, no. 3 (1993): 250.

25. Space-Age

Cooking

Unlike electric cooktops, which gener-ate thermal energy by the electricalresistance of the burner coils, magneticinduction cooktops generate thermalenergy by the magnetic resistance ofthe metal cooking vessel itself. The 60Hz AC current flowing in the induc-tion coil beneath the ceramic surfaceproduces an alternating magnetic fieldthan interacts with the Fe atoms—forexample, in the iron frying pan—tooscillate its magnetization 120 timesper second. The magnetization direc-tion changes have resistance, so muchenergy goes into thermal energy in themetal of the pan. Iron and stainlesssteel pans will work, but aluminum,copper, glass, and ceramic pans andpots will not. The advantages are nonoise and no hot cooktop exceptwhere the pan has been in contact.

Cooking with light is not done bylasers! Light is meant in the broadersense of the word, the infrared (IR)through the visible into the ultraviolet(UV) part of the electromagnetic spec-trum. Banks of 1500-watt halogenlamps in the oven walls put out about70 percent IR, 10 percent is visiblelight, and the remaining 20 percent issimply heat. The IR is not thermalenergy, but when IR is absorbed bymolecules, their random motions canbe increased. Thermal energy (“heat”

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587.5 590.0 592.5

Wavelength (nm)

PickleNaCl

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in the vernacular) is the randomkinetic energy of molecules and atoms.These frequencies penetrate meat onlyabout half an inch at most. Thermalconduction transfers some of this ther-mal energy farther inside.

However, these light ovens alsohave a microwave source to penetratewith microwaves to cook the interiorof the meat. So while the outside isbeing browned by the light, the insideis cooking via microwaves. The overallbenefit is much faster cooking than ispossible in conventional ovens.

Wolke, R. L. What Einstein Told His Cook:Kitchen Science Explained. New York: W. W. Norton, 2002, pp. 303–307.

Chapter 2Does Anybody

Really Know WhatTime It Is?

26. January Summer

Yes, the Northern Hemisphere enjoyssummer in January quite often (in thecosmic scheme of things), repeating,every 25,800 years, the period ofEarth’s precession. Just like a top withits axis precessing, Earth experiences aprecession of its axis with respect tothe stars with a 25,800-year period ofoscillation. So every 12,900 years the

North Pole will be alternating fromthe extreme of being pointed towardthe Sun and to being pointed awayfrom the Sun in January. At presentand for some years to come, the NorthPole points away from the Sun whenEarth is at the perihelion position in itsorbit on about January 5 each year.Gradually over the next 12,900 yearsthe North Pole will precess around toreceive more and more radiant energyin January.

However, we need not wait nearlyso long because the ellipse of Earth’sorbit is also precessing, so our summerwill coincide with perihelion in onlyabout 10,000 years!

27. Proximity of Winter

Solstice and Perihelion

The proximity of the two dates is anartifact of the particular century welive in. The date of perihelion does notremain fixed, but slowly moves laterinto the year at the rate of about one

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now

Earth

future

Sun

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full day every 58 years. It turns outthat the period from perihelion toperihelion (the anomalistic year) isabout 25 minutes longer than the yeardefined from equinox to equinox (themean tropical year). The date ofperihelion thus moves completelythrough the tropical year in about21,000 years. This slow change in thedate of perihelion may have a long-term effect on Earth’s climate. At thistime the temperature extremes aremoderated somewhat in the NorthernHemisphere, but that will change asthe perihelion shifts in the direction of summer.

28. Earth’s Speed

Since perihelion occurs in early Janu-ary, Kepler’s second law implies thatEarth is traveling faster during thewinter months. The time for Earth totravel from the autumnal to the vernalequinox, taken as a fraction of theyear (T = 178.83/365.25), can be usedto find an accurate value of the eccen-tricity of Earth’s orbit, ε = 0.5 π(0.5 – T) = 0.01632, about 2 percentaway from the precise value of ε =0.016713. A more accurate formulabased on T is found in the referencebelow.

Snyder, R. “Kepler’s Laws and Earth’sEccentricity.” American Journal of Physics57 (1989): 663–664.

29. The Equinox

Displaced

On the dates of the equinoxes, the dayis about seven minutes longer than thenight at latitudes up to about 25degrees, increasing to 10 minutes ormore at latitude 50 degrees.

The moment of the equinox occurswhen the geometric center of the Sun’sdisk crosses the celestial Equator. Butthe standard definition of sunrise is thetime when the Sun’s upper limb is justbreaking the horizon, and sunset whenthe Sun’s upper limb is just disappear-ing below the horizon. This adds oneSun semidiameter (about 16 arc min.)at both sunrise and sunset, extendingthe duration of daylight by a little overtwo minutes.

The other factor is atmosphericrefraction, which causes the rays tobend around the horizon. As a result,we see the Sun about 34 arc minuteshigher at both sunrise and sunset,adding roughly four minutes to thetime that the Sun is above the horizon.

In spring the days get longer as weapproach March 20, and the date ofequal day and night occurs severaldays before the March equinox, aboutMarch 17 at latitude 40 degrees.

Conversely, in the fall it takesseveral extra days for the time whenthe Sun is seen above the horizon to shrink to 12 hours. The date falls

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on about September 26 at latitude 40 degrees.

On their website, the U.S. NavalObservatory publishes excellent sun-rise and sunset tables for any location.

30. The Dark Days of

December

There are two effects that, together,determine the local times of sunrise andsunset. One is called the equation oftime; the other is the Sun’s declination.

Earth’s orbit around the Sun isslightly elliptical. As a result, the speedof the Sun’s apparent motion acrossthe sky is a bit faster in winter than insummer. Clocks, however, run at aconstant speed, so there is usually adiscrepancy—up to 16 minutes—between clock time and the solar timeshown by a sundial. We refer to thisdiscrepancy as the equation of time.

The Sun’s declination, its angulardistance above or below the celestialequator, determines the maximumheight of the Sun in the sky on anygiven day, thus causing our seasons. Inlate December, the daily rate of changeof the Sun’s declination is rather small.It is, in fact, exactly zero at the Decem-ber solstice (“solstice” means “sun sta-tionary”). Hence in late December, ormore precisely from about December8 to January 5 at latitude 40 degrees

north, the equation of time has thedominant influence over the changesin sunrise and sunset times. Prior toDecember 8, however, the declinationeffect is dominant, pulling the sunsetto its earliest time on December 8.Then the equation of time takes over,and during the two weeks before win-ter solstice all the shortening of theday comes from the later clock time ofsunrise. After winter solstice the dayslengthen, even as the sunrises continueto get later until January 5.

Steel, D. Marking Time: The Epic Quest toInvent the Perfect Calendar. New York:John Wiley & Sons, 2000.

31. Days of the Year

While the time interval to return to itssame point in the orbit is 365.2422days, Earth executes 366.2422 rota-tions on its axis. One can demonstratethis result by taking two coins, holdingone in place on a table, and rolling thesecond coin in contact with the fixedcoin without slipping. Therefore thenumber of solar days is 365.2422, butthe number of sidereal days (i.e., withrespect to the stars) is one more forone orbit of the Sun.

32. Leap Years

In years divisible by four, every fouryears is a leap year except years divisi-ble by 100. If the mean interval

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between vernal equinoxes, called thetropical year, lasts 365.2422 days,then in 100 years we should experi-ence 36,524.22 days. But there will be24 leap years in a century normally, sothere will be 0.22 day left over. Soevery 400 years is declared to be a leapyear with one extra day to approxi-mate the 0.88 day. The year 2000 wasthe first such leap year on a year divis-ible by 100 since the modern calendarbegan general use in the late 1600s. By the time the British were ready to go along with the rest of Europe inthe 1700s, the old Julian calendarrequired a correction of eleven days!The Gregorian calendar was adoptedin Britain in 1752, with Wednesday,September 2, 1752, being followedimmediately by Thursday, September14, 1752.

The famous physicist Isaac New-ton was born on Christmas Day, 1642,on the Julian calendar but on January4, 1643, on the Gregorian calendar inuse today. Therefore, Newton was notborn in the year of Galileo’s death,1642!

33. Full Moons

No, the orbital period of the Moon is27.554 sidereal days, and the averageinterval between full moons was29.535 days for the twentieth century.The difference between these two

time periods occurs because Earth ismoving with respect to the stars, so theMoon must travel slightly fartheraround its Earth orbit to reach its fullMoon position along the Sun-to-Earthradial line.

U.S. Naval Observatory, Nautical AlmanacOffice. The Astronomical Almanac for the Year 2000. Washington, D.C.: U.S.Government Printing Office, 2000.

34. Moon Time

The person is at the desk at 12:20 dur-ing the noon hour, not at night. TheSun must be at the upper left becausethe Moon is illuminated from thisdirection. This daytime Moon is sel-dom noticed because the sky is nor-mally bright, but the daytime Moon is up as often and as long as the Moonat night.

Perhaps the easiest way to appreci-ate the appearances of the Moon atnight and during daylight is to use alamp for the Sun and two spheres, onerepresenting Earth and the other theMoon. Fix the lamp position and theEarth position, but move the Moonaround Earth to observe its illumina-tion phases. Stop the motion at severalpoints in the orbit of the Moon toobserve its view from daytime loca-tions on Earth. You also might recon-struct the scene in the diagram.

Pryor, M. J. “Phases, Models, and Car-toons.” The Physics Teacher 3, no. 6(1965): 264, 288.

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35. Lunar Calendar

Modern farming methods tend to plantcrops at approximately the same timeyear after year, with minor adjustmentsfor quirks in the weather. Hence, a par-ticular crop is usually planted about365 days after its planting the previousyear, plus or minus about 10 days. Oneexample is spring wheat, usuallyplanted on about April 15 in the north-ern Plains states of the United States.

Rice is a different kind of organismthan wheat as far as its environmentalneeds. Rice planted at about the samedate every year will sometimes permittwo good rice crops per year, but inmost years the farmer will get only onegood rice crop. The cause is the some-times detrimental appearance at nightof the full Moon, which can interferewith the growth cycles of the rice plant.

By planting rice according to thesame date on the lunar calendarinstead of the solar calendar, farmerscan often harvest two good rice cropsevery year. The young rice shoots arevery sensitive to the light intensity atnight during their photoperiod-sensi-tive stage, so the timing of the Moon’sbrightness is essential for a good crop.Because the lunar calendar shifts withrespect to the solar calendar dates eachyear, the solar calendar provides badtiming for planting rice.

The photoperiod-sensitive stageoccurs before panicle initiation (where

the seed parts develop) and may varyextremely from one variety to the next,from days to months. Photoperiod sen-sitivity is a natural mechanism basedon the plant’s ability to distinguish pre-cise differences in the ratio of daylength to night length. The biologicalmechanism causing photoperiod sensi-tivity is quite complex and involves sev-eral genes. Essentially, some varietiesshould be planted only during certaintimes of year to ensure that prevailingday length/night length conditions willtrigger panicle initiation when desired.

Many varieties of rice are sensitiveto bright moonlight, which can inter-rupt their growth sequence. However,newer varieties have been bred andothers will be genetically modified todecrease their light sensitivity duringcritical photosensitivity times so thatthe moonlight will have a minimaleffect.

The use of the Moon for timing ofevents is not restricted to rice farmingand its related festivals worldwide. ForChristians, Easter Sunday is the firstSunday after the first full Moon afterthe vernal equinox!

University of the Philippines College of Agri-culture in cooperation with the Interna-tional Rice Research Institute (IRRI),comps. Rice Production Manual, rev. ed.Manila: compilers, 1970.

Yano, M., et al. “Hd1, A Major PhotoperiodSensitivity Quantitative Trait Locus in Rice,Is Closely Related to the Arabidopsis Flow-ering Time Gene Constans.” Plant Cell 12(2000): 2473–2484.

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36. The Sandglass

The hourglass shape ensures that thetime scale on the glass is uniform, withequal distances between scale divisionscorresponding to equal time intervals.If the sandglass didn’t taper, the top ofthe sand column would descend atincreasing speed. We can mathemati-cally determine the proper shape. Thespeed, V, at which the sand is escapingfrom the opening is given approxi-mately by Torricelli’s formula V ~√(2gy), where g is the local accelera-tion of gravity and y is the height ofthe sand column in the upper glass. LetA = πr2 be the circular cross-sectionalarea of the upper glass at the top of thesand column and v the speed at whichthis top is falling, then Av = aV, wherea is the area of the opening, becausethe sand is approximately incompress-ible. Substitution produces y ~ cr4,with constant c = π2v2/(2ga2). A plotof y versus r will produce the familiarhourglass shape.

Sandglasses without time intervalmarkings have been used since beforethe fourteenth century to time speechesat town meetings and other events.When the sand had run its course, thespeaker’s time was done. They are stillused today in some board games andas kitchen timers. Sandglasses withruled markings haven’t been so popu-lar, being replaced early on by mechan-ical watches and clocks.

Jargocki, C. P. Science Braintwisters, Para-doxes, and Fallacies. New York: CharlesScribner’s Sons, 1976, pp. 6, 70.

37. Old Watch

The old watch will run fast. The bal-ance wheel is the basic component that oscillates “exactly” 300 times foreach minute on the watch face—thatis, the wheel changes direction 10times each second! The moment ofinertia of the balance wheel dependson how much ambient air is draggedalong during each oscillation. Thesource of energy is a wound springthat is essentially unaffected by the airbecause of its extremely small changein configuration.

In the mountains, the viscosity anddensity of the air decrease slightly,allowing the balance wheel to oscillatefaster. Newton’s second law applied tothis rotational motion is required.From its momentary stop to changedirection, the balance wheel mustaccelerate to its maximum angularvelocity, then accelerate back to rest,etc. The net torque τ equals themoment of inertia I times the angularacceleration α—that is, τ = Iα. Themoment of inertia is determined by themass distribution with respect to therotation axis, and the air carried alongwith the balance wheel motion adds tothe moment of inertia of the wheelalone. That is why the watch must be

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recalibrated when the location of theowner is at a different elevation thanthe factory.

When the balance wheel dragsalong less air mass at a higher eleva-tion, the moment of inertia is less forthe same net torque, so the angularacceleration is greater. Less time isneeded to reach top angular speed,and less time is needed to come to restagain.

Jargocki, C. P. Science Braintwisters, Para-doxes, and Fallacies. New York: CharlesScribner’s Sons, 1976, pp. 6, 71.

38. Reading a Digital

Timer

For a digital timer that displays theelapsed time to one-hundredth of asecond, the minimum uncertainty inthe interval depends on the softwareand/or hardware method used to dis-play the last digit. Suppose that thehundredths digit fraction from 0.00through 0.49 is displayed as zero hun-dredths and from 0.50 through 0.99 isdisplayed as one hundredth. Likewise,1.00 through 1.49 is displayed as onehundredth, and 1.50 through 1.99 astwo hundredths. Seeing a 1 in the hun-dredths place then corresponds to therange from 0.50 through 1.49, so theminimum uncertainty in the elapsedtime value is ± 0.50, or one-half ofone-hundredth of a second. Thereported elapsed time should be given

as, say, 3.45 seconds ± 0.005 second,which is an awkward notation,because the display goes only to hun-dredths of a second, yet the uncer-tainty is smaller. Therefore, byagreement, the elapsed time is given as3.45 seconds ± 0.01 second so that thenumber of decimal places is the samefor the value and the uncertainty—that is, there is an uncertainty of plusor minus one digit in the smallest timeinterval position of the display.

39. Eternal Clocks?

The laser clocks and the atomic clocksmust maintain a vacuum within a rea-sonably small range of parameters tofunction accurately. The temperatureand pressure must be maintainedwithin a certain tolerance becausetemperature fluctuations or pressurefluctuations could bring about inaccu-racies. Even the outgassing of atomsand molecules from the containerwalls can create severe problems forsome designs. Certainly, improvementswill be made to ensure longer lifetimesand more robust timepieces. But main-taining vacuums, low temperatures,and so on, for decades and centurieswill be constant problems in thesesophisticated systems.

Whether a 10,000-year (or even a1,000-year!) mechanical clock willever exist and can stand the test oftime and environment is doubtful. A

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group of engineers and futurists arepresently developing such a clock con-taining a stack of rotating metal ringsconnected to a torsion pendulum. Peri-odic winding will be required, perhapsonce a year or so.

No special environment is needed,although the assumption seems to bethat a standard atmosphere with lim-ited pollutant content is a reasonableexpectation. We do know from a vari-ety of scientific research projects thatthe oxygen concentration in theatmosphere has been very nearly con-stant at 21 percent for millions ofyears, so the rusting rate of theexposed metal can be predicted. Butwe do not know what future chemistrywill bring. Even a local environmentaldisaster such as excess acidity in theair from volcanic eruptions, a chemi-cal explosion, or careless disposalcould shorten the life span of the clockdramatically.

Gibbs, W. W. “Ultimate Clocks.” ScientificAmerican 287, no. 3 (2002): 86–93.

40. Room Light

For a nanosecond flash, the light pulselength d = 30 cm is calculated with d =ct, where c is the speed of light and t isthe time interval. When summing allthe entering light, the photodetectordisplays an initial rise from the lightscattering from nearby walls withincreasing intensity until maximum,

and then a decrease to zero after thelight from the far-corner reflections isreceived. The detailed intensity curvecould be simulated on a computer.

With the photodetector array, theimage shows the six nearest spots—thecenters of each equidistant flat sur-face—which grow bigger and thenform rings of reflected light, thenmany arcs of light until eight cornersappear and disappear.

When the flash length is extendedto 1 microsecond, the light pulse is300 meters long. There will be an ini-tial detector response rise and the ringsof light from the walls will be seen fora very small fraction of the total imag-ing, then flooded, then decreased.

We are not accustomed to lightpulses lasting for milliseconds or less indaily life. But even nanosecond pulsesare very long in some research fields.For example, femtosecond (10–15 sec-ond) and shorter light pulses are usedin chemistry to watch molecular inter-actions in progress. The present recordfor subdividing the second with a laserstrobe light is a few hundred attosec-onds, an attosecond being a billionthof a billionth of a second!

41. Right to Left Driving

Switch

Yes, as long as the accelerations anddecelerations required are within the

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normal driving ranges, there should beno problems. One could check outeach roadway with physical trials, orone could make an aerial video andplay back the video in the reversedirection, essentially reversing time. Ifthe car accelerations appear unusualin the reverse sequence, there will bedriving problems.

Even nature at the most fundamen-tal level is cognizant of left versusright, a surprising discovery in the1950s related to the weak interaction,one of the four fundamental interac-tions, the other three being the gravita-tional, the electromagnetic, and color(also called the strong interaction). Forthe latter three, the interactionstrengths are the same for left-handspinning particles and right-hand spin-ning particles. But for the weak inter-action, the evidence shows that natureactually excludes any weak interactionbehavior for a right-hand spinning par-ticle! The origin of this behavior biasedtoward left-hand spinning particles isdescribed by the mathematics in thestandard model of leptons and quarks.

Bartlett, A. A. “A Simple Problem from theReal World That Can Be Solved throughTime Reversal.” American Journal ofPhysics 42 (1974): 416–417.

42. Light Clock

Yes and no. The clock will continue tokeep accurate time in both reference

frames, in the laboratory frame, and inthe rest frame of the clock. But the tickrates will be different. In the framemoving with the clock, the light flashesfollow the same path as before, reflect-ing perpendicularly to each mirror,keeping the same tick rate.

In the laboratory frame, the lightcontinues to reflect off each mirrorrepeatedly, but during transit from onemirror to the other the path length islonger, being the diagonal of a righttriangle. If the speed of light is thesame value in both reference frames,then the time interval between reflec-tions (and clock ticks) will be longer inthe lab frame now than for the clockat rest. Therefore, a moving clock ticksslower than an identical clock at rest.And this phenomenon is true for allclocks, no matter how they are made.

A more complicated situationoccurs when the light clock is acceler-ated parallel to the mirrors. We suggestthat you think about this case whenthere is ample time in your schedule.

Feynman, R. P., R. B. Leighton, and M.Sands. The Feynman Lectures on Physics.Vol. 1. Reading, Mass.: Addison-Wesley,1963, pp. 15-5–15-7.

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Mirror

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43. Time Reversal

Answer b: the acceleration is stilldownward. The reversed motion isupward, but the object is decreasing itsspeed because the acceleration is down-ward. A good example of this behavioris the flight of a ball tossed upward. Atall moments the acceleration is down-ward, toward Earth’s center. Yet theball moves upward with decreasingspeed, turns around, and moves down-ward. Even at the turnaround point itsacceleration is downward.

Quite often people become con-fused between velocity and accelera-tion. They are two different vectorquantities that should be separatedconceptually, but they are mathemati-cally related. Their directions can bethe same or opposite along the line ofmotion. Newton’s second law relatesforces and accelerations but says noth-ing about velocities, for example. Andwe know that Aristotle was wrongwhen he proposed that a force wasrequired to keep an object moving.The real world operates with just theopposite rule because no net force isrequired to keep an object moving in astraight line at a constant speed!

44. Molecular Clock

The changes in the DNA during theevolution of organisms do not occur ata common rate because any change in

a critical essential protein coding doesnot produce a viable organism eventhough a change in the nonessentialDNA might—that is, changes in theDNA that do not affect the biochem-istry critically will be tolerated. Thereare vast regions of DNA where suchineffective changes can occur, but anychange in the other regions pro-grammed for the production of essen-tial biomolecules will be disastrous.

If we assume the ideal case, that inprinciple the changes would be equallyprobable at any random location alongany DNA chain, and we assume thatthe organism will grow and reproducethe next generation, then we could havea molecular clock. However, as weknow, just as all genes are not equal invalue at any given time, all DNAsequences are not equal in value. In par-ticular, some DNA sequences code forproteins that control the expression ofother DNA genes themselves, turningthem on and off at appropriate times inthe development of cells in the organ-ism. The Hox protein in insects, forexample, will determine the structure ofseveral different body parts, and slightchanges in its amino acid sequence aremajor contributors to insect evolution.Therefore, both complementary DNAstrands at the critical location need notbe affected for the appearance of obvi-ous phenotype changes.

However, the whole DNA mecha-nism and its subsequent biochemistry

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in the cell are much more robust thanoriginally realized. The fact that manyof the amino acids have several DNAbase code triplets of nucleic acids fortheir selection is a built-in resiliencythat can produce a viable organismeven when the DNA has an error ofthis kind. In addition, if the erroneousamino acid substitutes at a locationthat is not critical for the 3-D shapeand the operation of the protein, onceagain there is a built-in resiliency. Let’sface the fact that Nature is much moreclever than we can ever hope to be!

Gibbs, W. W. “The Unseen Genome: Gemsamong the Junk.” Scientific American 289,no. 5 (2003): 46–53.

Ronshaugen, M., N. McGinnis, and W.McGinnis. “Hox Protein Mutation andMacroevolution of the Insect Body Plan.”Nature 415 (2002): 914–917.

45. SAD

Yes, if they suffer from SAD. At firstone might think that the variation inthe length of day and night changes solittle from January to June that no oneliving at the Equator would suffer fromSAD. This reasoning is true if everyonewent to bed at sunset and arose at sun-rise. The increasing light at sunrisewould trigger the start of another cir-cadian rhythm that brings about bio-chemical changes in our bodies.

But even people living near theEquator are tuned no longer into therise and setting of the Sun. There are

problems in their circadian rhythmsbecause the bright artificial lightingmeans that people stay awake longpast sunset, delaying and shifting themaximum in certain biochemicalcycles beyond their evolutionary timeof day. In particular, greenish lightfrom televisions and clock radiospasses into the eyes, even throughclosed eyelids while asleep, to triggerthe pineal gland to initiate some of thebiochemical circadian rhythm shifts.

Wright, K. “Times of Our Lives.” ScientificAmerican 287, no. 3 (2002): 59–66.

46. Two Metronomes

For the case of the periodic perturba-tion of one metronome by the other,the mode-locking occurs when the per-turbing frequency is sufficiently closeto the unperturbed frequency of themetronome. When a metronome isplaced on the skateboard, the move-ment of the pendulum causes theskateboard itself to move slightly, usu-ally in the opposite direction to thependulum swing, since the metronomebase is kept in place on the skateboardby static friction or could be bolteddown. Some of the energy of themetronome base motion is transferredto the skateboard, and this very smallamount of energy is further trans-ferred along the skateboard in severaldirections, with some amount reachingthe other identical metronome.

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If at first this energy arrives at somerandom phase point in the oscillationof the second metronome, eventuallyits regular energy delivery becomesmore and more effective in synchron-izing the pendulum oscillations. Ofcourse, the second metronome is actingon the first metronome in the same waysimultaneously. The synchronization isnormally in-phase, but antiphase syn-chronization can occur in special con-ditions. (See the second reference belowfor details.)

The behavior can be representedby two equations for two harmoni-cally driven oscillators with a signifi-cant amount of dampening. If thedampening were not significant, thenwe would see two coupled pendulumsalternating their swing behavior out ofphase from maximum amplitude tonearly zero amplitude. In the actualcase, the pendulums simply synchro-nize and keep nearly identical time.

For the case of one of the pendu-lums being driven by a force random intime, their fluctuating behavior canconverge to an identical response. Bothpendulums would exhibit the samerandom fluctuations eventually. Forboth periodic and aperiodic drivingforces, asymptotic stability results forlinear oscillators properly damped.That is, small changes in the parame-ters of the linear oscillator or the driv-ing force result in only small changes inthe asymptotic behavior. The equation

of motion for each oscillator is mathe-matically equivalent to describing a lin-ear spring in a viscous medium with afluctuating driving force.

According to the first referencebelow, the mode-locking can occuralso for a wide range of aperiodicallydriven nonlinear oscillators in thephysical and biological sciences, fromnonlinear electrical circuits to neuralsystems. As in the periodically drivensystems, the synchronization of ran-domly driven nonlinear oscillators wasfound to be structurally stable, whichmeans that even in the presence ofsmall amounts of noise an approxi-mate synchronization is achieved.

Jensen, R. V. “Synchronization of DrivenNonlinear Oscillators.” American Journalof Physics 70 (2002): 607.

Pantaleone, J. “Synchronization of Metro-nomes.” American Journal of Physics 70(2002): 992–1000.

47. Time Symmetry

No, nature does not need to obey timesymmetry at the most fundamentallevel. Equations sometimes have moresymmetry than the actual underlyingphysics behavior. For example, eventhough the tensor equations of generalrelativity are time-symmetric, they can be derived from a more funda-mental type of mathematical entitycalled a twistor. Twistor equations ofgeneral relativity are not time-symmet-ric. In applications to a black hole, for

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example, the tensor equations predicttime symmetry, but the twistor onesdo not. As a consequence, the forma-tion of a black hole and the time-reversed version cannot both representreal physical behavior.

One might think that the quantumtheory described by the Schrödingerequation is time-asymmetric, the equa-tion being first order in time. As RogerPenrose points out in the referencebelow, quantum theory and its equa-tions are indeed time-asymmetric. Thewave function can be used to calculatethe probability of a future state on thebasis of a known past state, but notthe other way—that is, one cannot cal-culate the probability of a past state onthe basis of a future state. You cannotretrodict the past!

Hilgevoord, J. “Time in Quantum Mechan-ics.” American Journal of Physics 70(2002): 301–306.

Penrose, R. The Emperor’s New Mind.Oxford: Oxford University Press, 1989,pp. 354–359.

Chapter 3Crazy Circles

48. Spider and Fly

To find the shortest path between anytwo points on a cube not on the sameface, one convenient method is to layout the six faces of the cube on a plane

and draw the straight-line path. Ofcourse, all parts of the path must be onthe faces, and the appropriate facessharing a common edge must retaintheir relative positions and orientations.

Steinhaus, H. Mathematical Snapshots, 3rded. New York: Oxford University Press,1983, pp. 173–176.

49. Moon Distance

The laser light pulse traveling from theEarth to the Moon and back willencounter the Earth’s atmospheretwice. The pulse will have an initialknown rise time and decay time, butthese times will be extended by pas-sage through the air medium. Weassume ideal reflection at the Moon’scorner reflector—that is, no pulsespreading in angle or in time.

First, the ideal case. We assumethat the laser source and the reflectoron the Moon are opposite each otheron the line connecting the centers ofEarth and Moon and that Earth’satmosphere does not affect the transit

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time. The major source of uncertaintywill be the ability of the detection sys-tem to locate the half-height point onthe rise time of the outgoing pulse andthe same point on the incoming pulse.If the system is good to about apicosecond in detecting this point,then a transit time of 2.56 seconds forthe 3.84 × 108 meter distance corre-sponds to a timing uncertainty of bet-ter than one part in 100 billion, withan uncertainty in distance of less than4 millimeters. That is, with the pro-posed laser system, one can measurethe distance to the Moon to almost thesame distance uncertainty as one canmeasure the length of a table with ameterstick!

Of course, the atmosphere willfoul things up a bit. The index ofrefraction and the change of this indexwith altitude will both alter the lightpulse speed and spread out the pulserise time and rise shape. Sophisticatedsignal processing techniques can elim-inate most of these atmosphericeffects. So the final uncertainty will bedetermined in the electronics creatingthe laser pulse and detecting thearrival of the pulse’s leading edge.

50. Ideal Billiards Table

On this ideal billiards table for whichthe incident and reflection angles at thecushion are equal, one simply consid-ers adjacent mirror image copies of the

table in a gridwork. Imagine placingthe ball in appropriate adjacent tablesat mirror image positions; then drawthe straight line to the pocket to findthe collision points on the cushions.

Normal billiards and pool tablesare marked around the perimeter toaccomplish precision bank shots. Usingthese markers takes practice. These realtables with their markings are not verysimilar to the ideal table discussedabove. Moreover, the rolling ball beforethe collision with the cushion may haveadditional spin—“English”—about anaxis not parallel to the table and per-pendicular to the travel direction. Theprofessional player uses all these quan-tities in the particular shot, but weamateurs simply enjoy the results as wepractice more of the many possibleimprovements to our games.

Steinhaus, H. Mathematical Snapshots, 3rded. New York: Oxford University Press,1983, pp. 61–64.

51. Wallpaper Geometry

Standing inside the cube, to your rightyou see your left side of your person inthe cube to the right. To your frontyou see your back. To the top you see

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the soles of your shoes. You see a 3-Darray of yourself from many differentviews, at many different distances, atmany apparent sizes, and at many dif-ferent image intensities. This view isnot like being inside a cube withreflecting mirrors on all sides becauseno image is reversed.

Cosmologists are trying to deter-mine whether our 3-D space is mathe-matically and physically discrete—thatis, compartmentalized into large cubesor regular dodecahedrons, each beingperhaps as large as 10 billion light-years or bigger. If so, seeing a galaxy inone direction could be complementedby seeing the same galaxy in the oppo-site direction. Of course, several prob-lems exist, such as the distance beinggreater in one direction than in theother, with the consequence that thegalaxy is being seen not only from the other side but also at a differenttime in its evolution of structure.There may be multiple copies of thegalaxies to confound things. Theremight even be multiple copies of eachof us! Any positive results will bringabout a revolution in our thinkingabout space and time in the universe.

Levin, J. How the Universe Got Its Spots:Diary of a Finite Time in a Finite Space.Princeton, N.J.: Princeton University Press,2003, pp. 132–155.

Thurston, W. P., and J. R. Weeks. “TheMathematics of Three-Dimensional Mani-folds.” Scientific American 251, no. 1(1984):108–120.

52. Space-Filling

Geometry

First consider a two-dimensional flatspace. A plane tesselation (or two-dimensional honeycomb) is an infiniteset of polygons fitting together tocover the whole plane once, with everyside of each polygon belonging to justone other polygon. A regular tessela-tion has regular polygons. There arethree regular tesselations of the plane:equilateral triangles, squares, and reg-ular hexagons. There are additionalplane tesselations with two or moreconvex polygon shapes. One also cancover the plane with Penrose tiles,polygon pairs with at least one poly-gon not being convex.

Now consider an additional spatialdimension. A three-dimensional honey-comb (or solid tesselation) is an infiniteset of polyhedrons fitting together to fillall space once, so that every face of eachpolyhedron belongs to one other poly-hedron. If we require all the polyhe-drons to be identical, then the onlyregular honeycomb is the one filled withcubes, eight at each vertex. If we allowtwo different regular polyhedrons, onecan fill the space with eight regulartetrahedrons and six regular octahe-drons surrounding each vertex. Thesespace fillings and others determinemany of the natural crystal structures.

From the apparent simplicity of a 3-D space filled with cubes, one may

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think that this solid tesselation wouldbe the most likely mathematically if realspace is discrete instead of continuous.However, mathematicians can showthat the most likely and interesting 3-Ddiscrete space is the non-Euclidean tes-selation by dodecahedrons, of whichthere are two kinds, depending on theangle of twist in relating one dodecahe-dron to the adjacent one. For furtherinformation see the Thurston andWeeks reference below.

Coxeter, H. S. M. Regular Polytopes. NewYork: Dover, 1973, pp. 58–74.

Levin, J. How the Universe Got Its Spots:Diary of a Finite Time in a Finite Space.Princeton, N.J.: Princeton University Press,2003, pp. 132–155.

Thurston, W. P. and J. R. Weeks. “The Math-ematics of Three-Dimensional Manifolds.”Scientific American 251, no. 1 (1984):108–120.

53. Archimedes’

Gravestone

Archimedes (287?–212 B.C.E.), perhapsthe greatest mathematician of ancienttimes, was the first to calculate the vol-ume ratio of the sphere inside the cylin-der. With a sphere and a cone insidethe cylinder touching top, bottom, andsides, Archimedes determined thattheir volumes are in the ratios 1:2:3!

The Roman general, Marcellus,tells of how he searched for and foundArchimedes’ gravesite with this

headstone. Archimedes was killed in212 B.C.E. during the capture of Syra-cuse by the Romans in the SecondPunic War after all his efforts to keepthe Romans at bay with his machinesof war had failed. Plutarch recountsthree versions of the story of his killingthat had come down to him:1. “Archimedes was, as fate would

have it, intent upon working outsome problem by a diagram, andhaving fixed his mind alike and hiseyes upon the subject of his specula-tion, he never noticed the incursionof the Romans, nor that the city wastaken. In this transport of study andcontemplation, a soldier, unexpect-edly coming up to him, commandedhim to follow to Marcellus; whichhe declining to do before he hadworked out his problem to a demon-stration, the soldier, enraged, drewhis sword and ran him through.”

2. “A Roman soldier, running uponhim with a drawn sword, offered to kill him; and that Archimedes,looking back, earnestly besoughthim to hold his hand a little while,that he might not leave what he wasthen at work upon inconclusive andimperfect; but the soldier, nothingmoved by his entreaty, instantlykilled him.”

3. “As Archimedes was carrying toMarcellus mathematical instru-ments, dials, spheres, and angles, by

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which the magnitude of the Sunmight be measured to the sight,some soldiers seeing him, and think-ing that he carried gold in a vessel,slew him.”Archimedes was buried in Syra-

cuse, where he was born, where hegrew up, where he worked, and wherehe died. On his grave there is aninscription of π, his most famous dis-covery. Also placed on his tombstoneis the figure of a sphere inscribedinside a cylinder and the 2:3 ratio ofthe volumes between them, the solu-tion to the problem he considered hisgreatest achievement.

His nicknames were, “the WiseOne,” “the Master,” and “the GreatGeometer.”

Plutarch. Lives of Noble Grecians andRomans. Translated by A. H. Clough. NewYork: Random House, Modern Library,1992, p. 517.

54. Brain Connections

A million neuron model of the brain isstill quite a formidable programmingtask for a computer simulation, butthere would be no information trans-fers from neuron to neuron. Why not?Because any neuron in this model ofthe human brain would have on aver-age nearly zero inputs. One calculatesas follows: if 1011 neurons have 1,000connections each, say, then the averageis 1 connection per 108 neurons.

Therefore, a model with 106 neuronswill not work as a useful scale modelof the real brain.

Of course, one could simply take asmall volume of the brain containing 1million neurons and ignore connec-tions to other parts. Or one could arti-ficially modify the unusable computermodel above by ensuring a few con-nections or more to each neuron.Whether the behavior that ensues isrealistic must be determined. The morepractical approach is to model a smallsection of the brain—perhaps tens ofthousands of neurons and all theirinterconnections—in a focused studyand simulation. A grid of computers,each representing one small section,could then be used to simulate a largerportion of the brain. Hopefully, whenquantum computers become a reality,they will be able to simulate the wholebrain. Whether the brain behavesquantum mechanically and requiresquantum superposition for its opera-tions is presently unknown.

There is the remarkable problemof information storage in the brain—that is, where exactly is informationstored? If each neuron stores only onebit of information, then the humanbrain is not large enough by many fac-tors of ten! In 1989 Roger Penrosesuggested that each neuron must becapable of storing many bits of infor-mation, in contrast to the prevailing

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ideas. Subsequently, the numerousmicrotubules in each neuron werefound to participate in the informationstorage game. There still remains thequestion of what each stored bit ofinformation represents.

Penrose, R. Shadows of the Mind. Oxford:Oxford University Press, 1994, pp.358–377.

55. Configuration

Space

There are many ways to approach thisproblem of describing the arm positionin physical 3-D space. We consider oneapproach only. In all approaches, theend of the rodlike hand must touch thespecified point, so three numbersdefine the end point of the hand.

Let’s start at the fixed shoulderposition. Two numbers will describethe upper arm position, the angle inthe vertical plane measured from afixed vertical axis through the shoul-der, and an angle about this verticalaxis. Two more numbers describe theforearm position, an angle in the verti-cal plane measured from a vertical axisthrough the end of the upper arm, andan angle about this vertical axis. Like-wise, two more angles are needed forthe hand.

At least six numbers are necessaryfor the robot to locate the particularpoint in the room. The program will

calculate the extent of the arm todetermine its end point distance, thusrequiring three more numbers, thelengths of the three parts. The space ofoperation is nine-dimensional and iscalled a 9-D configuration space todistinguish between physical space andcoordinate space. Of course, one couldhave determined this result by realiz-ing that each rod end point requiresthree coordinate values to be specified.

The movement of the arm to touchthe point is the next challenge. If feed-back exists in the robot, such as visualfeedback of the hand position and thedesired point location, the movementalgorithm can use a correction proce-dure that becomes finer and finer as thefingertip approaches the point, as wehumans tend to operate. If there is nocontinual feedback mechanism, thenthe algorithm must move the arm tothe point directly, somehow knowingwhere the fingertip is at all times. Asystematic error cannot correct itself ifno feedback exists. Many robotic armsoperate in both modes, first withoutfeedback for rapid deployment andthen with feedback for fine adjustment.

As humans, we learn to performmany tasks and do many of them sev-eral times daily. As a result, we oftenforget how we learned a particularprocedure and how much practice wasrequired. To relive that learning expe-rience, try using the “other hand” to

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punch in data in a calculator, or somesimilar task. The learning curve issometimes very steep!

56. Farmer Chasing a

Goose

If the farmer is restricted to chasingthe goose along the instantaneous line of sight to the goose, the farmerwill never be able to catch the gooseunless there is a head-on encounter.The best strategy for the goose is torun in a straight line, for then thefarmer’s velocity is eventually in thesame direction as the goose’s direction,and the relative distance remainsconstant. Note that even when thegoose changes direction often, thefarmer cannot close the gap com-pletely because the closer they are, themore toward being parallel are thevelocities!

In the real open-field experiencewithout the restriction, one strategythat might work if the goose is inexpe-rienced is for the farmer to anticipatethe position of the goose and to getthere at the same time the goosearrives. However, most geese “read”the game plan and change course inmidflight.

Behroozi, F., and R. Gagnon. “The GooseChase.” American Journal of Physics 47,no. 3 (1979): 237–238.

57. A Spooky

Refrigerator

Yes. Just as you could remove a dotfrom a piece of paper with an eraserbrought in from the third spatialdimension, a 4-D being could enter therefrigerator without needing to openthe door and remove a piece of food.That is, 3-D objects are open in thedirection of the fourth dimension.

Conceptually, visualizing a 4-Dobject is difficult in our 3-D world.Some mathematicians suggest lettingthe fourth coordinate direction be rep-resented by flashing color, such as thesequence of colors in the visible spec-trum from red to indigo. Take any 3-Dobject—a sphere, for example. As thesphere moves in the fourth coordinatedirection its flashing color changesfrom red to orange to yellow, etc. A 2-D sheet of paper moving in thefourth dimension would be changingits flashing color also to indicate itsfourth coordinate value. The inherentcolor of the object does not change, ofcourse.

Descriptions of 4-D objects inter-secting our 3-D world are quite fasci-nating. For example, a 4-D sphereintersecting our 3-D world would firstappear as a point, then an increasing3-D sphere, then a decreasing 3-Dsphere, then a point, then gone! Theanalog in fewer dimensions would be a

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3-D sphere intersecting a 2-D sheet ofpaper, being first a dot, then a widen-ing circle followed by a narrowing cir-cle, then a dot again, and then gone.

Although most people wouldexpect there to be more mathematicaldifficulty and complications in evenhigher dimensions than four, thisexpectation is false. The mathematicsactually simplifies with five dimensionsand more! Much geometry remains tobe worked out in a 4-D space, whereasthe mathematics is better understoodin the higher dimensions.

Gardner, M. The Colossal Book of Mathe-matics: Classic Puzzles, Paradoxes, andProblems. New York: W. W. Norton, 2001,pp. 137–149.

Peterson, I. The Mathematical Tourist. NewYork: W. H. Freeman, 1988, pp. 82–107.

Pickover, C. A. Surfing through Hyperspace:Understanding Higher Universes in Six EasyLessons. Oxford: Oxford University Press,1999, pp. 44–70.

58. Fractional

Dimensions?

Yes. Noninteger dimensions are knownas fractal dimensions. A pathwaytoward understanding fractal dimen-sions begins by considering duplicationsof well-known objects. A line segmentcan be duplicated to produce two linesegments. A square can be duplicatedin each direction to produce foursquares. A cube can be duplicated ineach of its three directions to produce

eight cubes. In each case we obtain thenumber 2 raised to an integer power.We can make a table and generalize to darbitrary dimensions.

Figure Dimension No. of Copies

Line segment 1 2 = 21

Square 2 4 = 22

Cube 3 8 = 23

Doubling figure d n = 2d

We can now determine the dimen-sion of an interesting but strange geo-metrical object, the Sierpinski triangle,named after the Polish mathematicianwho originally thought it up in 1916,shown here with its holes being gray.Double the length of the sides, and youget another Sierpinski triangle, similarto the first. For example, if the firstSierpinski triangle has one-inch sides,the doubled one has two-inch sides.How many copies of the original tri-angle do you have? Remember that thegray triangles are holes, so we can’tcount them.

Ignoring the hole in the center ofthe double-sized Sierpinski triangle,

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we learn that doubling the sides of theoriginal gives us three copies, so 3 =2d, where d = the dimension accordingto the scheme in the table. Using acalculator, one finds that its dimension d = 1.585 . . . , a noninteger!

In general, the mathematicalexpression for the dimension of thefigure is given by the ratio of twologarithms:

dimension = logarithm (number of self-similar pieces)/logarithm

(magnification factor).

For simplicity:1. A dimension between 0 and 1 is

supposed to correspond to thecapacity of a set of points to partlyfill a line without achieving it com-pletely, out of having the wholevalue 1 that is needed.

2. A dimension between 1 and 2 issupposed to correspond to thecapacity of a line to partly fill aplane, without achieving it com-pletely, out of having the wholevalue 2 that is needed.

3. A dimension between 2 and 3 issupposed to correspond to thecapacity of a surface to partly fill avolume without achieving it com-pletely, out of having the wholevalue 3 that is needed.There is a whole world of mathe-

matics to be learned with fractaldimensions and fractal geometry and

their applications to the familiar phys-ical world. One interesting question iswhether two or more objects with thesame fractal dimension must berelated in some fundamental way,either mathematically or physically.

Peterson, I. The Mathematical Tourist. NewYork: W. H. Freeman, 1988, pp. 114–142.

59. Platonic Solids

One must turn the second identical reg-ular tetrahedron upside down androtate by 30 degrees about the verticalaxis before mathematically pushingthem through each other to form a six-vertex regular solid. Obviously these sixvertices correspond to the vertices of aregular octahedron if the new objectwere placed inside one. However, thesides are not convex and flat. Our bi-tetrahedron does have twofold symme-try axes even though the two tetra-hedrons are rotated with respect to eachother.

We might have asked you to placethe two regular tetrahedrons face-to-face in congruence so that the com-bined object has five vertices defining

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a triangular bipyramid. There are nowthree twofold rotational symmetryaxes, each one being through an edgeof the joining faces. However, thisbipyramid is not a regular polyhedron.

If we were to extend the discussionto four spatial dimensions, there aresix regular solid convex objects analo-gous to the five Platonic solids in threedimensions. The number of regularconvex solids, called regular poly-topes, does not increase with morespatial dimensions. Instead, all dimen-sions beyond four have only three reg-ular convex solids.

The Platonic solids are special inmany ways, but perhaps their mostimportant mathematical property waspointed out by mathematician B.Kostant:

The ancient Greeks, especially theschool of Plato, had great reverencefor the regular polygons in the planeand regular solids in 3-space. Thelatter—the tetrahedron, cube, octa-hedron, dodecahedron, and theicosahedron—are often referred to asthe Platonic solids. The Greeksbelieved that these regular figureswere fundamental in the structure ofthe universe. If symmetry or itsmathematical companion—grouptheory—is fundamental in the struc-ture of the world, then one of thepoints of our lecture is the statement

that the Greeks were absolutelyright. That is, what we will be sayingin a very profound way, the finitegroups of symmetries in 3-space“see” the simple Lie groups (andhence literally Lie theory) in alldimensions.

One of us (F. P.) has proposed thatthe fundamental building blocks ofmatter, the leptons and quarks of theStandard Model of particle physics,are described mathematically by thespecific rotational symmetries of the 3-D Platonic solids for the leptons andby their 4-D analogs for the quarks.The key arguments are that themathematical symmetry groups forthese regular solids are subgroups ofthe Standard Model symmetry groupand that the lepton and quark massratios can be directly related to theratios 1:108:1728 of the invariants ofthese subgroups. Whether the naturalworld mimics this fundamental math-ematical behavior has yet to be deter-mined by experiments at particlecolliders.

Coxeter, H. S. M. Regular Polytopes. NewYork: Dover, 1973, pp. 41–57, 126–144.

Kostant, B. “Asterisque.” In Proceedings ofthe Conference “Homage to Elie Cartan,”Lyons, July 1984, p. 13.

Potter, F. “Geometrical Basis for theStandard Model.” International Journal ofTheoretical Physics 33 (1994): 279–305.

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60. Intersecting

Spheres

Two identical three-spheres can inter-sect in a point, a circle, a sphere (two-sphere), and a three-sphere. Nowbring in a third identical three-sphereto intersect with the former two inappropriate combinations of points,circles, spheres, and a three-sphere, the latter when all three are coinci-dent. With three intersecting identicalthree-spheres, a resulting single two-sphere can be obtained only when thethree three-spheres form a symmetrical configuration.

If the leptons and quarks of theStandard Model of particle physics arephysical manifestations of the finiterotational symmetries of the 3-D Pla-tonic solids and their 4-D analogs asproposed in a model by F. Potter (seethe reference below), then the intersec-tions of three-spheres will becomeimportant in fundamental physics. Aquark would be defined in a 4-Dspace, and its mathematical behaviorwould depend on the properties ofthree-spheres. The proton, for exam-ple, is a real particle composed of threequarks in our 3-D world—that is,three 4-D entities according to the pro-posed model. So three three-spheres(representing the quarks) must inter-sect to form a two-sphere that “lives”in our 3-D space.

Potter, F. “Geometrical Basis for theStandard Model.” International Journal ofTheoretical Physics 33 (1994): 279–305.

61. Arm Contortions

Yes, if one allows the arm to move over-head. This second rotation untwists thearm and brings the orientation of the book back to the initial one again.One can say that the arm-object pairrequires two 360-degree rotations toreturn to the initial orientation. Such anentity is said to mathematically corre-spond to a spin 1⁄2 system—that is,related to the continuous symmetry

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group SU(2). Any lepton or quark wave function, such as the electronwave function, behaves in this way with respect to rotations and angularmomentum.

Spheres, cubes, and other objectswith spatial symmetry also can be clas-sified as spin 1⁄2—that is, their rota-tions are described by symmetrygroups that are subgroups of SU(2)and SU′ (2) = SU(2) × Ci , where Ci isthe two-element inversion group. Forthe Platonic solids, the rotational sym-metry groups are discrete instead ofbeing continuous, and some of thesesymmetry groups are subgroups ofboth SU(2) and SU′ (2) because amongall the elements of both can be foundthe elements of finite order for the dis-crete subgroups.

As you know, our practical experi-ence is mostly with spin 1 entities—that is, those needing a 360-degreerotation to return to the initial orien-tation. Mathematically, these spin 1properties can be constructed from thespin 1⁄2 symmetry properties. Mathe-maticians know that even more funda-mental are the reflection groups fromwhich all spin 1⁄2 properties can bederived as two reflections in perpendi-cular planes. The two books listedbelow discuss these hierarchical rela-tionships and many more.

Altmann, S. L. Rotations, Quaternions, andDouble Groups. Oxford: Clarendon Press,1986.

Coxeter, H. S. M. Regular Polytopes. NewYork: Dover, 1973.

Rieflin, E. “Some Mechanisms Related toDirac Strings.” American Journal of Physics47 (1979): 379–381.

62. The Rotating

Cup

Done properly, you would see thesame sequence in both cases. The cupappears to rotate with changing rota-tion rates as time passes. We have herean example of Galilean relativity foruniform motion. At these familiarslow speeds compared to the speed oflight, the behavior of the rotating cupproduces no surprises. We can walkpast the cup, or the cup can move pastour stationary location.

In a later chapter, where we intro-duce the special theory of relativity(STR), we examine an object such as acube moving past a stationaryobserver at enormous speed, and wecould consider the opposite case, ofthe observer moving past the station-ary object. Of course, one sees thesame behavior in both cases, just likethe symmetry we observe in Galileanrelativity. However, in STR an effect,now called the Terrell effect, explainswhy a cup approaching and passing byat near-light speeds appears addition-ally rotated, the observer being able tosee the back side of the cup as itapproaches!

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63. Space and Time

Together

Using three real spatial coordinatesand one imaginary time coordinate forcalculations works correctly whencalculating the squares of space-timecoordinates and their sums and differ-ences. The important relationship isthe space-time interval τ defined by τ2

= c2 ∆t2 – ∆x2 – ∆y2 – ∆z2, where the∆x’s are the four “distances.” How-ever, physics textbooks that use aninterval τ defined by τ2 = + ∆x2 + ∆y2 +∆z2 – c2 ∆t2 are making a mathemati-cal faux pas in choosing three realspace coordinates and one imaginarytime coordinate—that is, the set (x, y,z, ict) with i being the imaginary and cbeing the speed of light—instead ofvice versa. Fortunately, this fundamen-tal error does not affect the calcula-tions of time intervals and spatialseparations because these calculationsinvolve the differences of squaredquantities. To be mathematically cor-rect in the (3 + 1)-dimensional space-time, one must use quaternions, whichare numbers in the form q = a + bi + cj+ dk, with i, j, and k being √–1 and a,b, c, and d being ordinary real num-bers, because they are the numbers infour dimensions that properly handlerotations, translations, and Lorentztransformations. Today, quaternionsare used everywhere in science to

describe the dynamics of motion inthree-space. Spinors are equivalentmathematical entities used in quantummechanical wave functions to describethe electron and other fermions in (3 + 1)-D space.

Quaternions were first “discov-ered” by W. R. Hamilton in the 1800s,and the quaternion q has one real com-ponent and three imaginary compo-nents. Just as complex numbers areformed from pairs of real numbers,quaternions are formed from pairs of complex numbers. Thus one shouldassign the time coordinate to the real component and the three spacecoordinates to the three imaginarycomponents of a quaternion. Hence,mathematically, we live in a quaternionworld with an imaginary 3-D physicalspace and a 1-D real-time clock!

Altmann, S. L. Rotations, Quaternions, andDouble Groups. Oxford: Clarendon Press,1986.

Pickover, C. A. Surfing through Hyperspace:Understanding Higher Universes in Six EasyLessons. Oxford: Oxford University Press,1999, appendix D.

64. Space > 3-D?

There are several arguments for whyspace is not larger than three dimen-sions. Planetary orbits are not stablewhen n > 3, except for a circular orbitfor n = 4, because the attractive forceand the centripetal force both do nothave the correct dependence on radial

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distance. In 1917 P. Ehrenfest showedthat one needs to consider the Poissonequation for arbitrary dimensions todetermine orbit stability. When the n ≥ 4 circular orbit for a body arounda central mass becomes slightly per-turbed, one can show that the com-parison of the central force to thecentripetal force for the orbit dependson the perihelion value r1 and theaphelion value r2 according to [1/2 –(n – 2)–1]/r1

2 < [1/2 – (n – 2)–1]/r22,

which cannot be true for n = 4 andlarger. In a 4-D space, a satellitelaunched from Earth toward the Sunwould either fly away to infinity or spi-ral into the Sun.

The hydrogen atom is not stablewhen n > 3 because there is no energyminimum for n ≥ 5, which is shownusing the indeterminancy principle—that is, the Heisenberg uncertaintyprinciple. For the case n = 4, the rela-tivistic energy equation must be exam-ined to show that no energy minimumis available and the atom is not stable.

Several other physical phenomenawould be unusual for n ≥4 dimensions.There is no satisfactory propagation ofsound waves or electromagnetic wavesfree of distortion and reverberation inspaces other than n = 1 and n = 3.Also, axial vectors such as the mag-netic field and the angular momentumvectors do not exist in even-dimen-sional spaces.

The considerations could beextended to a universe with more thanone time dimension! However, thismatter and others we leave for futurechallenges.

Büchel, W. “Why Is Space Three-Dimen-sional?” Translated by I. M. Freeman fromPhysikalische Blätter 19, no. 12 (1963):547–549. American Journal of Physics 37(1969): 1222–1224.

Ehrenfest, P. Annalen der Physik 61 (1920):440.

Pickover, C. A. Surfing through Hyperspace:Understanding Higher Universes in Six EasyLessons. Oxford: Oxford University Press,1999, pp. 202–205.

Chapter 4Fly Me To The Moon

65. Gunfight

One could do good classical physicshere, but the filmmakers have turnedthe scene into Hollywood exaggera-tion. The physics is determined by theconservation of linear momentum.Assume that the victim of the shootingis initially at rest, so the total momen-tum initially is all in the bullet (orbuckshot) of mass m and speed Vbefore hitting the victim, at mV. Afterthe collision, the final total momentumis in the backward “flying” personplus bullet. If the victim has mass Mand the combined victim-plus-bullet

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object has speed v, the total finalmomentum is (M + m)v. For all inter-actions, by the law of conservation oflinear momentum, the final momen-tum equals the initial momentum.

In the simplest case (assuming nofrictional drag at the feet and ignoringtransfer of momentum away to theearth, etc.), its application yields (M +m)v = mV. Solving for the velocity v ofthe victim afterward produces v =mV/(M + m). Substituting reasonablevalues of M = 80 kg, V = 400 m/s, andm = 0.03 kg yields a “blow-back”velocity maximum of v = 0.15 m/s.Most people can walk about 2 m/s(i.e., about 4 mph). So we can con-clude that any shooting victimdepicted as being blown backward bythe impact of the bullet (or shotgunblast of pellets) is ridiculous andbelongs in the fantasy world only!

A physicist wouldn’t actually needto calculate the velocity backwardusing linear momentum conservationexplicitly. Simply watching the behav-ior and movement of the shooter hold-ing the gun before and after the shotreveals approximately how muchmomentum is available by using New-ton’s third law. If the shooter isn’tblown backwards by the recoil forceof the shot, the victim won’t be either.Of course, someone will suggest thatinvoluntary muscle contraction in thestunned victim causes the “flying”

backward. Falling backward, perhaps,but not “flying”!

There is the story of a famousphysicist back in the 1950s who lovedto watch gunfights in Western movies.The bad guy always draws first, henoticed, but the good guy wins the gun-fight. How could this outcome happen?His hypothesis was that psychologyplayed an important role, slightly hin-dering the man who had to make theconscious decision to draw first. Thesecond man simply had to react.

Even today, the psychology ofchoosing a physical action is an impor-tant factor, particularly in sports.There are tennis coaches (and coachesin other sports) who preach the psy-chology of playing tennis, saying thatwhen you think too much on the courtinstead of simply reacting, as you learnto do in practice, then you are in trou-ble. You are letting self no. 1 (yourmind) control self no. 2 (your body),and your tennis game will suffer. Wewonder whether the concepts holdtrue for playing the physics game, too!

66. Body Cushion

We doubt whether landing on top ofanother body after such a long fallprovides much cushion! The impor-tant parameter here is the extent of thecollision time ∆t—that is, how longthe collision of the hero’s body with

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the object actually lasts. The longerthe ∆t, the better. We also need toknow the acceleration a versus timeprofile. In better words, what is themaximum acceleration to be experi-enced by the hero’s body? By defini-tion, the average a = ∆v/∆t, where ∆vis the velocity change during the timeinterval ∆t. Shorter ∆t’s make theexperience more painful.

Stunt professionals are often seenleaping off buildings or falling throughwindows in movies, but their colli-sions are with huge air-filled balloonsthat effectively extend the total colli-sion time to half a second or more. Wedo not see their collision with the bal-loon in the program because the edit-ing process substitutes the desiredbody lying dead on the concrete.

Back to the hero landing on theother body. The collision time herewill be less than one-tenth of a second,producing dangerous accelerations.For example, if the body falls from thetop of a two-story building, its speedwill be approximately 11 m/sec justbefore collision. The acceleration dur-ing the collision will be greater than110 m/s2, very dangerous. Even ifbones are not broken in bringing thehero’s body to rest, the internal organswill continue to move until they suffera collision inside the hero’s body. Andif the hero bounces back upward, theacceleration can be even worse,

because the change in linear momen-tum will be almost double, eventhough the collision duration may beincreased slightly. Automobile colli-sions provide plenty of evidence aboutthe damages done to internal organsby sudden collisions with very shortcollision times. We doubt whether ourhero will be able to walk away fromthe “body cushion.” In fact, our herowill be lucky to survive!

67. Cartoon Free Fall

When the cartoon character steps offthe cliff, the fall should begin immedi-ately, of course. The natural path isessentially a parabola, with approxi-mately free-fall acceleration downwardand a constant velocity horizontally.Even a cartoon character must havesome mass; otherwise the charactercould not exert a force on anything,including the ground being walked on.Unless the upward buoyant force of theair balances exactly the gravitationalforce downward, no cartoon characterstepping off the cliff would remain insuspension at the height of the cliff.Even if the buoyant force was suffi-cient, why would its upward push dis-appear suddenly to allow the characterto free fall?

We should see the character accel-erating downward with ever-increasingspeed unless the terminal velocity is

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reached or the buoyant force balancesthe weight. The collision at the bottomis also subject to analysis. To preventthis sudden collision, sometimesanother character is able to run downfrom the clifftop just in time to catchthe falling character. And sometimeswe even see another person falling witha greater acceleration downward toarrive in time to catch the victim! If thefall is nearly at the free-fall accelera-tion, the runner must be mighty swift!There are measured examples of skiersgoing down Mount Fuji with accelera-tions greater than the free-fall accelera-tion, but no runner has achieved thisfeat yet. And yes, an anvil always has agreater acceleration downward thanany other object (sure!)!

68. Silhouette of

Passage

The condensed matter physicist knowsa lot about the physical properties ofliquids and solids, so he or she proba-bly would say, “Wow! How was thatdone?” The only realistic possibility isthat the wall cut should be quite messyand the cartoonist cleaned the edgesfor heightened dramatic effect!

We can estimate how difficult thecookie cutter hole would be to achieveby considering a ball thrown at the wall. The impact surface area

increases rapidly in time as the balland wall both bend a little during thecollision. The initial kinetic energy ofthe ball just before the collisionbecomes distributed in the distortionsof the ball and the wall. The interac-tions between the molecules of thewall material change as some of theavailable energy from the collisionspreads from the immediate impactarea. If the total collision time isextremely short, the energy distribu-tion will be quite limited in distance,and much of the energy is available forripping. Otherwise, if the collisiontime is much longer, a big portion ofthe wall will respond by deformingjust a little, and ripping may not occur.

A bullet going through the paperat a practice target makes a fairly cleanhole for two reasons: (1) the contacttime is extremely short; (2) the bulletoffers a nearly round profile, so thereis symmetry about an axis perpendicu-lar to the hole. Even then, close exam-ination of the bullet hole reveals anirregular surface and additional tear-ing beyond the actual round hole.

You can check out the advantage ofa symmetrical object. Now, withappropriate safety precautions, try torapidly push any shaped profile otherthan round through a sheet of targetpaper. The lack of cylindrical symme-try perpendicular to the surface usuallycreates enormous problems for the

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material because slightly more energygoes into some directions. Moreover,tearing is required to occur at differentdistances along a noncircular profile,so there may be points where the trans-ferred energy density is significantlyhigher or lower than the surroundingpaper regions. All these factors andseveral more act against a very cleancut through the material. The cutthrough a thicker piece of paper or awall that has significant depth wouldbe even messier.

Of course, the cartoon characterhas several other options for gettingthrough the wall when time permits:(1) simply paint an exit onto the wallthrough which only he can pass, or (2)the character can carry around a holeto be affixed where needed!

69. Artificial Gravity

Space stations and spaceships havebeen created by authors and screen-writers in a vast array of shapes andsizes. A rotating dumbbell shape hasappeared in many space adventures,the rotation providing a pseudo-gravity force for Earth creatures. Therotation about the center of mass per-pendicular to the long axis provides apseudo-acceleration acting radiallyoutward called the centrifugal acceler-ation ac = v2/r, where v is the tangentialvelocity value and r is the radial dis-tance from the axis of rotation. The

resulting centrifugal force is called apseudo-force because the actual forceis acting radially inward toward theaxis of rotation to accelerate the objectfrom its inertial straight-line motion.We must assume that the structuralintegrity of the space station remainsintact—that is, the station wasdesigned for the rotation and for theallowed distributions of mass onboard.

Using the relation for the angularvelocity ω = v/r, we can express the cen-trifugal force as Fc = mrω2. The wholespaceship has the same angular velocityabout the rotation axis, so an object’sradial acceleration increases linearlywith distance r from the rotation axis.An astronaut at one end of the dumb-bell must climb—that is, walk up a lad-der and then down a ladder—from oneend to the other through the middle,where the radial acceleration is zero.The muscular effort required changesthroughout the climb, so the sensationsmust be wonderful!

70. Small Wings

The small wings on such alien beingsare probably much too small for a 20kg body. One could argue that theplanet’s gravitational force at its sur-face is much less than the value hereon Earth, so the alien being’s weight ismuch less also. That may be so, and

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the proposition is not unreasonable.However, we still require a sufficientair density for the wings to do theirwork and a breathing atmosphere forour Earthling on the foreign planet’ssurface. (After all, this example has thehuman standing there breathing with-out any special oxygen supply.)

We need to determine the requireddensity of the atmosphere of this alienplanet, assuming an adequate supplyof oxygen molecules for breathing byour visiting human. Earth’s atmos-phere at sea level has a total density ofabout 1.4 kg/m3, of which O2 com-prises about 20 percent by molecularcomposition. That is, 1 cubic meter ofair weighs 1.4 kilograms. The remain-der, of about 80 percent, is N2, whichhas a molecular weight of 28, com-pared to 32 for O2. For simplicity, weassume they have the same molecularweight, so we require an alien atmos-phere to have about 0.3 kg/m3 of oxy-gen available for breathing.

The gravitational acceleration g′ atits surface determines the air density atthe surface for a given molecular com-position and air temperature profile.Most planets will have an accelerationnot much different from the value of9.8 m/s2 here on Earth, as one cancheck out for the planets in the SolarSystem, for example. So the wingsmust be capable of exerting an upwardforce at least as great as the downwardgravitational force—in our example,

the weight F = g′ m of the alien beingwith small wings, or 200 N if g′ =10 m/s2.

We assume that a very strong 20kg individual can stretch out his armshorizontally to the side and pushupward against two supports withabout 200 N downward force to liftthe body. However, this same individ-ual will not be able to use small wingsof the same length and perhaps just alittle wider than the arms to beatagainst the air with equal effect. If youdoubt this hypothesis, put some arm-length wings on a strong person andobserve how easily he or she can liftoff and hover a few centimeters abovethe ground!

If the alien being were hollowinside so that its total mass is signifi-cantly less than expected for the bodysize, there may be no problem withhovering.

71. Shrunken People

We assume that the proposed shrinkingto one-hundredth scale can be accom-plished. Unfortunately, your weightwould remain the same (unless you getrid of molecules somehow), and yourdensity would increase a millionfold!The area of contact of your feet wouldbe 10,000 times smaller, so the pres-sure at your soles would be 10,000

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times greater, rising to about 20,000psi. Every step would break the con-crete, or you would sink into theground until the upward normal forcecould balance you. Among otherchanges, your metabolism must changeenormously, for your high ratio of sur-face area to volume will mean that therate of heat loss has increased at least100 times. Of course, we choose toignore any consequences inside thebody for simplicity.

Notice that if the opposite happensand you grow bigger and increase yoursize by a factor of 100 in all directions,without adding molecules, your den-sity decreases a millionfold. Youwould be blown away by practicallyany breeze! But your greater problemwould be that your density is nowmuch less than the density of air, so thebuoyant force upward would begreater than your weight. You are nowa giant balloon being pushed upwardtoward the upper atmosphere! Also,your metabolism would change dra-matically, but again, we ignore anyconsequences inside your body. Youcould probably make the journeyaround the world in 80 days withoutthe hot air!

72. Spaceship Designs

Landing a spaceship on Earth and thentaking off for space involve the same

forces, but the gravitational forcealways acts toward the center of Earth,sometimes being a help and sometimesbeing a hindrance. The major problemis the enormous energy requirement ingetting from the surface of Earth to areasonable distance away. Once thespaceship is more than a few Earthdiameters away, its nuclear engineoperation can be reasonably efficientin accelerating the vehicle. However, toget off the surface requires a tremen-dous amount of energy, and its rocketengines must throw out a lot ofmomentum in the exhaust gases athigh speeds to achieve “escape veloc-ity.” Newton’s third law dictates thismomentum requirement. The particlesejected backward act on the rocket in aforce pair, the rocket pushing particlesbackward while the particles are push-ing the rocket forward.

To reach outer space from Earth,the vehicle must provide a large supplyof energy and be able to eject a largeamount of momentum, usually byhaving a large supply of mass to eject.The energy needs can be accommo-dated by a variety of engineeringdesigns. However, the physics is quitedemanding on the amount of massejected per second. The fuel mass usedfor this propulsion is not consumedinstantly, so this fuel mass adds to themass of the vehicle at launch time.Consequently, even more propulsionfuel mass and energy are required for a

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launch than simply accounting for thepayload itself. The mass of the fuelsupply soon becomes many timeslarger than the actual payloadlaunched into space.

So when the spaceship leaves itsEarth spaceport and doesn’t eject a lotof stuff backward out of its rocketengines, the film is expressing a modeof operation that is not achievablewith present technology. But perhapsthe propulsion will be different in thefuture, say the disbelievers. So let’snow go to the extreme propulsionlimit. The most efficient process wouldbe particle-antiparticle annihilation,conversion of fuel and antifuel com-pletely to energy in the form of high-energy photons according to Einstein’sfamous E0 = mc2. If we ignore manyproblems such as a source for antipar-ticles, radiation exposure, and so on,and also assume that all the photonsare eventually directed rearward, eachkilogram of fuel could provide 3 ×1016 joules of energy and 3 × 108 kg-m/sec of linear momentum. To acceler-ate upward at about 10 m/s2, akilogram of this fuel can provide a mil-lion-kilogram spaceship with 30 sec-onds of thrust. If one requires 3,000seconds of thrust, simply use 100 kg ofmatter-antimatter fuel. We look for-ward to the future of space travel withantimatter engines, but for the presentwe can enjoy the entertainment pro-vided by space travelers in films.

73. Warp Speed

Spaceships with a warp drive to accel-erate beyond the local speed of lightcannot be ruled out just yet! Oneexample is the expansion of the uni-verse, which carries everything along,and speeds can exceed the speed oflight. Distant quasars with recessionvelocities greater than c are a reality.The analog on Earth may be to have a100-meter race on a stretchable trackthat can change length during the race.

However, if the spaceship has thetechnology to be able to distort space-time itself in its local vicinity, thenthere is no need for enormous speeds.Simply contract the space in the frontto bring distant points closer, warpingspace-time directly. That starbase thatformerly was light-years ahead of youis now close to you, reachable by nor-mal propulsion in minutes!

Krauss, L. The Physics of Star Trek. NewYork: HarperPerennial, 1996, pp. 56–58.

74. North Pole Ice Melt

There would be no change in the sealevel if all the ice at the North Polemelted. Why? Because this ice is float-ing on water. Upon melting, the watermolecules in the ice simply occupy thespace of the liquid displaced by the iceoriginally. Of course, we need to bemore careful on defining the extent ofthe North Pole. If we include ice on

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some landmass, then this ice will addextra water molecules to the liquidseas and slightly raise the sea level. Incontrast, most of the ice at the SouthPole is several kilometers thick and ispredominantly on the Antarctica land-mass, so its melting could significantlyraise the sea level. Some films depictoceans that have risen hundreds ofmeters from the worldwide ice melt.Simple estimates easily reveal that thisconjectured amount of sea levelchange is ridiculous.

Another concern might be the lin-ear expansion of water when its tem-perature is above 4°C of about 70 ×10–6/°C. Even if the water temperaturerose by 10°C throughout the first 10kilometers of ocean depth, theexpected water level rise for a columnof water would be no more than 7meters if the surface area remainedconstant. But the surface area willexpand, so the actual water level willrise about 2 meters of so. For a 1°Cincrease in temperature, the oceanlevel rise will be several centimeters.

Sea level changes played majorroles in the migrations of our humanancestors. Some Bushmen left theirhomeland and their cave dwellingsabout 40,000 years ago, seeking betterclimates and less arid lands. A mini iceage had developed, so seawaterbecame locked into the ice at the poles.Their caves, which were originally nearthe sea, were now several hundred

kilometers inland from the sea, and thearid climate made living off the landvery difficult. So they migrated alongthe eastern coast of Africa all the waythrough the Middle East and India toAustralia. The Aborigines in Australiaare descended directly from these peo-ples from Namibia and South Africaand together with the Bushmen are theoldest civilizations on Earth.

75. Lightning and

Thunder

Let’s consider identical explosions onthe battlefield at two different dis-tances from the observer (i.e., the cam-era) but seen simultaneously. Thesound intensity emanating from thefarther explosion should be less loudcompared to the closer one, and thesound of the farther explosion shouldbe delayed more with respect to itslight flash than for the closer explo-sion. The extra distance affects boththe sound intensity and its time delay.Depending on the distance and thetemperature gradient in the air, therecould be additional effects, such as dif-ferent frequencies having slightly dif-ferent speeds and/or paths en route.

For example, if the closer explo-sion is one-fifth of a mile away and thefarther one is two-fifths of a mileaway, the differences in arrival timesshould be clearly heard. The sound

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from the closer one begins to be heardone whole second after the light flashand the weaker sound from the fartherexplosion should be heard one secondlater, two seconds after the apparentlysimultaneous light flashes. Count thesetime delays aloud and you will imme-diately realize that most battlefieldscenes do a false portrayal of the audiotiming. But who really cares? The dra-matic license enhances our viewingexperience of the fantasy world of thecinema! But experienced military peo-ple know the difference.

Many more audio violations can berecalled from the movies. We simulta-neously hear and see airplane crashes inthe distance, cars falling and crashinginto a chasm far below, jet airplanespassing overhead with no sound delay,etc., all for the benefit of a moviegoingpublic who should know the difference.

76. Explosions in Outer

Space

The colors of the explosion are proba-bly okay, and perhaps one can appre-ciate the beautiful outward-goingstreamers isotropically distributed.However, very seldom does a realexplosion, even in a vacuum, distrib-ute the stuff isotropically. One alsowould expect bits and chunks of vastlydifferent sizes, with a large chunk ortwo left over near the explosion origin.

As a dramatic example, in 1987Supernova 1987A in the Large Magel-lanic Cloud, bound to our Milky Waygalaxy, blew apart, dumping practi-cally all its energy into neutrinos, butthere was still enough energy to pro-duce a lot of photons for a bright flashof visible light that was first seen byamateur astronomers in Japan. Todaythat light flash continues to expandwith decreasing intensity, and the gascloud of particles also continues tostream outward, impacting moleculesand other stuff in various directions tohelp “paint” the region in beautifulcolors. Data seem to indicate that theremains at the origin consist of twosmall, massive objects orbiting a com-mon barycenter.

The real problem with hearing anexplosion in space is that there is nomedium to carry the sound waves, sono sound from the original explosionshould be heard by the spaceship crewor by the audience in the theater. Thelight travels at nearly 3 × 108 m/s, andthe sound would transit at the snail’space of 3 × 104 m/s or slower. The lightflash comes before the sound for anysafe distance. Of course, any debrisfrom the explosion hitting the space-craft would make an impact soundcarried by the walls of the ship and bythe internal artificial atmosphere.Then there is the noise of the otherspaceship going past, but that isanother story.

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77. Space Wars

Don’t try to learn your physics fromspace battles. Practically everything isincorrect, except the ability to causesomething to explode if your weaponcan dump enough total energy into thetarget in a short time!

The laser beams, no matter howpowerful they are, would not be seenen route. Seeing them requires that areasonable amount of light be scat-tered back to you along their paths.But there is practically nothing in thevacuum of space to scatter off. A fewhydrogen atoms per cubic meter areall there is out there.

The explosion would not be heardat all. Sound requires a medium for itstransport, and the vacuum of space isnot a material medium able to conductsound waves. You will feel something,the impact of the bits of the explodingdebris from the enemy battlecruiserbecause they travel unimpeded fromthe explosion volume to your space-ship. Their speeds may be enormous,so they could do significant damage ifyour shields are not in place.

78. Security Lasers

As the director of the theft scene, youwould know that the laser light wouldnot be visible in the air in a normalroom because there is not enough

scattering of the light to your eyes bythe molecules in the air. Spots wherethe beams strike walls, mirrors, or anyobjects could be seen, but theirstraight paths to the spots are invisi-ble. Therefore you have two ways tomake the laser beams visible so youraudience can see the sequence ofmaneuvers required to succeed in theheist: (1) put something in the air itselfto scatter laser light such as chalk dust,smoke, or a fog of liquid nitrogendroplets or water mist, or (2) artifi-cially put in the light beams duringediting, after the scene is done.

We suppose that some real-worldlocations may use an arrangement ofcrisscrossed laser beams for securityaids, but we do not know of any. Cer-tainly, infrared lasers are muchcheaper than visible lasers, and theywould not be seen by injecting a foginto the space. Nor would their spotson the wall be seen.

Some heist scenes show the substi-tution of a mirror to fool the securitysystem while the protagonist steals the item. Indeed, the mirror substitu-tion may reflect the beam in the cor-rect direction, but the tenths of asecond disturbance in the originalbeam would be very easy to detect bythe security system electronics, whichcan sense disturbance spikes quicklyand sound an alert. Of course, thehuman security operator may choose

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to ignore the alert, which so often hap-pens in movies!

79. Bullet Fireworks

Normal bullets are copper-clad leadand do not spark upon impact withsteel or any other surface. Just go to agrinder to check out the properties ofcopper versus other metals in the pro-duction of visible sparks. The grindingof steel produces sparks everywhere,seen even in sunlight. The many small,hot particles of steel are actually burn-ing. Now grind a piece of copper tub-ing. No sparks. You might see anoccasional spark due to contaminationon the grinding wheel or in the copper.The copper bits ground off do reactwith oxygen, but they do not get verywarm. Please do not grind lead becausetoxic particles will be released into theair and, besides, lead is known not toproduce sparks. So the conclusion isthat practically no bullets produce abrilliant flash of light on impact.

In support of the film depictions isthe fact that the military does havemachine gun bullets containing whitephosphorus so that the point of impactcan be seen by the gunner. These bul-lets are used also to ignite fuel tanksand other possible containers of explo-sives by producing sparks to ignite thevapors. But phosphorous bullets arevery rare outside the military.

80. Internet Gaming

The Internet is usually not the culprit,being extremely fast compared to thehand-eye coordination of the player.The actual travel time between themajor switching stations along theInternet is extremely short because the data packages travel at nearly thespeed of light in a vacuum. So a datapackage can go 20,000 kilometers inabout 70 milliseconds with no delays.Delays at the switching stations alongthe Internet are typically in hundredsof milliseconds, with your local Inter-net service provider (ISP) contributingmost of the delay time, on the order of300 milliseconds or so.

In contrast, your local computerand cable modem, for example, usu-ally will be slower in responding toyour input and getting the data out toyour ISP and onto the Internet. Thefaster the equipment at your end, up toa certain speediness, the quicker yourresponse will appear in the game. Ifyour equipment delay time becomesless than the Internet delays, there’snothing further to be improved byspending more money on a faster com-puter system.

81. Cartoon Stretching

There are many ways to approach theproblem of determining the speed of

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sound in the cartoon character’s body.We consider one approach only. Startwith the application of an externalforce to the surface of the body atsome location—the character’s foot,say. We see the stretch region progressup the leg over a period of one to twoseconds.

“Wait just one minute!” youexclaim. How is the speed of sound inthe material related to its speed ofstretching in response to an appliedforce? The answer is that bothprocesses require communication fromone molecule to the next molecule out-ward, from the applied force region tothe far reaches. Usually the muchsmaller energy in the sound applicationproduces a very tiny stretch followedby a relaxation and overshoot, thenanother stretch, etc., repeatedly atsome frequency above 14 Hz or so.The stretch produced by the tug of amuch larger applied force shows amuch larger displacement of the mole-cules that may or may not relax whenthe applied force is lessened. The largerdisplacement between molecules forthe stretch process may require slightlymore time if the process cannot bemodeled by a collection of linear har-monic oscillators, but the speed ofstretch will be very close to the value ofthe speed of sound for most materials.

The stretched cartoon character’sbody exhibits a communication speedof about a meter per second, a very

slow speed of sound indeed when com-pared to most materials, which have aspeed of sound of about 300 meters persecond. So we conclude that cartooncharacters are made of very unusualmaterials. Perhaps designer materials inthe future will be able to mimic a car-toon material. According to the world’sgreatest detective, the game is afoot!

82. Infrared Images

The actual converted infrared imagewould be a blurred black-and-whiteimage, not a sharp one. The physicsplaces a limit on the resolution ofimages we see in the infrared com-pared to the visible. When we lookthrough binoculars or any lens systemin the visible, these systems have a res-olution limit that depends on the qual-ity of the optic elements. No matterhow much the image is enlarged orenhanced by dithering, etc., the origi-nal resolution is not improved eventhough the image looks cleaner. Butthe physics is even more restrictivewhen comparing an infrared image toa visible image because the visible lighthas been coherently scattered from theobject, whereas the infrared light hasnot, as explained below. Of course,one cannot exceed the Rayleigh crite-rion for resolution of approximatelyone wavelength of the light, except byusing interference techniques, whichwe do not consider here.

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In the visible part of the electro-magnetic spectrum, atoms absorb andemit the photons of light in a two-stepprocess, usually absorbing and emit-ting in about 10–16 second. During thistime interval, the molecule holdingthat atom moves very little. Nearbyatoms also scattering this impingingbeam of numerous photons tend toremain in place during their scatter-ings. In effect, during the scattering ofeach photon there always will be afixed phase relationship among all theatoms scattering light from the objectto your light sensors. Hence, photonsscattering from different areas on theobject’s surface carry detailed phaseinformation with fixed phases toachieve nearly maximum resolution. Ifthe phases actually varied randomlyfrom one location to another, the visi-ble image would become blurred.

In the infrared, the image is blurredbecause most of the infrared isabsorbed and emitted by molecularvibrations and rotations that have ran-dom phases over the object’s surface.This scattering process takes muchlonger—about 10–12 second—sufficienttime for the molecule to move consid-erably during the scattering, so therewill not be a fixed phase relationshipamong neighboring molecules on thesurface of the object. The same surfacethat appeared well resolved in the visi-ble will now be quite blurred in theinfrared. No magical digital techniques

will be able to take an infrared imageand make a sharpened black-and-white image true to the original object.

In the ultraviolet, the scatteringtime is very short, but so is the wave-length, so the extent of the coherentscattering area also is very short; thusthe image is blurred compared to thevisible one. Nature has given us avision range in the visible that ensuresthe best resolution of detail.

83. Light Sabers

Yes, the light sabers would passthrough one another as if nothing werethere! The photons of the light arebosons, which do not repel each other.The clashing of the light sabers is pureartifact and far beyond artistic license.It’s an outright lie to the public!

There are two general categories ofparticles in the universe: fermions andbosons. Two identical fermions (e.g.,think electrons, or any other funda-mental spin 1/2 particle) cannot existin the same quantum state defined byits 4-momentum and spin. The exis-tence of all matter, including ushumans, critically depends on theinability of two identical fermions toget together in the same state, so mat-ter occupies a volume. That is, objectscan be bigger than a point!

In the case of bosons, one can notonly put as many identical bosons(think photons or any integer spin

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fundamental particles) into the samequantum state, they also prefer to bein one, with the probability to do soenhanced by the number N of bosonsalready in the state. There is no repul-sion experienced. So two light sabersintersecting at an angle would passright through one another with nochange in either. If the light beamswere powerful enough, though, theycould cut material objects made out offermions—that is, the ordinary stuffall around us—whenever this materialstuff intercepts the beams because suf-ficient energy could be absorbed tochange the physical state of the mate-rial and result in vaporization.

84. Force Fields

We don’t know why we can see thegood guys through the battlefield forcefield via visible light at the same timethat the visible laser beams cannot getthrough! This result might be due tosome dramatic nonlinear natural effectnot yet experienced in research labs, orthe phenomenon is a pure artisticfalsehood. We would bet on the latter.Obviously, the light transmissionproperties of a transparent materialcan change dramatically on absorbingenergy, but the usual effect is hole-burning, not reflection.

Then there is the problem of whythe laser beams can be seen along theirpaths, unless the dust in the air above

the battlefield scatters enough of thelight. But when the laser beam origi-nates in space where there is no air (orother scatterers in significant density),one can still see the laser beams. Isthere no correct laser physics in thesemovies? Is there any correct physics at all?

85. Cold Silence of

Space

Yes and no. The very good vacuumbetween Earth and Venus, for exam-ple, certainly does not conduct sound,so the space is silent. But what is thetemperature of outer space? This ques-tion is an improper question, for wemust instead ask: “What temperaturewould be recorded by a thermometerplaced in space between the Venus andEarth orbital distances?” Whateverkind of thermometer we use, if the sen-sor is not rotated, one side will beexposed always to the direct rays fromthe Sun and the other side will be inshadow. At thermal equilibrium, theamounts of outgoing radiation energyand incoming radiation energy in alldirections will balance.

The equilibrium temperature Tfor an object at Earth’s average dis-tance from the Sun is about 280 K, or+7°C, the actual value being lowerbecause there will be some energyreflected away and not absorbed by

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the thermometer. One calculates Tfrom this equation: flux absorbed =flux emitted. If we assume for simplic-ity that the sensor is a sphere of radiusR, then the equation becomes S (1 – A)πR2 = σT4 (4πR2), where S = 1.4 kWs–1 is the solar constant at Earth dis-tance and σ = 5.67 × 10–8 W m–2 K–4.The parameter A is the reflectivity,which we take as zero in the ideal caseof a perfect absorber and radiator.Real materials will have an A valuebetween 0 and 1.

If you are orbiting at the Earth dis-tance from the Sun, you may desire torotate your spacecraft slowly in spaceso that all sides cook evenly! This pas-sive heating can be augmented byactive heating from within to maintaina cozy environment.

If you are orbiting closer to theSun than Earth, the equilibrium tem-perature will be higher, the solar fluxincreasing as the ratio of the squareddistances. Near the orbit of Mercuryyou may be too hot! If you are fartheraway, the temperature decreases, soyou may need artificial heating. Somerotation may produce a system thatrequires less fuel for heating, but thedetails need to be worked out.

86. Nuclear Submarine

Submariners love to dive in their sub-marines. A dive to several hundredmeters under the surface may be

effective in limiting the initial spreadof the debris from the thermal explo-sion. No nuclear explosion wouldoccur, or else everything around wouldbe vaporized by the energy released, inwhich case the depth in the waterwould help very little in preventing thespread of energy in many forms. Thethermal explosion in a nuclear reactorin the sub releases the fuel andcoolant, so radioactive particles anddebris will be sent out in all directions.Some of this stuff would be slowedeffectively by the water, and someprobably would make the surface andescape into the air.

When the Chernobyl nuclear reac-tor broke its containment vessel in the1980s, its nuclear particles weredetected around the world withinhours to days. At the University ofCalifornia at Irvine, the air filteringsystem at the local nuclear reactorrecorded the radioactive cesium andiodine particles in parts per billionfrom the Chernobyl incident 10 daysafter the chemical explosion.

87. Plutonium vs.

Uranium

A plutonium bomb would be muchsafer to handle. Weapons-grade pluto-nium (Pu-239) emits primarily alphaparticles and low-energy gamma rays,both being easy to shield. The trace

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amounts of even-numbered Pu iso-topes have spontaneous fission reac-tions that emit neutrons. However,neutron detectors would need to bewithin a few meters to detect theseneutrons above background. So a lostplutonium bomb could be very diffi-cult to find.

The plutonium usually is coatedwith beryllium or another appropriatesealant because the exposed elementwill react chemically with the oxygenin the air or in water and increase itstemperature considerably. The personholding a plutonium bomb wouldprobably feel that the protective casingis warm, because the alpha particlesdeposit their kinetic energy in the cas-ing material.

Of course, inhaled or ingested Pu isone of the worst carcinogens known.Any explosion releasing Pu into the aircreates a hazard for all life that wouldremain for a long time.

In contrast, weapons-grade U-235releases gamma rays at several ener-gies, the most intense at 186 KeV. Sothe detection of a device containing U-235 is much easier than trying to findplutonium. Some films portray the dif-ferences correctly, while other filmsdramatize any nuclear device and itspossible dangers with remarkably ado-lescent scare techniques.

88. Nuclear

Detonation

We know of no nuclear explosivedevice that does not require at leastvery good spherical symmetry to bedetonated. The simpler atomic deviceshave either two hemispheres that mustbe rapidly moved together into asphere, or two spherical sections heldapart until a slab of nuclear material isshot into the gap to start the fissionreaction. The more complicated hydro-gen devices require a strong, spheri-cally symmetrical implosion from theperimeter shell to initiate the fusionreaction.

Dropping the weapon from anyheight would damage the casing asym-metrically. Even shooting pellets orbullets, etc., through the casing intothe warhead would create an asym-metrical result but no explosion. Initi-ating the nuclear reaction and keepingthe reaction going are not easy. Theprocess is certainly not worth worry-ing about to the extent portrayed inthe movies. One should be more con-cerned about whether the propersafety precautions are being practicedagainst accidentally dropping the thingon one’s toes.

By the way, the smallest practicalnuclear weapon tips the scales at onlyabout 9 kilograms, about 20 pounds,and is small enough to fit into a bulky

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attaché case. For the physics detailsand an estimate of the smallest device,see the reference below.

Bernstein, J. “Heisenberg and the CriticalMass.” American Journal of Physics 70(2002): 911–916.

89. Fabric of

Space-time

Space-time is not a piece of cloth, nordoes space-time behave like a real mate-rial. The metaphor “fabric of space-time” allows one to visually modelspace-time with its coordinates of spaceand time in a way similar to the simplertwo-dimensional construction of wovencloth. There is no way for space-time torip or tear, although mathematicallythere can be singularities of variousdimensions and other mathematicalproperties that some people havestretched into real physical propertiesin the name of dramatic license.

As far as being able to go back intime as a time traveler, the time dimen-sion of space-time is not like the spatialdimensions of space-time. Mathemati-cally, the passage of time operator inquantum mechanics is antiunitary,while the spatial displacement operatoris unitary. In addition, no one hasshown convincingly that a fundamen-tal particle can progress either forwardor backward in time differently thanwe now experience and understand—

that is, time passes at its normal ratefor all particles and collections of par-ticles. One possible explanation for allparticles behaving the same withregard to the passage of time relies onthe direction of time being built intothe quantum state definition of a fun-damental particle, with the oppositedirection of time for its antiparticle.

The phrase “speed of gravity” ismeaningless in the simplest interpreta-tion, being an error in stating theacceleration of gravity. Most filmsconfuse the concepts of speed andacceleration in the same manner thatmost people do. Unfortunately forsociety, the scriptwriters did not learntheir beginning physics, and we allcontinue to suffer from their intuitivemistakes in describing the behavior ofnature. Or perhaps a more positiveview should be taken in that the enter-tainment value is improved by ignor-ing the laws of physics!

Chapter 5Go Ask Alice

90. Spotlight

Yes, a spot of light from the lighthousebeacon when moving across your fieldof vision can move faster than c. Butthe actual light itself (i.e., the photons

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in the beam) moves from the source to the reflecting spot in the sky at c, no faster.

A good example from astrophysicsis the radio wave beam from the pulsarin the Crab Nebula, which sweepsacross our Earth observatory thirtytimes a second from a distance of afew thousand light-years. The electro-magnetic waves coming to us from thedistant source travel at light speed, butthe sweep across our planet movesfaster than c.

Occasionally one encounters othersuggested examples of spots travelingfaster than light speed, such as theintersection edge in a very long pair ofscissors progressing outward whenclosing. Unfortunately, this intersec-tion edge’s speed is limited by thespeed of sound in the metal of the scis-sors, which is quite slow compared toc. However, the spot of light on anoscilloscope trace can move across ascreen faster than c even though thisspot is produced by the slower-movingelectrons striking a phosphor.

Bergmann, P. G. “Can a Spot of Light MoveFaster than c?” The Physics Teacher 19(1981): 127.

Rothman, M. A. “Things That Go Fasterthan Light.” Scientific American 203, no. 1(1960): 142–152.

———. “Not So Fast.” Scientific American269, no. 6 (1993): 10.

91. Quasar Velocity

The special theory of relativity saysthat information cannot be transmit-ted faster than light. The photonsalways travel at light speed in the localreference frame, but the space in whichthe photons travel may be expanding.An analogy would be a 100-meter racewith the track lengthening during therace. The elapsed time to reach the100-meter tape depends on the modelfor the expansion rate for the runnerand for the photons from that distantquasar. Under these expansion condi-tions, there can be recessional veloci-ties greater than c!

Chown, M. “All You Ever Wanted to Knowabout the Big Bang.” New Scientist (April17, 1993): 32–33.

Peebles, P. J. E., D. N. Schramm, E. L.Turner, and R. G. Kron. “The Evolution ofthe Universe.” Scientific American 271, no.4 (1994): 52–57.

———. “Out with the Bang.” ScientificAmerican 272, no. 3 (1995): 10.

Stuckey, W. M. “Can Galaxies Exist withinOur Particle Horizon with Hubble Reces-sional Velocities Greater than c?” Ameri-can Journal of Physics 60 (1992): 142–146.

92. Spaceship

Approach

The observer sees the highly relativisticobject approaching back side first!Therefore, the spaceship seems to beapproaching tail first! What is oftenreferred to as a contraction in the

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direction of motion for a rela-tivistically approaching object isactually a rotation known as theTerrell effect.

We need to discuss some aspects of the Terrell effect to explain thebehavior of the approaching space-ship. Consider a solid, opaque cubeapproaching. At low speeds, the lightrays emitted off the back side of theapproaching cube cannot pass throughthe box to reach the observer. Athigher speeds nearing the speed oflight, however, enough of the boxmoves out of the way for light emittedfrom part of the back side to reach theobserver. When this behavior happens,the observer will not see all of thefront side because some of the lightrays from the front are intercepted bythe extremely fast moving box. Thebox appears rotated, with the awayside of the front hidden, and the nearside of the back visible. The rotationangle increases with increasing speed,nearing c and with proximity to thetrajectory. Additional complicationsalso occur, such as nonrigidity, whichwe ignore in this simple explanation.

So the spaceship approaching atnear light speed will appear rotated sothat the back end is almost totally visible and the front end is almosttotally hidden from view. J. Terrell in1959 was the first to recognize thatwhat physicists had been calculating as a Lorentz-Fitzgerald contraction is

actually a rotation for a real three-dimensional object. What we havedescribed above is a snapshot of thespaceship (and cube)—that is, whatphotons from different parts of theobject would imprint on a camera sensor simultaneously. E. Sheldon (seethe reference below) discusses thestereoscopic appearance of a three-dimensional object that involves shear-ing and other distortions in addition torotation, all these effects first discussedby J. Terrell.

Sheldon, E. “The Twists and Turns of theTerrell Effect.” American Journal of Physics56 (1988): 199–200.

Terrell, J. “Invisibility of the Lorentz Con-traction.” Bulletin of the American PhysicalSociety 5 (1960): 272.

Weisskopf, V. F. “The Visual Appearance ofRapidly Moving Objects.” Physics Today13 (1960): 24–27.

93. Mass and Energy

The answer to both questions is equa-tion 1, although the majority of physi-cists seem to prefer equations 2 or 3!Their choices probably are caused bythe confusing terminology widely used

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in the physics literature that says that a body at rest has a “proper mass” or “rest mass” m0, and a body inmotion has a “relativistic mass” m =m0/ √(1 – v2/c2) .

There is only one mass in physics,m, which does not depend on the ref-erence frame. This mass m is the rela-tivistic invariant quantity in E2 – p2c2

= m2c4, whereas the energy is differentin different reference systems. There isno need to place the index 0 with themass. However, the total energy Eneeds the 0 index if the particle has nomomentum in that reference frame—that is, E0 = mc2.

For a complete and stimulatingdiscussion of these ideas and theirhistory see the L. V. Okun referencebelow.

Einstein, A. “Zur Elektrodynamik bewegterKörper.” Annalen der Physik 17 (1905): 891.

———. “Ist die Trägheit eines Körpers vonseinem Energieinhalt abhängig?” Annalender Physik 18 (1905): 639. Translated byW. Perrett and G. B. Jeffery in The Principleof Relativity. New York: Dover, 1923, p.71.

———. “Zur Theorie der BrownschenBewegung.” Annalen der Physik 19 (1906):371.

Okun, L. V. “The Concept of Mass.” PhysicsToday 42, no. 6 (1989): 31–36.

94. Strain Gauge

The strain gauge continues to show azero value. What I interpret as a lengthcontraction when I run past is really

the measurement of the length compo-nent along my direction of motion ofthe metal bar that appears to berotated. The atoms do not move closerto one another, so the strain gaugeremains at zero.

The apparent rotation is called theTerrell effect: if a snapshot is taken ofa moving object, the object does notappear contracted, but rather rotated.A snapshot is understood to be a two-dimensional, nonstereoscopic photo-graph. The stereoscopic appearance ofa three-dimensional object is morecomplicated because shearing andother distortions can be present. Infact, there is no such thing as a rigidobject in relativity!

DeCampli, W. M. “A Gedanken Experimentto Demonstrate Lorentz Contraction.” ThePhysics Teacher 13 (1975): 420–422.

Terrell, J. “Invisibility of a Lorentz Contrac-tion.” Bulletin of the American PhysicalSociety 4 (1959): 294.

———. “Invisibility of a Lorentz Contrac-tion.” Physical Review 116 (1959):1041–1045.

———. “The Terrell Effect.” AmericanJournal of Physics 57 (1989): 9–10.

95. Mass/Energy

Mass is energy. There is no distinctionto be made. There is no conversion! In1905, Einstein states explicitly: “Themass of a body is a measure of itsenergy content . . . ”. What Einsteinwas stating is that mass and energy are

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equivalent, that they are possibly twodifferent aspects of the same physicalquantity, only their units have beenchosen to be different. One does notconvert one to the other if they areequivalent.

We can imagine a conversationwith a student: “Does a photon havemass?” asks the student. “Yes, becausethe photon has energy.” The studentcounters, “But for a photon E = pc, sothe relation E2 – p2c2 = m2c4 becomesE2 – p2c2 = 0. Therefore m = 0 for the photon.” Can you complete thisdialogue?

Einstein, A. “Ist die Trägheit eines Körpersvon seinem Energieinhalt abhängig?”Annalen der Physik 18 (1905): 639. Trans-lated by W. Perrett and G. B. Jeffery, in ThePrinciple of Relativity. New York: Dover,1923, p. 71.

———. Relativity: The Special and GeneralTheory. New York: Crown, 1961, p. 47.

96. System of

Particles

No and yes! Except in the special cir-cumstance described below, the answeris no. Energy and momentum are addi-tive, but not mass. Mass is a measureof the magnitude of the energy-momentum 4-vector. From the totalenergy E and the total momentum Pcan be determined the mass M of thesystem: M2c4 = E2 – P2c2. Therefore themass M of the system is greater thanthe sum of the masses of its particles by

the amount equal to the total kineticenergy of all the particles as seen in theframe in which the total momentum iszero. The exception “yes” occurs whenall the particles move in the same direc-tion with the same speed—that is, havethe same velocity.

The value of defining the mass inthis relativistic fashion means that Mdetermines the system’s inertia, itsresistance to acceleration by a forcethat acts on the system as a whole. Abox with a hot gas of particles hasmore mass than the same box after thegas has cooled. Also, the box of hotgas exerts a greater gravitational pullon a test particle. In addition, a box ofphotons exerts a gravitational pull ona test particle, and vice versa.

Taylor, E. F., and J. A. Wheeler. SpacetimePhysics: Introduction to Special Relativity.San Francisco: W. H. Freeman, 1992, p.135.

97. Light Propagation

According to the special theory of rel-ativity (STR), (1) no object can moveat the speed of light, (2) the speed oflight is the same for all observers, and(3) the space-time interval τ betweentwo events defined by τ2 = c2 ∆t2 – ∆x2

– ∆y2 – ∆z2 is the same for allobservers, but the ∆t and ∆x may bedifferent, for example.

For one-dimensional motion τ2 =c2 ∆t2 – ∆x2. The driver has ∆x ≠ 0, so

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her ∆t must be greater for the twoevents than the elapsed time for theobserver on the ground. Therefore thedriver measures the longer time inter-val between events A and B.

Suppose the car makes a secondrun at a great speed. The nearer to thespeed of light the car goes, the smalleris ∆t for the ground observer, and τ = c ∆t is smaller in this case also. Butagain, as expected, the time interval islonger for the driver. The nearer to thespeed of light the car goes in theground frame, the difference will bethe difference in arrival times asobserved in the two frames.

Taylor, E. F. “Light Propagation.” ThePhysics Teacher 25 (1987): 252.

98. Sagnac Effect

No, they do not tick at the same rates.Their tick rates are different becauseEarth is rotating with respect to aninertial reference frame such as the dis-tant stars. The clock moving eastwardhas a higher velocity with respect tothe inertial frame than the clock mov-ing westward at all moments. Accord-ing to the STR, the higher the velocity,the slower the clock ticks. That is, aclock ticks fastest when at rest in anSTR inertial reference frame.

The difference in the elapsed timefor the two clocks can be calculated byconsidering a light clock following acircular light path around the Equator.

One also could use a regular n-gon offlat mirrors to reflect the light aroundthe Equator and then take the limit asn becomes infinite. The light leavesfrom point P on the Equator of therotating Earth and returns to point Pin time T. The light going eastward hastraveled the distance 2πR + ωRT in theinertial system, where ω is the angularfrequency of rotation with respect tothe inertial reference frame. The pointP has traveled ωRT. The ratio of pointspeed to light speed is ωR/c =ωRT/(2πR + ωRT), from which T =2πR/(c – ωR). For the system at rest, T = 2πR/c. Hence, when ω ≠ 0, defineδT = T – 2πR/c as the extra timerequired. Substitution for T gives δT =2πωR2/[c (c – ωR)]. Upon returning topoint P on the Equator after one cir-cuit, the clocks will differ by 2δT forthe measured elapsed times.

Schlegel, R. “Comments on the Hafele-Keating Experiment.” American Journal ofPhysics 42 (1974): 183–187.

99. Light Flashes

The observer on planet A sees theflashes 20 minutes apart. From theSTR postulate, we know that noobservations of the light flashes onlycan discern which inertial frame is atrest. If the flashes sent out by thespaceship at 10-minute intervals areseen at planet B separated by 5 min-utes, then flashes sent out from B at

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10-minute intervals will be seen on thespaceship at 5-minute intervals.

One also realizes that if there were alight flash every 5 minutes from planetA, the observer on planet B would seethem at 5-minute intervals. What is theinterval for these flashes from A as seenby the spaceship observer? The answeris every 10 minutes, by invoking theSTR postulate above. So the spaceshipsees planet A’s flashes to be spaced twiceas much apart as the interval at thesource on A; likewise, the 10-minuteflash intervals from the spaceship mustbe twice the interval at planet A, or 20minutes apart.

Hewitt, P. G. Conceptual Physics, 6th ed.Glenview, Ill.: Scott, Foresman, 1989, pp.650–656.

100. Forces and

Accelerations

No. In the STR, all contact forces willproduce an acceleration in a directionnot parallel to the applied force! Forexample, a rigid sphere is movingalong the plus x-direction of an iner-tial reference frame. Now let anapplied contact force act in the plus y-direction to increase the speed of the sphere in the y-direction. Whathappens to the speed in the x-direc-tion? The x-component of the speeddecreases—the object slows down inits original direction, corresponding toa negative acceleration!

To understand why the objectslows in the x-direction when theapplied contact force is in the y-direc-tion, we begin with the space-timeinterval: (interval)2 = c2 ∆t2 – ∆x2 – ∆y2

– ∆z2. For real objects traveling atspeeds less than c, the time term ismuch larger than the spatial terms,and the interval is called the propertime τ. The linear momentum px in thex-direction in Newtonian physics isdefined as px = m dx/dt (for an objectthat is not changing its mass, i.e.,excluding objects such as a leakingbucket of water). The correct STRexpression simply substitutes propertime τ for Newtonian time t so that px

= m dx/dτ. For a low-velocity object,dτ ~ dt. But the actual relationshipbetween τ and t depends on the mag-nitude of the object’s total velocity, avector quantity, not just the speedcomponent in the x-direction. There-fore, as the object speeds up in the y-direction, its speed in the x-directionmust decrease to maintain a constanttotal velocity magnitude, otherwise itsx-component of linear momentumwould change, forbidden by the law ofconservation of linear momentum.The pertinent relationship is dt/dτ =1/ √(1 – v2/c2) . Remember, the mass isfixed in value.

One could write down the relativis-tic momentum px = mvx// √(1 – v2/c2)and argue that since m is constant, thecomponent of velocity in the original

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direction must decrease to keep themomentum component constant.

Ficken, G. W. Jr. “A Relativity Paradox: TheNegative Acceleration Component.” Amer-ican Journal of Physics 44 (1976):1136–1137.

González-Díaz, P. F. “Relativistic NegativeAcceleration Components.” American Jour-nal of Physics 46 (1978): 932–934.

Tolman, R. C. Philosophical Magazine 22(1911): 458.

101. Uniform

Acceleration

In the STR, the velocity in the labframe is no longer V = a ′t for a uni-form acceleration a ′ in the movingframe. However, in the moving frameat each instant the expression V ′ = a ′t ′continues to be true. To convert fromthe moving frame to the lab frame, wemust essentially convert the clockreadings and time interval using dt/dτ= 1/ √(1 – v2/c2) . Here, τ is the propertime—that is, the clock reading on awristwatch worn by an observer onthe spaceship, say, and dτ is the propertime interval between two events atthe same location. In the example, τ isthe elapsed time on the wristwatch ofthe person on the moving frame.Hence, on the moving spaceshipframe, V ′ = a′τ.

Before we determine the answer for the velocity of the object in the

lab frame, let’s review the simpler prob-lem of how velocities are added in rela-tivistic frames. If an object movesforward with the velocity V ′ in thespaceship frame, then the object’s veloc-ity V in the lab frame is determined bythe law of addition of velocities V/c =(V ′/c + Vs/c)/(1 + V ′Vs/c

2), where Vs isthe uniform velocity of the spaceship inthe lab frame. One can check the limit-ing case for low velocities, whenV ′Vs/c

2 is very small, to verify agree-ment with Galilean relativity—that is,the two velocities simply add.

To relate the acceleration of theobject as seen by both observers, theaddition of velocities expression isdifferentiated with respect to the timein the lab reference frame to obtain a = a′ /{(1 + V′Vs/c

2) √(1 –Vs2/c2))3 }, a

messy expression. The velocity of theaccelerating object in the lab frame isfound by substituting V ′ = a′ τ. There-fore a ≠ a′ and V < c.

An alternative mathematical tech-nique using a velocity parameterdefined in terms of hyperbolic func-tions is given in the Taylor andWheeler reference below.

Taylor, E. F., and J. A. Wheeler. SpacetimePhysics: Introduction to Special Relativity.San Francisco: W. H. Freeman, 1966, pp.47–58.

Tipler, P. A. Modern Physics. New York:Worth, 1978, p. 27.

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102. Long Space

Journey

The 7,000 light-year journey with 40-year aging is possible in STR physicsbut not in Newtonian physics!

Define V/c = tanh θ, where tanh isthe hyperbolic tangent. Substitute intothe law of addition of velocities toobtain tanh θ = (tanh θ′ + tanh θs)/(1 +tanh θ′ tanh θs). Some checking of themathematics of hyperbolic functionswill reveal that the θs are additive, justas velocities are additive in Newtonianphysics with Galilean relativity. Thatis, θ = θ′ + θs. Some people call θ thevelocity parameter.

Back to the problem at hand: Howmuch velocity V in the lab frame doesthe accelerating spaceship have after agiven time? We need three frames ofreference: the lab frame, the spaceshipframe, and an instantaneously comov-ing inertial frame that for an instanthas the same velocity as the spaceship.With respect to the instantaneouslycomoving frame, the velocity parame-ter changes from 0 to dθ in wristwatch

time dτ. In the same astronaut time thevelocity parameter of the spaceshipwith respect to the lab frame changesfrom θ to θ + dθ. But dθ = a dτ/c. Thatis, each time interval dτ on the astro-naut’s wristwatch is accompanied byan additional increase dθ = a dτ/c inthe velocity parameter of the space-ship. Since the spaceship starts fromrest, we get θ = aτ/c, telling us thevelocity parameter θ of the spaceshipin the lab frame at any time τ in theastronaut’s frame.

Our solution is V = c tanh (aτ/c).There is no limit to the product aτ,which can be much greater than c, buttanh ≤ 1, so the lab velocity V onlyapproaches c after a long wristwatchelapsed time. The distance traveled inthe lab frame is dx = tanh (aτ/c) cdt. Inthe lab frame, the astronaut’s wrist-watch seems to be ticking slower thanthe lab clock, so dt = cosh θ dτ, with θ= aτ/c. Therefore dx = c sinh (aτ/c) dτ,which can be integrated from zeroastronaut time to the final time T toproduce the distance traveled x = (cosh(aT/c) – 1) c2/a.

The journey would be done byaccelerating to the halfway distance at3,500 light-years, then decelerating tothe 7,000-light-year distance. Substi-tuting 3,500 light-years in units ofmeters, g as 9.8 m s–2, and the speed oflight, one calculates a journey dura-tion for the space traveler’s wristwatch

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co-moving frame co-moving frame

Astronaut time τ Astronaut time τ + dτ

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of T ~ 8.62 years. The round-tripwould require about 34.5 years. So the space travelers would age less than40 years!

Are there any plans to make thisjourney? Assuming human volunteersare available who want to achieve thisfeat, other factors, such as a reliablefood supply, sufficient health care, andan energy source for the constant 1-gacceleration for 40 years would be dif-ficult to provide with present technol-ogy. And, of course, more than 14,000years would have passed for civiliza-tion here on Earth. Who or whatwould be here to greet them on theirreturn?

Taylor, E. F., and J. A. Wheeler. SpacetimePhysics: Introduction to Special Relativity.San Francisco: W. H. Freeman, 1966, pp.47–58.

103. Head to Toe

Yes, your feet and toes age slower thanyour head. That is, whenever you arestanding or sitting, a clock at the alti-tude of your head will tick faster thanan identical clock at the altitude ofyour toes. The ambient gravitationalfield affects the tick rate of all clocks inthe same way. A clock will tick fastestat rest in an inertial reference frame.The difference between clock rates indifferent gravitational environments isnormally minuscule but measurable

and, to a first approximation, the timeinterval between ticks differs by (δr/r)GM/rc2 ∆T, where δr is the altitude dif-ference, M is Earth’s mass, r is theradial distance from the center ofEarth, G is the gravitational constant, cis the speed of light, and ∆T is the timeinterval between ticks on the referenceclock. Substituting r = 6.37 × 106 mand dr ~ 1.5 m produces a value of 1.6× 10–16 ∆T, an incredibly small changein rate. Over a lifetime of about 80years, the head becomes about 0.4microsecond older than the toes.

To understand the effect of gravi-tation on the clock rate, we can utilizethe equivalence between an accelerat-ing rocket frame and being in a uni-form gravitational field. Consider twolight flashes sent from the bottom ofthe accelerating rocket to its top, asshown in the animation diagram fromthe view of our inertial reference framewith respect to the stars. The two lightflashes are one second apart in ourframe but arrive at the top of therocket three seconds apart. Why?Because the top has moved away from the approaching light flash withthe appropriate acceleration value.Therefore the frequency of arrival islower than the starting frequency. In astroke of genius, Einstein realized thatthe only reason for different flash fre-quencies would be if the clock at thetop ticked at a different rate than the

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identical clock at the bottom. There-fore, gravitation makes time run slow.

Is there a place where one can puta clock so that the time intervalbetween ticks becomes infinite? Yes,near a black hole, at the event horizon.

Hewitt, P. G. Conceptual Physics, 6th ed.Glenview, Ill.: Scott, Foresman, 1989, pp.671–678.

Taylor, E. F., and J. A. Wheeler. SpacetimePhysics: Introduction to Special Relativity.San Francisco: W. H. Freeman, 1966, p.154.

104. Neutrino Mass

For a change in a system to occur—such as the change of a muon neutrinoto an electron neutrino, for example—time must elapse. That is, the referenceclock must tick in the rest frame of themuon neutrino. We know that thegreater the velocity of a real clock inour laboratory reference system, theslower is its ticking rate. In the speedlimit of a massless particle such as a

photon traveling at light speed, theclock would not tick. As a photontraverses the universe, no time elapsesin its reference system. The photon canbe absorbed by an atom and disap-pear, but the photon cannot changedirectly into another photon. Like-wise, if all three neutrino types did nothave any mass, none could oscillateinto another neutrino type becausethey do not experience the passage oftime. Therefore, for neutrino oscilla-tions to occur, at least two neutrinotypes must have mass. The data indi-cate that the sum of the three neutrinomasses cannot exceed about 1 eV/c2,very much smaller than the 0.511MeV/c2 mass of an electron.

105. Spaceship

Collision

The method of determining positionand clock reading for the three eventsfirst before answering the question is a good one. However, the valuesinserted already are not all correct forthe observer. Simultaneous measure-ments at both the origin X1 = 0 and atX2 = L cannot be made by the methodassumed since they are not equidis-tant. Therefore, if the notation (X, T)is correctly (0, 0) for event 1, thenevent 2 is labeled by (L, –L/c) becausethe light from event 2 takes L/c sec-onds to travel the distance L to the

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observer. Event 3 is not at position L/2between the two spaceships at T = 0because spaceship B has already trav-eled for L/c seconds. Therefore thedistance between the two spaceships isL – VL/c. Thus T3 = L(1 – V/c)/2V. Wecan summarize the events as:

Event 1: X1 = 0 T1 = 0

Event 2: X2 = L T2 = – L/c

Event 3: X3 = L(1 – V/c)/2 T3 = L(1 – V/c)/2V

These same events can be specifiedin the inertial frame (primed) of space-ship A as:

Event 1′: X1′ = 0 T1′ = 0

Event 2′: X2′ = γL(1 + V/c) T2′ = – γL(1 + V/c)/c

Event 3′: X3′ = 0 T3′ = γ–1 L(1 – V/c)/2V

We have defined γ = √√√(1 – V 2/c2

and have used the normal Lorentztransformations x′ = γ (x – Vt) and t′ =γ (t – Vx/c2) of the STR.

Now, finally, we can determine theclock reading—that is, the elapsedtime—for the observer who sees thecollision a distance L(1 – V/c)/2 awayas T = L(1 – V/c)/2V + L(1 – V/c)/2,which reduces to T = L(1 – V2/c2)/2V.The observer on spaceship A has anelapsed time of γ –1 L(1 – V/c)/2V.

Chai, A.-T. “Some Pitfalls in Special Relativ-ity.” American Journal of Physics 41(1973): 192–195.

106. Twin Paradox

Peter experiences actual accelerationsduring his spaceship journey that will

result in less aging than for his twinbrother, who has remained at home onEarth. Even if the acceleration wassimply an immediate turnaround atthe farthest distance, the spaceshipvelocity vector reversed direction from+V to –V, a change of 2V, in a timeinterval T. Peter felt the acceleration.Therefore, all observers will agree thatPeter was the traveler and that hisclocks ran slow, so he ages less thanhis stay-at-home twin.

Feynman, R. P., R. B. Leighton, and M.Sands. The Feynman Lectures on Physics.Vol. I. Reading, Mass.: Addison-Wesley,1966, pp. 16-3 to 16-4.

Chapter 6Start Me Up

107. Air-Driven

Automobile Engine

Yes. Many companies worldwide havebeen operating compressed-air-drivencars using a standard gasoline four-cylinder engine but replacing the gaso-line fuel input with compressed airfrom a tank. Of course, there is nocombustion, so the electrical supplyfor the spark plugs is not needed, norwill there be any need to change the oilvery often. The compressed-air tank isstored in the trunk.

The piston upstroke compresses

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and heats the atmospheric air in thecylinder chamber until just about topdead center, when cool compressed airis injected to drive the piston downand turn the crankshaft. The processrepeats itself until the compressed airis depleted. The exhaust is just cool air.The horsepower rating is about 35horsepower for some models, but thevalue will increase to more practicalvalues with further development.Using traditional electricity sources tocompress the air, there will be somecarbon dioxide air pollution for theoverall process, but only about a fifthor less that of conventional autos.

Perhaps the best-known air-pow-ered car is that designed by Frenchinventor and engineer Guy Nègre forMotor Development International(MDI) in France. The car has a maxi-mum speed of about 110 kilometersper hour and can travel about 300kilometers at a cost of less than a centper kilometer. (Details can be found onthe Internet.)

108. Coin Tosses

You should be able to pick out theexperimentally obtained sequenceswith about 98 percent accuracy! In arandom sequence of 256 fair cointosses, you would expect to find at least1 run of 6 heads or 6 tails with a prob-ability of 98.2 percent. If the sequencesimagined by students unfamiliar with

the characteristics of randomness donot contain long runs, you should beable to distinguish them reliably.

The actual estimate of the numberof runs with 6 or more heads or tails is4, meaning that you should be able tofind about 4 of these long runs. For arun of at least k heads in n tosses,where k ≥ 1, the mean number of runsis ~ n/2(k+1); thus 2 (256/27) = 4. Thefollowing table contains actual datafor 256 coin tosses, with a 1 repre-senting heads. You can count the num-bers of the different run lengths.

10111110101101101011010010001100

11011110100001001001110010100100

11001100111110001000001011111000

10110010001111100110111001110010

11111000011011100000001011111000

11110110110000001010000010111110

11111100111011001011100010111110

01110110111100001111111000001100

Silverman, M. P. A Universe of Atoms, anAtom in the Universe. New York: Springer-Verlag, 2002, pp. 284–291.

109. More Coin Tosses

Most people would expect to return tothe lamppost quite often—20 or moretimes during the 1,000 tosses. How-ever, returning more than 2 times isunlikely! There will be a long driftaway from the lamppost for most ofthe coin-tossing time.

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One can do the actual coin tossingto experience the drift away from theorigin for long time periods, or onecan run a computer simulation. Theexpected distance after N tosses willbe √N times the unit step distance, the random walk distance in onedimension.

110. Brownian Motor

As long as the ratchet potential is off,there can be no net movement to theright or the left because the particleswill move diffusively according to a(biased) random walk, leading to avariance in position of δx = √(2Dτ) anda mean position of <x> = fτ/γ, whereD = kT/γ is the diffusion constant.When the ratchet potential is switchedon, one or more particles get trappedin one of the potential minima. If αL ≥δx ≥ (1 – α)L for the variance holds,the particle on average gets trappedinto the minimum left of the startingpoint. The maximum flux is obtained

if the switching time t is large enoughto assure that the particle can adjust inthe trapping minimum (adiabaticadjustment time) and also is smallenough to fulfill the above require-ment for the variance. Roughly, onecan say that a net flux to the leftalways occurs when thermal energy issignificantly smaller than the potentialmaximum, the external force chosen isnot too big, and the driving frequencymatches the adiabatic adjustment timeneeded for the particle to move into apotential minimum.

Where does the energy come fromleading to a drift against the externalforce? The energy does not come fromthe heat bath but from the ratchetpotential when it is switched on. Atthat moment the potential energy ofthe particle will suddenly be increased.In a simulation, this can be seen by asudden increase of the energy. Butmost of the energy pushed into the sys-tem will just be dissipated into the heatbath due to the relaxation of the parti-cle into a potential minimum. Only atiny portion will be used for doingwork. Thus a Brownian motor doesnot violate any law of thermodynam-ics because it only turns one type ofwork into another one. Nevertheless,the fluctuating force due to the heatbath is essential for a Brownian motor.

For more details and possibleapplications in biology and chemistry

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40

30

20

10

0

-10

-20

-30

-40

0 200 400 600 800 1000 1200

Toss Number

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read the following review articles. Fora simulation, there are Java applets onthe Internet.

Astumian, R. D. “Thermodynamics andKinetics of a Brownian Motor.” Science 276(1997): 917 –922.

———. “Making Molecules into Motors.”Scientific American 285, no. 1 (2001):57–64.

111. Magnetocaloric

Engine

The ferrofluid is cycled around theloop by the stationary permanentmagnet. A small volume of ferromag-netic material has less energy wherethe magnetic field density is greater,just like iron filings are pulled to thepoles of a magnet. So the ferrofluidapproaching the magnet becomesmagnetized and drawn into the loopvolume between the magnetic poles.But the heat source nearby warms theferrofluid to partially randomize themagnetic dipoles in the ferrofluid, sothe energy of the system can be low-ered again by drawing in some more ofthe cooler magnetized ferrofluid,which pushes out the warmed fer-rofluid. The heat put in by the heatsource is deposited at the heat sink,and the cycle repeats.

To use this engine for the solarheating of buildings, two avenues ofoperation are possible. One couldhave all the piping contain ferrofluid,

which probably would be costly. Thealternative would be to have a smallclosed loop of ferrofluid in contactwith a large loop of piping containingthe water to be heated in a heatexchange device. The advantage overtypical systems would be no movingmechanical parts in the solar heatingsystem.

Rosensweig, R. E. “Magnetic Fluids.”Scientific American 247, no. 4 (1982):136–145.

112. Magnetorheological

Fluid

The flow properties of the fluid changeso radically that the fluid becomes gel-like and can be pushed to one side ofthe beaker where no relaxation mayoccur. The degree of solidificationdepends on the inherent properties ofthe fluid and the strength of the mag-netic field. Of course, its solidificationmay vary within the gel itself becausethe magnetic field may vary with posi-tion in the beaker. Practical applica-tions of these materials with theirunusual properties are being devisedand tested. Perhaps automobile brak-ing systems may someday use thesetypes of fluids to replace solid materi-als that wear away.

Klingenberg, D. J. “Making Fluids intoSolids with Magnets.” Scientific American269, no. 4 (1993): 112–113.

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113. Binary Fluids

Both phase diagrams can representactual binary fluids, although the dia-gram to the right is quite rare. Tounderstand these phase diagrams, bothenergy and entropy must be consid-ered. The energy part involves the vander Waals interaction between adjacentmolecules, an induced dipole-dipoleelectromagnetic interaction. In general,this attractive force between unlikemolecules is much weaker than theattractive force between like molecules.The stronger the force of attractionholding the molecules together, thelower the energy of the system. Hence,when most of the molecule’s neighborsare of the same chemical species, thesystem energy is lowest and immiscibil-ity is favored. Even the increased ran-dom tumbling about of the moleculesat higher temperatures doesn’t disruptthis clustering of like molecules.

However, energy considerationsalone do not explain the behavior ofbinary liquids. Why are they miscibleat all? The miscibility occurs at lowertemperatures because the system tendsto minimize not its energy but ratherits free energy, Efree = Esys – TS. Thefree energy is the energy of the systemminus the product of the temperatureT and the system’s entropy S. At agiven T, the free energy can bedecreased by decreasing the system’senergy or by increasing its entropy. At

low T, changing the entropy has a min-imal effect because the product TSmay be small. But at high T, the prod-uct can be large. So systems at high Ttend to maximize their entropy, thatis, their randomness or disorder.

We now have a good argument forruling out the diagram to the right,with its reappearing miscible phase atlow temperatures. Not so! For somemolecules, hydrogen bonding occurswith its very small angular spread,locking two molecules together. Thishydrogen bonding occurs primarily at lower temperatures because of the orientation dependence, with“orientation” entropy lost in formingthe hydrogen bond being greater thanthe “compositional” entropy gained.Therefore both energy and entropy arelowered, and the lowered energy of thehydrogen bond has a large effect onthe free energy. Water and butyl alco-hol is one example of a binary liquidwith the rare phase diagram.

Walker, J. S., and C. A. Vause. “LatticeTheory of Binary Fluid Mixtures: PhaseDiagrams with Upper and Lower CriticalSolution Points from a Renormalization-Group Calculation.” Journal of ChemicalPhysics 15 (1983): 2660–2676.

———. “Reappearing Phases.” ScientificAmerican 256, no. 5 (1987): 98–106.

114. Baseball Bats

The main source of drag on the swingof a baseball bat is not air friction but

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the retarding force produced by thepressure difference across the bat fromfront to back. As the bat carves itsswath, the air in front gets separatedinto two boundary layers that passaround the bat and recombine behindthe bat. In the wake of the bat,between the two separated boundarylayers, the “lack of air” means a lowerpressure immediately in back of thebat, with a resulting backward forcedue to the pressure difference. There-fore, some of the energy of the swingdoes work against this backwardforce.

A dimpled bat sends the boundarylayers tumbling in turbulent eddiesinto the space behind the bat, reducingthe pressure difference and cutting thedrag. More swing energy is now avail-able to accelerate the bat and to trans-fer to the ball, so the ball’s exit velocitywill be increased.

Gibbs, W. W. “To Fenway, with Love.”Scientific American 271, no. 1 (1994): 98.

115. Old Glass

Many people have suggested that theglass experiences some flow downwardin response to the gravitational pull ofEarth. Contrary to popular conjecture,there is no evidence that any of this oldglass could flow enough during thetime interval of centuries to create thedifference from top to bottom.

Another factor against the flowhypothesis is the actual profile, whichis essentially a linear relationship ofthickness to vertical distance. As asimple model, assume that the proper-ties of the glass are identical at eachvertical position along the pane. If afixed amount of glass material flowsfrom position 10, say, the sameamount would replace this amountfrom position 11, slightly higher upthe glass. The major changes over along time interval would be a thickbuildup at the very bottom and adepletion at the very top, with practi-cally no thickness change between, incontrast to the linear dependence ofglass thickness to height.

In the old days, window glass pro-duction made panes that variedslightly in thickness from one end tothe other because the flat support sur-face had a slight tilt. The installerssimply put the thicker end on the bot-tom. Quality control must have beenmarginal in some areas of the world,because we have seen some large

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Low-pressureregion

High-pressureregion

Airflow

Directionof bat

Boundary layer

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differences in glass thickness betweenthe two ends!

Glass is normally elastic at temper-atures below about 1000 K, and glassmay break but never deform perma-nently because the solid is crystalline.Delicate telescope and camera lenseswould reveal such creep by changingtheir optical characteristics in obviousways.

Pasachoff, J. M. “Comment on ‘MagneticFluids.’” American Journal of Physics 66(1998): 1021.

Zanotto, E. D. “Do Cathedral GlassesFlow?” American Journal of Physics 66(1998): 392–395.

116. Ferromagnetism

Many atoms and molecules have aninherent magnetic dipole moment.When we assume that each dipolebehaves independently of its neighborsexcept for its alignment, the magneticfield direction next to the dipole isopposite to the direction in which thedipole itself points. In paramagnetic

substances the dipoles are far enoughapart to behave approximately inde-pendently, and when no applied field ispresent, these dipoles have randomorientations. Each dipole is affected bythe applied magnetic field but not byits neighbors. The applied magneticfield competes with the random ther-mal motion to cause a net magnetiza-tion that increases nearly linearly withthe strength of the applied field, theratio being known as the magneticsusceptibility.

When the density of magneticdipoles becomes high enough forneighbors to affect each other, onlyneighbors in the head-to-tail configu-ration will tend to align one another.The side-by-side neighbors will beoppositely aligned because all thefields from its neighbors are oppositeat its location. Thus every other dipolein each layer of the crystal will bealigned and form one sublattice, likethe white squares on a checkerboard,and the remaining dipoles (on theblack squares) will form a second sub-lattice of dipoles pointing in the oppo-site direction. The two sublatticesinteract strongly or ferromagnetically,but they cancel each other’s magneti-zation. Therefore, when magneticdipole moments are crowded together,they are more likely to disalign theirnearest neighbors than to align them.So ferromagnetism is rare.

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Then how can ferromagnetic sub-stances exist at all? Via a cooperativeeffect when the dipoles are very closeand no longer behave independently.In these conditions a state of lowerenergy can form if groups of dipolesalign each other into magneticdomains that themselves point in ran-dom directions. With an applied field,these domains will change their sizesto find the lowest energy state. Ofcourse, domain formation cannotform above a certain temperaturecalled the Curie temperature, becausethe thermal agitation interferes withthe dipole interactions. Above theCurie temperature the substancebecomes paramagnetic.

Kolm, H. H. “Why Are So Few SubstancesFerromagnetic?” The Physics Teacher 20(1982): 183–185.

117. Coupled Flywheels

The overall angular momentum of thesystem must be conserved, so includingjust the change in angular momenta ofthe flywheels leads to an incomplete cal-culation. The tension is different in thetwo sides of the belt, so the belt exerts adownward force on pulley 2 and anupward force on pulley 1. These forcesare counteracted by reactions at thebearings, in addition to the reactions tothe weight of the components. Theseadditional reactions produce a torque

that accounts for the change in angularmomentum.

If the pulleys are the same size, thisadditional torque does not exist unlessthe belts are crossed.

118. Superconductor

Suspension

The demonstrated superconductorsuspension does not illustrate theMeissner effect. Instead, this demon-stration depends on the persistenteddy current in the zero resistivitysuperconducting material induced bythe magnet. The eddy current direc-tion is determined by Lenz’s law toproduce a magnetic field that ends upcausing a repulsion between the super-conductor and the permanent magnet.

To show the Meissner effect, thesequence of events must be different.Place the superconductor on the mag-net at room temperature first, and thencool the superconductor below its crit-ical temperature Tc. Then the magneticflux will be “expelled” by the Meiss-ner effect and the superconductor willbecome suspended above the magnet.

Wake, M. “Floating Magnet Demonstration.”The Physics Teacher 28 (1990): 395–397.

119. Nanophase Copper

With smaller grain sizes, one wouldexpect there to be many more grain

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boundaries in the nanophase coppermetal than in normal copper. Theextra grain boundaries would stop orimpede any moving dislocation,thereby making the nanophase coppermuch harder. However, the surpriseturned out to be that nanophase cop-per is mostly dislocation-free! Lackinglarge numbers of moving dislocations,these nanophase metals are muchstronger.

Siegel, R. W. “Creating Nanophase Materi-als.” Scientific American 275, no. 6 (1996):74–79.

120. Head of a Pin

Experiments show surprising results.Any fraction of the fundamentalcharge, such as +0.5 e or –0.1 e, canexist on the head of a pin! The con-ceptual argument goes as follows. Themetal pin is an electrical conductor. Ingeneral, an electric current flows in theconductor because some free electronsmove through the lattice of atomicnuclei. Any particular volume of theconductor has virtually no chargebecause the negative charges are bal-anced by the positive charges of thenuclei.

So the important physical quantityis not the electric charge in any givenvolume but instead how much chargehas been carried through the conduc-tor—that is, the “transferred charge,”which can have any value, even a

fraction of the charge of a single elec-tron. This “transferred charge” is pro-portional to the sum of the shifts of allthe electrons with respect to the latticeof nuclei. These electrons in the con-ductor can be shifted as little or asmuch as desired, so the sum canchange continuously, and therefore socan the “transferred charge.” The pin-head can have any charge value, notjust integer multiples of the fundamen-tal charge.

Likharev, K. K., and T. Claeson. “Single Elec-tronics.” Scientific American 266, no. 6(1992): 80–85.

121. Coulomb Blockade

No, the current across the junctionwill not be a steady current. There willbe single electron tunneling (SET),with the voltage across the junctionchanging periodically with a frequencyequal to the current divided by thefundamental unit of charge e.

The tunnel junction is a conductor-insulator-conductor device, so trans-ferred charge flows through theconductor to accumulate on the sur-face of the electrode against the insu-lating layer of the junction. Anopposite surface charge of equalamount accumulates on the other elec-trode across the junction. The actualamount of surface charge has a con-tinuous change in value as the chargeaccumulates, including fractional

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values such as +0.8642 e, because theelectrons near this surface can adjusttheir positions slightly.

However, only discrete amounts ofcharge can tunnel through the insulat-ing layer—that is, each electron tun-neling through changes the surfacecharge by +e or –e, depending on thedirection of tunneling. The tunnelingprocess is energy-dependent. If thecharge at the junction is greater than+e/2, an electron can tunnel through toreduce the surface charge by e, thusreducing the electrostatic energy of thesystem. And if the surface charge isless than –e/2, an electron can tunnelin the opposite direction to decreasethe energy. But if the surface chargevalue is greater than –e/2 or less than+e/2, tunneling would not occurbecause the system energy wouldincrease. This tunneling suppression isknown as the Coulomb blockade, firststudied in the 1950s.

The tunnel junction connected to aconstant current source begins in theCoulomb blockade condition, thenreaches tunneling for the one-electroncondition, then back to the Coulombblockade, then one-electron tunneling,etc. The analogue may be a drippingfaucet.

Many electronic devices are beingmade with SET operation. For exam-ple, an SET transistor can switch on oroff the flow of billions of electrons persecond when the charge on the middle

electrode is changed by only half thecharge of an electron!

Likharev, K. K., and T. Claeson. “Single Elec-tronics.” Scientific American 266, no. 6(1992): 80–85.

122. Deterministic

Competition

The time evolution here depends onthe value of r. One finds that Nt = 1 isa stable equilibrium only when r liesbetween 0 and 2. If r = 2.3 with N0 =0.5, then successive Nt will oscillatebetween about 1.59 and about 0.40 asa stable 2 cycle. For r > 3.102, no cycleis stable, all cycles are possible, etc.

In the chaotic regimes, the equa-tion results are deterministic, but thetime evolution is indistinguishablefrom that governed by probabilitylaws. One really needs to see the cal-culations proceed to appreciate theamazing behavior of this simple-look-ing equation.

Gleick, J. Chaos: Making a New Science.New York: Penguin, 1987, pp. 166–186.

123. Two Identical

Chaotic Systems

Yes, the two identical chaotic systemsdescribed can be synchronized.Chaotic systems are very useful for sev-eral reasons: (1) Chaotic systems are acollection of many regular, ordinarybehaviors, none of which dominate.

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(2) The proper perturbation canencourage the chaotic system to followone of its many regular behaviors. (3)Chaotic systems are very flexiblebecause they can rapidly switch amongdifferent behaviors. (4) Chaotic sys-tems are deterministic and, althoughno one can say which output willresult, two identical chaotic systems ofthe appropriate type will produce thesame output in response to the samesignal input.

To synchronize two identicalchaotic systems each with the stablesubpart behavior, one can apply theappropriate pseudoperiodic signal(one type is called a Rössler signal) tocoax them into step. For the reasonslisted above, the outputs will be thesame. The details can be learned in thereference below, where the chaoticattractor and the Poincaré section arediscussed. Applications to secure com-munications and to biological systemsare included also.

Ditto, W. L., and L. M. Pecora. “MasteringChaos.” Scientific American 269, no. 2(1993): 78–84.

124. Tilley’s Circuit

The galvanometer does nothing! Thereis no induced potential because nowork was done (assuming frictionlessswitches). This result appears to vio-late Faraday’s law V = dΦ/dt, where Vis the potential difference induced by

the rate of change of magnetic flux Φ.But work must be done for V to begenerated because the change in thework dWork = V dt.

Nussbaum, A. “Faraday’s Law Paradoxes.”Physics Education 7 (May 1972): 231–232.

125. Thermal Energy

Flow

The classical flow of thermal energytoward the cooler region occursbecause the free energy of the com-bined system Efree = Esys – TS becomesless, where T is the temperature and Sis the entropy. If the free energy is thesame at two temperatures, one can seethat for a given amount of systemenergy there is more disorder at thelower temperature. Assuming that thetwo-block system initially simplytransfers thermal energy from thewarmer to the cooler block, with noother energy transfers, then a coolersystem is preferred.

Dyson, F. J. “What Is Heat?” ScientificAmerican 191, no. 3 (1954): 58–64.

126. Cadmium Selenide

The wavelength of visible light is com-parable to nanophase cluster sizes. Forexample, greenish light has a wave-length of about 580 nanometers, fiveto ten times the nanophase clustersizes. Clusters behaving as particlesranging from about 1 nanometer to 50

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nanometers in diameter are too smallto have any significant scattering of vis-ible light, so these materials are effec-tively transparent. Clusters of sizescomparable to particular wavelengthranges of visible light are subject toquantum confinement restrictions.

Quantum mechanics predicts thecorrect behavior at the small clustersizes. The smaller the nanophase clus-ter size becomes, the greater are theenergy spacings for the electron states.Which colors of light are absorbed andemitted are determined by these energyspacings. If the energy spacings are toogreat, the incoming light will not beabsorbed, and light of that wavelengthand longer will not be scattered. Forexample, a typical semiconductor iscadmium selenide. When the size of thecluster is 1.5 nanometers, the cadmiumselenide appears yellow, but when thesize is 4 nanometers, it will appear red.And larger clusters appear black.Therefore, the observed color of theclusters in the nanophase depends ontheir actual sizes.

127. Optical Solitons

Under the right conditions, the twoeffects—dispersion and the Kerr effect—can be made to cancel exactly. Thenonlinearity of the Kerr effect candelay the “fast” carriers relative to the “slow” carriers, bringing themtogether to counter the dispersion.

These pulses conserving their shapeand integrity are exhibiting solitonbehavior. Optical solitons were firstobserved in fibers in 1980 and are nowfundamental components in opticaltransmission systems.

Desurvire, E. “The Golden Age of OpticalFiber Amplifiers.” Physics Today 47 (1994):20–27.

128. Ceramic Light

Response

Certain ceramic materials will changetheir shape upon exposure to lightbecause some molecules in the mate-rial have changed their shape uponabsorption of particular frequencies oflight. If the responses of many mole-cules are coordinated, the overalleffect can be a macroscopic shapechange. Called the photostrictiveeffect, research began in the 1990s,and some practical devices are begin-ning to be developed, such as directconversion of light to mechanical dis-placement for speakers instead of con-version to an electrical signal first. Atelephone speaker could be one of thefirst products.

These ceramics are examples of anew type of “smart” material. Thefour most widely used classes of smartmaterials are piezoelectrics, electro-strictors, magnetostrictors, and shape-memory alloys. The resulting changesin the shapes of these materials are

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large enough to make them useful asactuators. A sensor receives a stimulusand responds with a signal; an actua-tor produces a useful motion oraction. By definition, smart materialsare both sensors and actuators,because they perform both functions.

Photostrictive materials such asPLZT—a combination of lead, lan-thanum, zirconium, and titanium—someday may be used to controlrobots and machines. Engineers atPennsylvania State University, forexample, are exploring applicationsfor devices that move when lightshines on them and have created atwo-legged stand that walks veryslowly when illuminated.

Dogan, A., et al. “Photostriction of Sol–GelProcessed PLZT Ceramics.” Journal ofElectroceramics. 1, no. 1 (1997): 105–111.

Newnham, R. E., and A. Amin. “Smart Sys-tems: Microphones, Fish Farming, andBeyond.” ChemTech 29, no. 12 (1999):38–46.

129. Random

Movements

Wobbles in any system can be fol-lowed with fast cameras. For mosthuman actions, from balancing a stickvertically on a finger to balancing on atightrope, wobbles occur that lastfrom seconds to tens of milliseconds.Usually the shorter the fluctuation, the

more of them there are. But the typicalhuman reaction time for such balanc-ing acts is about 100 milliseconds, somost of the wobbles are faster thanhumans can react. Mathematical mod-eling of human balancing acts matchthe measured fluctuations only whenthe person or object is on the verge offalling. Then the random fluctuationscancel each other out and the objectremains upright.

Related research has found thatelderly people and others with balanceproblems showed signs of better bal-ance when they stood on a pair of bat-tery-operated, randomly vibratinginsoles. The idea is that these vibra-tions amplify balance-related signalsfrom the feet to the brain and viceversa that may have become reducedby age or illness. When people walk,then turn or reach, they are most vul-nerable to a fall. When a person leansor sways to one side, the pressure onthe sole of that side increases, and thenervous system senses the change inpressure and sends a message to thebrain so that the posture can beadjusted. In many people, those mes-sages can be altered by age, stroke, orconditions such as diabetes. Furthertesting is under way to optimize thesehelpful insoles.

Cabrera, J. L., and J. G. Milton. “On-OffIntermittency in a Human Balancing Task.”Physical Review Letters 89 (2002): 158702.

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Chow, C. C., and J. J. Collins. “PinnedPolymer Model of Posture Control.”Physical Review E 52 (1995): 907–912.

Priplata, A. A., et al. “Vibrating Insoles andBalance Control in Elderly People.” Lancet362, no. 9390 (2003): 1123–1124.

130. Gravitational Twins

The traveling twin actually returnsmuch younger than her stay-at-homesister. The argument given was cor-rectly stated but incomplete. The localgravitational tidal effects are not thesame for the twins—that is, the rate ofchange of gravitational potential expe-rienced was different. These tidaleffects contribute to the clock ratesand, when included in the calcula-tions, contribute enough to change theresult so that the stay-at-home twinages faster and is older upon return ofher sister. For a calculation, see the ref-erence below.

Bradley, M., and J. Higbie. Physics Teacher22, no. 1 (1984) 34–35.

131. Photon Engine

We can analyze the operation of thequantum Carnot engine in the samemanner in which we would analyze aclassical Carnot engine. Let Qin be theenergy absorbed from the bath atomsduring the isothermal expansion andQout be the energy given to the heat sinkduring the isothermal compression.

Then the Carnot engine efficiency η = (Qin – Qout)/Qin.

If the bath atoms are assumed to betwo-state systems that absorb and emitradiation at the same photonfrequency, then we need the thermo-dynamic properties of a photon gas inorder to determine the theoretical effi-ciency of this photon engine. Assumingthermal equilibrium for the photongas, the average number of photons n2

with energy ε coming in from the heatbath at temperature T2 is given by n2 =1/(exp[ε/kT2] – 1), while the averagenumber of photons n1 leaving at tem-perature T1 is n1 = 1/(exp[ε/kT1] – 1).Since Qin ∝ n2 ε and Qout ∝ n1 ε, theefficiency of the quantum Carnotengine is η = 1 – T1/T2, exactly thesame as for the classical Carnot engine.When there is only one heat bath, withT1 = T2, no work can be done.

A different quantum engine occurswhen the bath atoms have three statesinstead of two, bringing in quantumbehavior called quantum coherence,with a nonvanishing phase differencebetween the two lowest atomic statesinduced by a microwave field. One caneliminate the photon absorptionprocess (analogous to laser operationwithout a population inversion). Thetemperature T2 becomes altered to adifferent effective temperature, Tφ. Theefficiency ηφ = (Tφ – T1)/T1 can exceedthe efficiency of the classical Carnot

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engine. This quantum engine canextract work from a single heat bath,even when T1 = T2! For the details ofthe three-state quantum engine’s oper-ation, see the reference below.

Scully, M. O., et al. “Extracting Work from aSingle Heat Bath via Vanishing QuantumCoherence.” Science 299 (2003): 862–864.

Chapter 7A Whole New World

132. Grain of Sand

If one assumes that the grain of sandhas a diameter that is a reasonablefraction of 1 millimeter, then the lineof atoms would be about 1010 meterslong, about thirty times the distance tothe Moon!

133. Forensics

Until the mass production of paintsbecame available in the late 1800s andearly 1900s, each paint used by anartist is known to contain atoms inparticular characteristic amounts,depending on the source. Paints wereoriginally made from natural materi-als, so when an artist mixed his or herpaints, there was usually a uniquemixture of atoms and molecules foreach color and color combination.

Different atoms absorb and emittheir unique characteristic frequenciesof light in the visible and the ultravio-let. The types of atoms present and theintensity of the characteristic spectrumfrom each atom type will create a“spectral fingerprint” for each artist.As you know, some artists simply laidout the design of the painting, forexample, and lesser painters filled inthe regions, with the master artistcompleting the final touches. Eventhese paintings have their own finger-print of spectral colors.

With a tunable laser capable ofscanning from the infrared frequenciesto the ultraviolet frequencies, the“spectral fingerprint” of any region ofthe painting can be recorded andcompared to other paintings by thesame artist or even other artists,including fraudulent painters. Thislaser approach is normally combinedwith other approaches to achieve thecomprehensive evaluation.

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Original

400

Wavelength (nm)

600

Copy

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The laser technique also permitsthe identification and removal of envi-ronmental coatings on top of the paintbeneath, such as dust and grime, andensures that no harm to the paintingoccurs. Famous paintings such asRembrandt’s 1642 De Nachtwacht(The Nightwatch) in Amsterdam’sRijksmuseum have had the soot andgrime safely cleaned off to reveal amarvelously brighter background offaces when compared to the somewhatobscure dull background that hadexisted for centuries.

134. Doppler

Elimination?

Yes. First consider the emissionprocess. Normally, a typical electricdipole emission occurs with a singlephoton exiting the atom as a result ofan allowed transition within the atomthat conserves energy and angularmomentum—that is, the angularmomentum of the atom changes by ±1unit of Planck’s constant h/2π. Theprobability for all other emissionprocesses is lower by a factor of 1/137,or by a higher power of this factor.

A two-photon electric quadrupoleemission process is possible betweentwo atomic states with angularmomentum quantum numbers differ-ing by zero or two units of h/2π. Thereis a broad continuous spectrum of

possible energies for the two photonsemitted in this quadrupole emissionprocess. A very small fraction of thesetwo-photon emissions will spit outtwo photons of the same energy, go offin opposite directions, and produce norecoil of the atom. The two-photonemission from hydrogen was the firstatom to be measured and the first to becalculated by quantum electrodynamics(QED) in the 1940s. Two-photon emissions after laser excitations havebecome commonplace for many uses intoday’s optics research.

Likewise, simultaneous two-photonabsorption is possible. A container ofsingle atoms is placed between twocounterpropagating laser sources,shining two identical frequency laserbeams on an atom so that energy andangular momentum will be conservedand recoilless absorption can occur.First achieved in the 1970s, the preciseenergy-spacing values within atomshave been determined. Today, two-photon absorption with nonidenticalenergies plays a critical role in theupconversion of laser light to higherfrequencies to achieve coherent beamsin the UV and for providing lightsources of precise frequencies.

At the nuclear level, recoillessgamma-ray emission and absorptionare possible if the whole crystal recoilssimultaneously with the photon emis-sion or absorption. This Mössbauer

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Effect transition, discovered in the1950s, relies on the inability in princi-ple of identifying the single nucleusinvolved and includes an exponentialfactor proportional to the negativeratio of the temperature of the crystalto its Debye temperature.

As an interesting historical note,Albert Einstein in 1917 was among thefirst to recognize that classical electro-magnetism cannot explain sponta-neous emission of light from atoms. Inparticular, he inferred that an atommust recoil upon spontaneous emis-sion, in conflict with the symmetric-field distributions produced byelectromagnetic theory based onMaxwell’s equations. According toEinstein, “. . . outgoing radiation in theform of spherical waves does not exist. . .” for if an atom radiated a classicalspherical wave it could not recoil.

Einstein, A. “Zur Quantentheorie derStrahlung.” Physika Zeitschrift 18 (1917):121–128.

135. Light Tweezer

Yes. A focused laser beam can exert atrapping force perpendicular to thebeam direction of 2 × 10–12 Newtonsor more to keep cells confined in amicroscope at the optical axis. Theintensity gradient across the lightbeam is the source of the force.

In the simplest geometry, consider asemitransparent object with a diameter

greater than the wavelength of the inci-dent light but smaller in size than thediameter of the incident light beam. Letthe light source be a parallel beam oflight rays all of the same frequency,such as in a laser beam focused to thepoint f by a symmetrical lens. Theobject tends to focus the light rayssomewhat, changing the direction ofthe light rays. The sideways recoil ofthe object occurs to simply conserve thelinear momentum. If the light beam hasan intensity gradient, brighter in thecenter than near the edge, the objectwill receive a net push back toward theoptical axis in the center. There mustalso be a recoil of the object in thedirection of the original light beam,which usually is taken up by the appa-ratus and Earth because the object is ona horizontal platform. A one-celledparamecium remains well trapped in amicroscope via this light tweezer tech-nique, begun at Bell Labs in the 1970s.

When the object is smaller than thewavelength of the incident light, amore detailed analysis is required tounderstand the 3-D trapping and thequantum interference effects.

Optical tweezers have been widelyused for several decades in applicationsas diverse as experiments on molecularmotors in biology and the movement ofBose-Einstein condensates in physics.The capabilities of single optical tweez-ers have been greatly improved and

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extended by the development oftailored beams and by schemes for gen-erating large numbers of trapping sitesand shapes simultaneously.

Block, S. M. “Making Light Work with Opti-cal Tweezers.” Nature 360 (1992):493–495.

Chu, S. “Light Trapping of Neutral Parti-cles.” Scientific American 266, no. 2(1992): 70–76.

MacDonald, M. P., et al. “Creation andManipulation of Three-Dimensional Opti-cally Trapped Structures.” Science 296(2002): 1101–1103.

Ulanowski, Z., and I. K. Ludlow. “CompactOptical Trapping Using a Diode Laser.”Measurement Science and Technology 11(2000): 1778–1785.

136. Fluorescent Lights

Today, artificial illumination requiresmore than 25 percent of the electricitygenerated worldwide. There are twotrends in “energy saving” technolo-gies. The first trend is using improvedlamps, such as fluorescent, mercury,sodium, metal halide, and halogenlamps. The second trend is improvingthe electronic circuit design for suchlamps.

Although fluorescent lights arefour to six times more efficient thanincandescent lamps, there now existmany other types of light sources thatare even more efficient. For the fluo-rescent lamp, its efficient productionof the UV is extended into the visibleby a powder coating inside the tube.

This powder absorbs the UV light andfluoresces in the visible. Very littleheating of the fluorescent lamp occurs,so the efficiency occurs before the pro-duction of the visible light, with verylittle electrical energy being convertedinto thermal energy. The conversionprocess in the powder makes the tubeuseful for room lighting.

So why is the incandescent lamp soinefficient, converting only about 4percent to 12 percent of the electricalenergy to visible light? The incandes-cent lamp is simply a resistor whosefilament temperature rises until it getsrid of thermal energy at the same ratethat thermal energy is being generatedin the filament. In a standard 100-watt, 120-volt bulb, the filament tem-perature is roughly 2550°C, about4600°F, so that the thermal radiationfrom the filament includes a significantamount of visible light.

The output is 17.5 lumens perwatt, compared to a maximum of 240lumens per watt if all the energy couldbe converted to visible light. The rea-son for this poor efficiency is the factthat tungsten filaments radiate mostlyinfrared radiation at any temperaturethat they can withstand. An ideal ther-mal radiator produces visible lightmost efficiently at temperatures ofabout 6300°C (about 6600 K or11,500°F). Even at this high tempera-ture, a lot of the radiation is either

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infrared or ultraviolet, and the theo-retical luminous efficiency is 95lumens per watt.

Most fluorescent lights predomi-nantly emit light in the visible part ofthe spectrum and they do emit someUV light, but only in a narrow rangeof the UV spectrum. Unfortunately,their UV emission range does not over-lap the two small ranges of UV lightneeded by humans for the best func-tioning of certain internal organs,which receive some of the UV lightthat passes through the skin, as well asvitamin D production from 7-dehy-drocholesterol in the skin.

Special fluorescent lights morebenevolent to human needs are avail-able and mimic sunlight to produce aUV spectrum better matching theneeds of these internal organs. Indeed,the lack of the required UV parts ofthe ambient light spectrum can lead tocertain illnesses. Of course, the lack of

vitamin D production in the skin canlead to rickets and other problemsassociated with the calcium and inor-ganic phosphate metabolism. Eskimosand other indigenous peoples obtainplenty of vitamin D from the fish oilsin their diets.

Porter, J. P., ed. How Things Work in YourHome (and What to Do When They Don’t).New York: Henry Holt, 1985, p. 158.

137. Phase Conjugation

Mirror

Yes, the light can return undisturbed ifthe light wave retraces it original pathas its time-reversed twin and themedium retains its previous integrity.The phase conjugate of a wave pos-sesses exactly the same spatial proper-ties as the original wave, but it is saidto be reversed in time. This means thata phase conjugate wave exactlyretraces the path of the original beam.This method has the useful propertythat if a light beam propagatesthrough a distorting medium, then thephase conjugate is produced andexactly retraces the path through thedistorting medium, enabling the unfa-vorable effects of the distorting mediato be reduced or eliminated. Phaseconjugation is the general term for aprocess in which both the direction ofpropagation and the overall phase fac-tor of a wave function are reversed.

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Intensity

400 Wavelength (nm) 700

SunlightStandard FluorescenceSpecial Fluorescence

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Some laser sources come with opti-cal phaser conjugators to remove dis-tortion in the laser beam. Opticalphase conjugation occurs also whenthere are four waves mixing with allfour waves of the same frequency.Another useful application of a phaseconjugate mirror might be to put onein one reflecting path of an interfer-ometer as a reference for detectingchanges in the other path.

Blaauboer, M., D. Lenstra, and A. Lodder.“Giant Phase-Conjugate Reflection with aNormal Mirror in Front of an OpticalPhase-Conjugator.” Superlattices andMicrostructures 23 (1998): 937.

Brignon, A., and J.-P. Huignard. Phase Con-jugate Laser Optics. New York: John Wiley& Sons, 2004, chap. 1.

138. Stationary States

In the Bohr model of the hydrogenatom, one would calculate the fre-quency f = 2πr/v of the electron’sorbital motion. The virial theorem

states that twice the kinetic energyplus the potential energy add to zero, so mv2 = ke2 r, from which theelectron’s frequency of orbit is f =n3h3/(4π2me4). The actual Bohr energyE = –2π2me4/(n2h2) is clearly a differ-ent quantity, and for an electron jumpbetween two energy states, E2 – E1 ≠hf2 – hf1.

Spielberg, N., and B. D. Anderson. SevenIdeas That Shook the Universe, 2nd ed.New York: John Wiley & Sons, 1995, p.248.

139. Angular Momentum

We take the space quantization ofangular momentum as given, so therewill be (2j + 1) components in the z-direction from j to –j, decreasing byan integer each step. Since there is nopreferred direction, J2 = Jx

2 + Jy2 + Jz

2,that is, J2 = 3 <Jz

2>avg , where avg rep-resents the average value given by [ j2 +(j – 1)2 + . . . + (–j + 1)2 + (–j)2] h2/(4π2

[2j + 1]). Using a math table or findingthe sum of the series of squared inte-gers directly, one can verify that J2 =j (j + 1) h2/4π.

Feynman, R. P., R. B. Leighton, and M.Sands. The Feynman Lectures on Physics.Vol. II, Reading, Mass.: Addison-Wesley,1965, p. 34-11.

140. Kinetic Laser

The explosion of the lasing materialcreates many free electrons, some of

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Phase conjugate mirror

Conventional mirror

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which have been blown out of low-lying atomic states, creating theneeded population inversion for possi-ble lasing action. Practically any mate-rial can be used. During an extremelyshort time interval after the explo-sion—on the order of nanoseconds—stimulated emission may occur asphotons from the exploding materialexit the expanding blast volume.These photons pass through regions ofthe expanding cloud of ionized debrisand can stimulate the emission ofmany more photons into the samequantum state at the same wavelength.The resulting coherent radiation atmany frequencies, including the softX-ray region, will show intensityspikes in particular directions.

Some of the first kinetic laserexplosions were done at LivermoreNational Laboratory in the 1970s and1980s with exploding foils and theNova laser system. Since the firstdemonstration of soft X-ray lasing—emissions at about 10 nanometers ormore—using the collisional excitationmechanism in neonlike selenium,many other neonlike ions, rangingfrom copper (Z = 29) to silver (Z =47), have been made to lase. However,attempts to produce lower-Z neonlikeX-ray lasers have been unsuccessful.

In the effort to develop a tabletopX-ray laser that would require smallerhigh-energy laser drivers than Nova

and that could be used for applicationssuch as biological imaging, nonlinearoptics, holography, and so on, a pre-pulse technique has been developed.This technique has been used success-fully to produce lasing in many lower-Z neonlike ions such as titanium (Z =22), chromium (Z = 24), iron (Z = 26),and nickel (Z = 28). The use of thisprepulse technique has opened up anew class of neonlike X-ray lasers forinvestigation.

Chapline, G., and L. Wood. “X-ray Lasers.”Physics Today 6 (1975): 40.

Dunn, J., et al. “Demonstration of X-rayAmplification in Transient Gain Nickel-likePalladium Scheme.” Physical Review Let-ters 80, no. 13 (1998): 2825–2828.

Nilsen, J. “Reminiscing about the Early Yearsof the X-ray Laser.” Quantum Electronics33, no. 1 (2003):1–2.

141. Noninversion Laser

Yes, lasing without inversion (LWI)can occur whenever absorption can-cellation is established. Light amplifi-cation is then possible even when theupper-level population is less than thelower-level population. This cancella-tion can be set up in a three-level sys-tem in an atom in which the twoabsorption transitions to the samefinal state interfere and cancel, makingthe absorption probability zero.

In the diagram, upper-level state | a > is connected to lower levels | b >and | c >. Use incident photons of the

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appropriate energies E1 and E2, whichcorrespond to transitions | a > to | b >and | a > to | c >, respectively. Theuncertainty in these atomic transitionsleads to interference, since the transi-tions end in the same final state. Thereis no way to determine which absorp-tion transition to the final state actu-ally occurred, so like the Youngdouble-slit experiment, one must havethe interference. There is no interfer-ence between the emission paths, sincethey have different final states. Byarranging the phases of the twoincoming light rays properly, one canmake the interference completelydestructive for absorption. Then stim-ulated emission is the only process left.For details of the probability calcula-tions, see the references below.

Narducci, L. M., H. M. Doss, P. Ru, M. O.Scully, S. Y. Zhu, and C. Keitel. “ A SimpleModel of a Laser without Inversion.”Optics Communications 81 (1999): 379.

Scully, M. O., and M.S. Zubairy. QuantumOptics. Cambridge, Eng.: CambridgeUniversity Press, 1997.

142. X-ray Paradox

The index of refraction n for a mate-rial is normally stated with regard tothe phase velocity, unless indicatedotherwise. The phase velocity is vph =c/n(k), where the index is a function ofthe wave number k. If n(k)<1, then thephase velocity is greater than the speedof the light in the crystal. There is noalarm that the energy is being trans-ported faster than c, for the groupvelocity is still less than c.

Essentially, travelling harmonicwaves in all physical examples requirewave packets or groups because of thenon-infinite extent of space and/ortime. There are two velocities associ-ated with these wave packets orgroups: the phase velocity and thegroup velocity. Harmonic waves orharmonic components have a phasevelocity vph = ω/k, where ω = 2πf and fis the frequency. This phase velocity isthe velocity at which the wave frontstravel. A group of harmonic waves orwave packet has a group velocity vg =dω/dk, the velocity at which the packetshape or envelope travels—that is, thevelocity at which information orenergy is transported.

On the atomic level, the slowing oflight passing through a material maybe considered as a continuous processof absorption and emission of photonsas they interact with the atoms of the

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State a

State b

State c

3-state system

E1 E2

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material. One assumes that betweeneach atom, the photons travel at c, asin a vacuum. As they impinge on theatoms, they are absorbed and nearlyinstantly re-emitted, creating a slightdelay at each atom, which (on a largeenough scale) seems to be an overallreduction in the speed of the photons.Quantum mechanically, the scatteringis a two-step process of absorbing theincident photon and emitting a newphoton.

Experiments in other ranges of theelectromagnetic spectrum, particularlyin the visible, have shown that by stor-ing the phase information of the inci-dent light beam in a gas vaportemporarily, one can even claim thatthe light pulse can be brought to rest!

Addinall, E. “The Refractive Index of X-rays.” Physics Education 6 (1971): 77–78.

143. Benzene Ring

The benzene ring has six-fold rota-tional symmetry about an axis perpen-dicular to the plane of the ring. Onesimply requires a wave function solu-tion of the Schrödinger wave equationthat has this six-fold symmetry, andsuch a solution is easy to find. Onewould expect that knowing this solu-tion would allow one to calculate theenergy levels.

However, we are not done! Thereare two possible configuration basestates, as shown in the diagram.

Both states should have the sameenergy, and they do. Therefore wereally have a two-state system, analo-gous to the hydrogen molecular ion orthe ammonia molecule, so the analysisshould be for a two-state system. Therewill be the possibility that configura-tion A changes into configuration B. Asa result, quantum mechanics willreveal that two new stationary stateswill occur, one state (the new groundstate) with energy below the ground(lowest) state determined before, andone state with higher energy. The newground state will be neither of the twoconfiguration states shown but will bea linear combination of these two con-figuration states. Only this state isinvolved in the chemistry of benzene atroom temperatures.

Understanding benzene was one ofthe first verifications of the linear super-position of states that is at the heart ofquantum mechanics and also indicatedthat quantum mechanics will be suc-cessful at larger scales than atomic.

Feynman, R. P., R. B. Leighton, and M.Sands. The Feynman Lectures on Physics,Vol III, Reading, Mass.: Addison-Wesley,1965, pp. 10-10 to 10-12.

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144. Graphite

Place identical atoms into a diamondcrystal structure. First, one wouldmathematically find a wave functionfor the four bonding electrons usingthe Schrödinger equation, resulting inwhat are called sp3 orbitals. Then onewould represent the periodic symmetryin the crystal. Each carbon atom willmake four orthogonal bonds withtetrahedral symmetry if it can to itsnearest neighbors. This diamond struc-ture is one way to do this bonding.

Another way to have four carbonbonds is for six carbon atoms to forma regular hexagonal ring with twobonds in the ring for each carbon, andthe other two bonds extending per-pendicular to the ring, one upwardand the other downward. Upon calcu-lating the energy states for the fourcarbon binding states, one learns thatthe two perpendicular binding statesare held less securely than the ones inthe ring that form a plane. The struc-ture makes graphite, a layered crystalthat slips easily between the planes.Pencil writing surfaces have beenmade from graphite for several thou-sand years.

Carbon in the fullerene structure is even more interesting. The structureof 60 carbon atoms that resultsdepends on many factors, includingthe velocity distribution of the freecarbon atoms before collision, the

formation of intermediate structures,and so on. Fullerenes tend to form by“rolling up” a graphite sheet andadding carbon pentagons to achievecurvature. If you just roll the sheet intoa cylinder and cap off the ends withpentagon-curved hemispheres, youmake a carbon nanotube. These nan-otubes are quite different from the tra-ditional fullerene-type materials (i.e.,roundish cages), so they have quite dif-ferent properties.

Collins, P. G., and P. Avouris. “Nanotubesfor Electronics.” Scientific American 283,no. 6 (2000): 62–69.

Pauling, L. General Chemistry. New York:Dover, 1988, pp. 168–170, 207–210.

145. Ozone Layer

Ozone plays two important roles withregard to the energy balance for Earth.As a minor greenhouse gas in all partsof the atmosphere, including near thesurface, ozone helps maintain Earth’saverage temperature at about 13°Cinstead of a frigid –17°C. The concen-tration of ozone in the upper atmos-phere, however, regulates the UVintensity in sunlight reaching the sur-face. All organisms need some UVlight to maintain a healthy existence,but any reduction of ozone in theupper atmosphere might allow dan-gerously large amounts of UV to reachthe surface.

The two polar regions are extremely

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susceptible to ozone depletion, particu-larly by chlorofluorocarbon (CFC) mol-ecules and others, because the icecrystals in the air provide these fluoro-carbons with platforms for rapid ozonedissociation. Already, as a result ofozone depletions in the upper atmos-phere above the polar regions, particu-larly above the South Polar region,there has been an increase in eye prob-lems in land animals such as sheep inthe southern parts of South Americaand in Australia and New Zealand.

Allègre, C. J., and S. H. Schneider. “The Evo-lution of the Earth.” Scientific American271, no. 4 (1994): 66–75.

Newchurch, M. J., et al. “Evidence for Slow-down in Stratospheric Ozone Loss: FirstStage of Ozone Recovery.” Journal of Geo-physical Research 108, no. D16 (2003):4507.

146. Greenhouse Gases

The greenhouse gases trap most of theinfrared, and this additional energyhelps heat Earth to its present average

equilibrium temperature of about13°C. Without the greenhouse effectin our atmosphere, Earth’s averagesurface temperature would be about256 K, or about –17°C, much too coldfor many life forms. The greenhouseeffect involves the influx of sunlight,its absorption by the atoms and mole-cules of the stuff on Earth, and theattempted emission of light andinfrared energy back into space.

Although carbon dioxide receivesthe most attention in the press, HOHvapor is the most important green-house gas because the HOH moleculeabsorbs energy over practically thewhole range of visible and infrared fre-quencies, while carbon dioxideabsorbs in a small region of the nearinfrared only. Water vapor controlsabout 60 percent of the greenhouseeffect, carbon dioxide about 20 per-cent, and the other trace gases in theatmosphere the remainder.

Additional greenhouse gas concen-trations added to the atmospherewould be expected to trap even moreinfrared radiation and probably raisethe temperature further. However, aconvincing comprehensive model ofthis process has not been achieved.There are many complications to anymodel of Earth, including the trans-mission and reflection of light fromclouds, the movements of the oceancurrents, the effects of human-madesources and sinks, the perturbations

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CO2

HOH

HOH

280 K blackbodyRadiance

400 1600Wavelength (cm-1)

O3

Absorption curve

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by vegetation, land animals, and seaorganisms such as plankton, the ther-mal energy input from additional heatsources such as mantle transport ofthermal energy from the interior ofEarth, and the effects of the bombard-ment by cosmic rays from the galaxyand beyond.

Many natural temperature recordshave been mined in the past fewdecades that provide the history oftemperature changes, so fluctuations inaverage temperature, a vaguely definedconcept, are not new. The claim seemsto be that the rate of increase of theaverage temperature is among thegreatest ever experienced on Earth.Whether this hypothesis is verified inthe near future will take better models,meaning greater computing capabilityand more included physical andchemical processes and/or a definitive,unambiguous example.

Gillett, N. P., F. W. Zwiers, A. J. Weaver, andP. A. Stott. “Detection of Human Influenceon Sea-Level Pressure.” Nature 422 (2003):292–294.

Herzog, H., B. Eliasson, and O. Kaarstad.“Capturing Greenhouse Gases.” ScientificAmerican 282, no. 2 (2000): 72–79.

147. LED vs. LCD

We assume that they all have the sameresolution, and we know that all threetypes of display—LED, LCD, andplasma—require energy to operate.But the majority of the energy for the

LCD is provided by the ambient light,whereas all the energy for the LED andplasma displays must be provided bythe electronic power source itself, suchas a battery or the AC supply. In addi-tion, considerable thermal energy canbe produced in a plasma display, anenergy requirement beyond simplyproducing a picture on the screen. Ofcourse, there are LCD displays thatmust provide their own ambient lightif they are to be used in a dark envi-ronment, so these displays have addi-tional energy requirements whenoperated in this manner.

So LCDs consume much lesspower than LED and gas-display mod-els because LCDs work on the princi-ple of blocking light rather thanemitting it.

148. Sonoluminescence

The light produced by sonolumines-cence must originate in atomic transi-tions, electrons in excited states inatoms jumping down to lower energylevels and emitting photons to con-serve energy and angular momentum.The apparatus consists of distilledwater with an admixture of a littlehelium or other inert gas in a sphericalflask surrounded by a piezoelectriccrystal or two to send in sound wavesat practically any frequency. Thedetails of the apparatus can be foundat many sites on the Internet.

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The sound energy creates bubblesin the water that rapidly collapse andemit a flash of light from their centralregion. Instead of sound waves, apowerful laser pulse also can createthe bubbles for the pulse of light. Thespectrum of the emitted sonolumines-cent light pulse is similar to a black-body spectrum of an object at about8,000 K, hotter than the Sun’s surfacetemperature of about 6000 K! And thepulse of light lasts for picoseconds,with such an intensity that it can beseen by the unaided human eye.

The reference below providesexperimental results that support thepopular theory that a plasma insidethe bubble causes sonoluminescence.The research team fitted their pulses’spectra to a blackbody radiation curveand found the correspondence toplasma temperatures at about 8000 K.The gas in the bubble becomes apartially ionized plasma, and the radi-ation is emitted by an energy cascadefrom ions to electrons and finally tophotons.

More details will be understoodeventually as faster optical responsesystems become available to better fol-low the time development of the lightemission process. In fact, how quicklya state-of-the-art photodetector systemoperates is measured against what ini-tial parts of the sonoluminescent pulseof light can be discerned!

Baghdassarian, O., H.-C. Chu, B. Tabbert,and G. A. Williams. “Spectrum of Lumi-nescence from Laser-Created Bubbles inWater.” Physical Review Letters 86 (2001):4934.

149. Siphoning Liquid

Helium

At temperatures near absolute zero,normal liquid He I becomes superfluidliquid He II by undergoing a second-order phase transition. Its He atoms can move without viscosity in thesuperfluid. Superfluidity is a quantummechanical phenomenon, with amacroscopic volume (centimeter dimen-sions) of liquid acting like a singlemacroscopic particle and described by asingle-particle Schrödinger equation.

Immediately, superfluid He II in anopen beaker will form a film thatcrawls up the walls, over the top, anddown the sides until the beaker is emp-tied. Normal fluids also can besiphoned out of containers, but only iftheir motion is started externally! Thesolid surfaces in contact with He II arecovered with a film 50 to 100 atomsthick along which frictionless flow ofthe liquid occurs. Supposedly, masstransport flow in the He II film takesplace at a constant rate that dependsonly on temperature.

As the atoms of liquid He II moveup the wall, they gain potential energy.What process provides the energy?

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The answer lies in the ability of heliumatoms to wet any surface—that is, nor-mal liquid He I atoms cling to thewall. The helium-helium force is theweakest force in nature because the Kshell of electrons is complete and thehelium zero-point motion is signifi-cant, so the helium-anything force isstronger. Hence helium atoms wouldrather be next to anything other thananother helium atom. So He atomsquickly form a film when presentedwith the wall of the container becausethe helium–anything attraction lowersthe potential energy and so on, whilethey gain gravitational potentialenergy. These He atoms clinging to thewall are no longer in the superfluidphase because their flow velocities arenow lower than a critical velocityvalue.

The thickness of the film is usuallylimited to a few hundred atomic diam-eters because at some thickness theadvantage of being near to the wall iscanceled by the increase in gravita-tional potential energy. Then, whilethe normal fluid is clamped to thewall, the superfluid He II flows freelyas the He atoms on the wall act as asiphon.

Goodstein, D. L. States of Matter. Engle-wood Cliffs, N.J.: Prentice-Hall, 1975, p. 327.

150. Quantized Hall

Effect

In a two-dimensional metal or semi-conductor, the standard Hall effect isobserved, but at low temperatures, aseries of steps appear in the Hall resist-ance as a function of magnetic fieldinstead of showing the typical monoto-nic increase. By confining the electronsystem in the third dimension to con-fine the electron gas to two dimen-sions, only specific electron wavefunctions meet the boundary condi-tions, so only certain quantized energylevels are available for the electrons.These steps in the Hall resistance occurat incredibly precise values of resist-ance, which are the same no matterwhat sample is investigated—that is,the resistance is quantized in units ofh/e2 divided by an integer. This amaz-ing result is the quantized Hall effect.

Recall that electrons have a spin

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0 10 20 30

Magnetic Field (Tesla)

RH

(h/e2)3

2

1

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1/2 and obey the Pauli exclusion prin-ciple. As electrons are added to anenergy band, they fill the availableenergy band states, just as water fills abucket. The states with the lowestenergy are filled first, followed by thenext higher ones. At absolute zerotemperature T = 0 K, the energy levelsare all filled up to a maximum energycalled the Fermi level. At higher tem-peratures one finds that the transitionregion between completely filled statesand completely empty states is gradualrather than abrupt and described bythe Fermi function, which has a valueof 1 for energies that are more than afew times kT below the Fermi energy,equals 1/2 if the energy equals theFermi energy, and decreases exponen-tially for energies that are a few timeskT larger than the Fermi energy.

Consider the ideal case of a fixedFermi energy and a changing appliedmagnetic field. In the presence of themagnetic field, the density of electronenergy states in 2-D is no longer con-stant as a function of energy andbunches into discrete energy levels,called Landau levels, of finite widthseparated by the cyclotron energy,with energy regions between theLandau levels where there are noallowed electron states. As the mag-netic field is swept to higher values, theLandau levels move relative to theFermi energy.

When the Fermi energy lies in agap between Landau levels, there areno available states to scatter into, sothere is no scattering, and the electricalresistance falls to zero. The Hall resist-ance for the Hall current cannotchange from the quantized valuewhenever the Fermi energy is in a gapbetween Landau levels, so one meas-ures a plateau. Only when the Fermienergy is in the Landau level can theHall voltage change and a finite resist-ance value appear.

Kivelson, S., D.-H. Lee, and S.-C. Zhang.“Electrons in Flatland.” Scientific American274, no. 3 (1996): 86–91.

151. Integrated Circuits

Heat dissipation is the biggest prob-lem in ICs. Simple, old-fashioned ther-mal energy limits the density ofelectronic components. The ability tominiaturize continues to improve, butunless thermal energy production pervolume is decreased or new geometri-cal paths for thermal energy transportaway from the sources are devised, thegame is lost. At present, 3-D ICs offera temporary reprieve, but even theywill find their limit.

Some short time scale solutionsmay be possible. The best thermal con-ductor among crystalline materials isdiamond, so going to a diamond sub-strate may be a solution. However, the

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technology of diamond is not yet com-petitive with silicon technology. Also,components on these substrates thatrequire significantly less energy tofunction as a gate would delay theoverwhelming impact of thermal prob-lems. Optical information transferbetween components would eliminateelectrical currents and their thermaleffects, but silicon does not have theright optical properties; hence theactive research into doping silicon tomake the desirable optical properties.

At the longer time scale of severaldecades, perhaps the silicon and semi-conductor technology will simply fadeaway in favor of some other technol-ogy on the time horizon that seemsunachievable today but that wouldbecome viable then. Or the newertechnology hasn’t even been dreamedabout yet!

For any solid or liquid material,quantum disturbances from cosmicrays may decide the ultimate limit inelectronic component density unlessredundancy can solve this problem.For optical systems based on lightinterference, and so on, who knowswhat is possible? Whatever wins in thedecades ahead will be numerousorders of magnitude smaller andfaster, as well as more robust thanwhat we have today.

152. Atomic

Computers?

Yes. One can use electron spin direc-tions as binary holders, for example.Even the nuclear spins can join in thegame. Quantum computers alreadyuse nuclear spins for storage. On a big-ger scale, DNA molecules are beingused for a DNA computer.

Several difficulties in makingatomic computers exist, but all of thedifficulties can be overcome by clevertechniques. Putting information in andreading information out of theseatomic systems have been done in thelaboratory already. Maintaining theirfixed states is a different kind of prob-lem that depends on which type of sys-tem is being used. Nuclear spin systemshave been used quite successfully sincethe 1940s with the development ofnuclear magnetic resonance (NMR).Electron spin systems also are con-trolled quite nicely in labs. If isolationof the system is required, then vacuumchambers work well for long enoughtime periods of particle isolation.

At the other extreme are proposedquantum computers utilizing the caf-feine molecules in a cup of java. Theyare being bombarded constantly by theother molecules in the liquid, so theliquid environment brings about arapid decoherence of the system. How-ever, there are an awful lot of caffeine

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molecules in the cup, at least 1020. Thequantum computer requires probablyonly a million or so to retain their iso-lation for the duration of the computa-tion time—microseconds, perhaps—sothe numbers may win out.

Like the limits to integrated circuitcomponent density caused by thermaleffects and by cosmic ray bombard-ment, atomic computers also may facesimilar limits. The type of atomic com-puter devised will determine how hos-tile the environment can be.

153. X-ray Laser?

The mechanism for the intense X-raysource appears to be the followingaccording to K. DasGupta, the origi-nator of this unique X-ray source. TheW X-rays from the Cu-W X-ray tubeknock out K shell electrons and othersin the Cu atoms in the external Cucrystal to produce a temporary (about10–15 second) population inversion,and the Cu X-rays coming simultane-ously from the same tube then stimu-late transitions in these Cu atoms toproduce the Cu Kα1 line at the Braggangle to the Cu(111) atomic planes.This mechanism is very selective, theline being so narrow and intense andthe process being so efficient that onedoes not detect any of the competingCu Kα2 emission to the available 1sstate. The single-frequency intense

X-ray line has been used to analyzematerials in minutes that formerlyrequired hours to days to accumulateenough data.

Whether the population inversionfor the 2p–1s transition in the externalCu atoms actually occurs is unknown.The emission X-ray line is uncharac-teristically narrow and intense, and theabsence of the other competing lineindicates that whatever the selectionprocess is doing must be very efficient.Other element sources such as nickel,based on the same mechanism, alsohave been made.

DasGupta, K. “CuKa1 X-ray Laser.” PhysicsLetters A 189 (1994): 91–93.

154. Bose-Einstein

Condensate

A Bose-Einstein condensate is formedat the coldest temperatures, whichmeans that the atoms have beenslowed in their motion to be almoststationary. By the de Broglie relation,each atom of mass m has a de Brogliewavelength λ = h/p, where p is itsmomentum mv and h is Planck’s con-stant. As the velocity v is furtherreduced to cool the atoms, the deBroglie wavelength increases accord-ingly. Eventually temperatures arereached for which the wavelengths ofadjacent and nearby atoms begin tooverlap in space considerably. Further

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cooling places all the atoms in intimatecontact in one collective quantumstate. Individual atoms can no longerbe discerned because they act like onebig “atom.”

The first Bose-Einstein condensatewas achieved in 1995, even though thephysics principles have been knownsince Einstein and Bose proposed themin the 1920s. About 2000 rubidiumatoms in the gas were cooled to 170nanoK when they formed a Bose-Ein-stein condensate less than 100 micro-meters across. The condensate lastedfor about 15 seconds and was cooledfurther, all the way down to 20 nanoK.

Anderson, M. H., et al. “Observations ofBose-Einstein Condensation in a DiluteAtomic Vapor.” Science 269 (1995): 198.

Castin, Y., R. Dum, and A. Sinatra. “BoseCondensate Make Quantum Leaps andBounds.” Physics World (August 1999): 37.

Cornell, E. A., and C. E. Wieman. “TheBose-Einstein Condensate.” ScientificAmerican 278, no. 3 (1998): 40–45.

Townsend, C., W. Ketterle, and S. Stringari.“Bose-Einstein Condensation.” PhysicsWorld (March 1997): 29–34.

155. Quantum Dots

Quantum dots are crystals, essentiallymetal or semiconductor boxes, con-taining only a few hundred atoms and a well-defined number of elec-trons. The number of electrons can be controlled by the electrostaticenvironment. The trick is to adjust

how many electrons end up in eachquantum dot.

An electron in a 3-D box is con-strained to have a quantum mechanicalwave function that matches the bound-ary conditions for the Schrödingerwave equation, producing discreteenergy levels that are inversely pro-portional to the square of the wave-length. As the box is made smaller, theenergy levels become farther apart. Ifthe quantum dot diameter—that is,box diameter—is made small enoughin fabrication, only a few energy levelswill exist inside for the electron. Henceone can make quantum dots smallenough to allow only one fluorescencetransition possible in the visible part ofthe spectrum.

The data from the first quantum dotspectrum showed a rich harmonic seriesof transitions between electron energylevels. Subsequent tweaking of the elec-trostatic potential was shown to reducethe dot size and increase the energyspacings. Later researchers have beenable to magnetically link together quan-tum dots with the hope of makingarrays of them for quantum computing.

Flügge, S. “Particle Enclosed in a Sphere.” InPractical Quantum Mechanics. Vol. I. NewYork: Springer-Verlag, 1974, pp. 155–159.

Reed, M. A. “Quantum Dots.” ScientificAmerican 268, no. 1 (1993): 118–123.

Whitesides, G. M., and J. C. Love. “The Artof Building Small.” Scientific American285, no. 3 (2001): 38–47.

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Chapter 8Chances Are

156. Schizophrenic

Playing Card

According to the rules of QM, the finalstate should be the superposition ofthe two alternative falling directions,with equal amplitudes ψ1 for left andψ2 for right. But we never see a cardfall both ways simultaneously. Any airmolecule colliding with the card isequivalent to an observation, a meas-urement process, so QM rule 3 appliesand the outcome reduces to the classi-cal one, with equal probabilities P1 tofall to the left side and P2 to fall to theright side.

The term describing this reductionof the wave function to the classicalprobabilities that have no QM inter-ference is often called decoherence.The Schrödinger equation, which isdeterministic, controls the entireprocess.

Tegmark, M., and J. A. Wheeler. “100 Yearsof Quantum Mysteries.” Scientific Ameri-can 284, no. 2 (2001): 68–75.

157. Schrödinger’s Cat

In QM, it is irrelevant whether youactually peek or not. If in principle youcould have determined the status of thecat, QM reduces to the classical result

by rule 3. The cat is now either alive ordead, not both. The two QM alterna-tives reduce to just one possibility.

Note that this example with the catbrings the connection between the non-intuitive behavior of Nature on themicroscopic scale up to the macroscopicscale of our everyday experiences.There has been an enormous amount ofcontroversy over this example and itsinterpretation. Some of the issues arediscussed in the references listed below.

Albert, D. Z. “Bohm’s Alternative to Quan-tum Mechanics.” Scientific American 277,no. 5 (1994): 58–67.

Loeser, J. G. “Three Perspectives onSchrödinger’s Cat.” American Journal ofPhysics 52 (1984): 1089–1093.

Wick, D. The Infamous Boundary: SevenDecades of Heresy in Quantum Physics.New York: Copericus Books, 1996, pp.149–152.

Yam, P. “Bringing Schrödinger’s Cat to Life.”Scientific American 276, no. 6 (1997):124–129.

158. Wave Functions

No. Beyond three dimensions there isno direct one-to-one correspondencebetween many-dimensional configura-tion space coordinates and the three-dimensional coordinates of positionspace.

The misconception referred to hereshows up in discussing the wave func-tion for two-particle systems, espe-cially when the discussion refers to thetwo-particle wave function reducing to

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the classical result. One often encoun-ters questions about how the wavefunction can reduce instantaneously tothe result, as if there has been somefaster-than-light information transfer.Fortunately, the two-particle wavefunction reduces in configurationspace, not in position space!

Hilgevoord, J. “Time in Quantum Mechan-ics.” American Journal of Physics 70 (2002):301–306.

Mermin, N. D. “Is the Moon There WhenNobody Looks? Reality and the QuantumTheory.” Physics Today 38, no. 4 (1985):38–47.

Styer, D. F. “Common MisconceptionsRegarding Quantum Mechanics.” Ameri-can Journal of Physics 64 (1996): 31–34.

Wick, D. The Infamous Boundary: SevenDecades of Heresy in Quantum Physics.New York: Copericus Books, 1996, pp.162–166.

159. Wave Function

Collapse?

The original wave function Ψ = ψ1 +ψ2 + ψ3 + . . . will change. The probephoton did not scatter off the electronin particular imaginary boxes, so weknow immediately that the wave func-tion should not include their ampli-tudes. One could say that there hasbeen a partial collapse of the wavefunction even though there has beenno interaction! We believe that thisgedanken experiment was discussedfirst by physicist Robert H. Dicke inthe reference below.

Dicke, R. H. “Interaction-Free QuantumMeasurements: A Paradox?” AmericanJournal of Physics 49 (1981): 925–930.

Hilgevoord, J. “Time in Quantum Mechan-ics.” American Journal of Physics 70 (2002):301–306.

160. Quantum

Computer

A quantum computer relies on main-taining its linear superposition ofquantum states—that is, Ψ = ψ1 + ψ2 +ψ3, its coherence during the calcula-tions—so that all the states participatein the calculation. Quantum decoher-ence is a bad thing for a quantumcomputer. A collision with the wall ofthe chamber or with another moleculewill ruin the coherence because anobservation has been made. By QMrule 3, we no longer sum over theamplitudes ψi. This decoherence thenruins the quantum computationbecause only one state will be partici-pating in the computations.

Maintaining coherence in a realphysical system has been progressingslowly for the past decade, with coher-ence times of tens of nanoseconds forthree identical subsystems working asa quantum computer. No one knowswhat type of physical system will com-pose the first 18-subsystem quantumcomputer in the future, but this com-puter probably will outdo all the otherclassical computers combined in com-puting speed.

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Awschalom, D. D., M. E. Flatté, and N.Samarth. “Spintronics.” Scientific American286, no. 6 (2002): 67–73.

Lloyd, S. “Quantum-Mechanical Comput-ers.” Scientific American 273, no. 4 (1995):140–145.

Nielsen, M. A. “Rules for a Complex Quan-tum World.” Scientific American 287, no. 5(2002): 67–75.

161. Cup of Java

Quantum Computer

Coffee contains caffeine molecules,which may be useful as quantum sub-systems for a quantum computerbecause they contain two rings in aplane with many attached hydrogenatoms. The nuclear spin states of the Hatoms attached to the rings can beused for information storage à laNMR. That is, a nuclear magnetic res-onance (NMR) system is a collectionof nuclear spin states in an externalmagnetic field that tend to align thespins. In the simplest ideal case at tem-perature T, the external magnetic fieldB is uniform and there are two spinstates, up and down. Let’s say that Baligns most of the spins to the up state,with the ratio of down to up spinsbeing determined by the exponentialfactor Exp(– µB/kT), where µ is thenuclear magnetic moment and k is theBoltzmann constant. An externalradiofrequency pulse of the properfrequency v and energy hv = 2 µB can

flip a down spin to an up spin for astimulated absorption transition orcan cause a stimulated emission of aphoton by a spin flip from up to down.

Now for some coffee. The liquidcontains about 1020 caffeine molecules.Even if we assume that all of them par-ticipate initially in bunches as coherentstates of many quantum computers inthe cup just before the calculation,most bunches will experience colli-sions during the calculation time of ananosecond, say, and drop out fromthe collection of coherent states of thesystem. However, a significant numberof bunches of coherent states may beparticipating still when the calculationsare done, and these will provide astrong signal above the backgroundnoise. At least that’s the hope!

Gershenfeld, N., and I. Chuang. “Bulk SpinResonance Quantum Computation.” Sci-ence 275 (January 17, 1997): 350.

162. Bragg Scattering

of X-rays

Bragg scattering requires λ < d; there-fore there will not be any collectivescattering from a group of scatterers atdifferent atoms within one wave-length. The actual scatterers of the X-rays are the electrons at each atomin these planes of the crystal. Coherentscattering requires fixed phase rela-tionships, but there is no fixed phase

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relationship between electrons at dif-ferent atoms nor between the electronsdoing the scattering at any moment.Therefore, the X-rays scattered intothe Bragg angle have a multitude ofrandom phases and not fixed-phaserelationships. The scattering probabil-ity is proportional to N, the number ofscatterers, and not N 2 , as it would befor coherent scattering.

Here is the QM argument mathe-matically. Let ψi represent the proba-bility amplitude to scatter an X-ray atthe i th atom. We know from QM rule 2that Ψ = ψ1 + ψ2 + ψ3 + . . . , for alter-native ways to go from the X-raysource to the crystal to the X-ray detec-tor. Each ψi represents one atom, andwe assume single scatterings on theway to the detector for simplicity. Eachψi = exp[iδ] φi , which includes a phasepart exp[iδ] and the identical scatteringamplitude φi at the identical atoms inthe crystal. If the phase part at eachscattering atom is identical, then wewould have Ψ = N ψ1 and the proba-bility P = N 2 |ψ1|

2, giving us coherentscattering proportional to N 2.

However, there is no correlatedmotion between electrons on differentatoms, so their phases are random. Ifthe phase differences between scatter-ers—that is, the electrons on differentatoms—are not fixed differences, thenthe sum is over random phases and,like the random walk problem, the

total amount is proportional to √Ninstead of N. Therefore Ψ = √N ψ1, soP = N |ψ1|

2. The Bragg scattering of X-rays is not a coherent scatteringprocess.

163. Beautiful Faces

Coherent scattering of light by theatoms in the skin is the reason for ourability to see details of a face. Theambient incident light is scattered bythe molecules of the skin. Two factorsare significant for this two-step scat-tering process: the time intervalrequired and the number of coherentscatterers. In the visible region of theelectromagnetic spectrum, this scatter-ing process occurs in atoms in lessthan 10–8 second over an area of theskin involving about a million atomswithin a circle with a radius of aboutone wavelength of the light. The wave-length of greenish light is about 500nanometers.

Consider scattering one incidentphoton at a time. During the scatteringtime of a single photon by these onemillion alternative paths there isalmost no movement of the scatteringatoms in the molecules, so alternativepaths have essentially fixed phase rela-tionships. By QM rule 2, ψ = ψ1 + ψ2 +ψ3 + . . . , and ψ = N ψ1 with probabil-ity P = N2 |ψ1|

2, giving us coherentscattering proportional to N 2. With

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incoherent scattering we would not seemuch detail.

In the UV, both factors are smallerthan for light in the visible spectrum—the scattering occurs in less time, andthe area for each scattering is less andinvolves fewer atoms because thewavelength is much less. The face seenin the UV would appear grainier withless detail because the adjacent coher-ent scattering areas are smaller and the shorter time interval means thatthey will have some effects of almost-random phases.

In the IR, most of the scatteringinvolves molecular transitions, whichare relatively slow processes, so thescattering process involves a muchlonger time interval. But each mole-cule itself is completely involved in thescattering. So even though the wave-length is large, involving many morescattering centers, the molecular scat-terers move significantly during the IRscattering process, producing randomphases everywhere and a smearing ofthe image.

Organisms of many different typessee in the UV and/or in the IR to findtheir nourishment, as well as in the vis-ible. However, we humans evolvedwithout being able to see either the UVor the IR, our vision being confined tothe visible part of the electromagneticspectrum. Why our eye-brain systemevolved in this way is not known.

164. Gravitational

Waves

Yes, the coherent scattering of gravita-tional waves is expected to occur, withthe scatterers being mass quadrupoles—that is, mass pairs in the antenna. J. Weber, the same physicist who firstcalculated the classical cross section forgravitational wave scattering in 1959,proposed in 1981 that the coherentscattering of gravitational waves wouldenhance the scattering cross section forcertain detectors by a factor of 106 ormore. The larger cross section mightexplain the large responses of his twoindependent one-ton cylindrical alu-minum bar gravitational wave detectorsevery time either end faced the nucleusof the Milky Way galaxy, approxi-mately twice per day. If his proposal fora coherent scattering response is cor-rect, then solid bar antennas would bemuch more sensitive to gravitationalwaves than large interferometers withtheir small masses at the mirrors such asLIGO and VIRGO.

The QM calculation can be out-lined as follows. With wavelengths inthe kilometer range being much longerthan the size of the Al bar antenna inthe lab, all the mass pair quadrupolesin the antenna are within this onewavelength. Hence, their responses areapproximately in phase, and each masspair offers an equivalent alternative

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scattering path. By QM rule 2, Ψ = ψ1

+ ψ2 + ψ3 + . . . , and Ψ ~ N ψ1 withprobability P = N 2 |ψ1|

2, giving uscoherent scattering proportional toN 2, where N is the total number ofmass pairs in the bar, about 1024.However, the bar is actually composedof many microcrystallites, so one reallysums the QM amplitudes over thenumber of mass pairs within eachmicrocrystallite, then sums the proba-bilities over all the microcrystallites.The coherent scattering probability isstill more than 10 million times larger(after accounting for the crystallinedefects) than the classical non-coher-ent scattering response that Weber firstcalculated in 1959.

Whether any bar antenna for grav-itational waves behaves as a coherentscatterer has not been unambiguouslydemonstrated. Instead of the classicalresult with the bar oscillating at its res-onant frequency and its harmonicswhen hit by a pulse of gravitationalwaves, the coherent scattering barwould essentially have an almost equalresponse to a wide range of frequen-cies. The actual experimental barresponses are complicated and requireelaborate methods to find gravita-tional wave scattering signals buriedin background noise.

If the Weber bars were reallydetecting gravitational waves from thegalactic nucleus, there is an enigma

when the original classical responsecross section is used. The rate of con-version of mass to energy at the galac-tic nucleus should have devoured thewhole galaxy by now! I suppose thatwe must wait for LIGO and VIRGOto detect and calibrate gravitationalwaves before we truly know whethergravitational waves can scatter coher-ently in Weber bar antennas.

Gibbs, W. W. “Ripples in Spacetime.” Scien-tific American 286, no. 4 (2002): 62–71.

Preparata, G. “Superradiance Effect in aGravitational Antenna.” Modern PhysicsLetters 5 (1990): 1–5.

Weber, J. “Gravitons, Neutrinos, and Anti-neutrinos.” Foundations of Physics 14(1984): 1185–1209.

165. Coherent Neutrino

Scattering

In 1984, so the story goes, J. Weberproposed to build a detector for thecoherent scattering of neutrinos in aproposal for research monies. Theproposal review committee challengedhim to write up the neutrino coherentscattering idea and publish the paperin a reputable physics journal. InDecember 1984 he submitted thepaper, “Method for Observation ofNeutrinos and Antineutrinos,” toPhysical Review C, and the paper wasaccepted by a referee within eight daysof the December 12 reception date!

The paper triggered an enormous

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response in parts of the physics com-munity. Numerous rebuttals of hisarguments appeared in the physics lit-erature within months after this publi-cation, but all of these rebuttals can berefuted. Every paper erroneouslyassumes that the nuclear scatterers actas potentials. Wrong! Weber shows inthe first section of the paper that suchan assumption cannot lead to coherentscattering for neutrino wavelengthsless than the spacing between nuclei.However, everyone seems to ignore thedetails presented by Weber, who cor-rectly explains why the nonrelativisticcalculation does not predict coherentneutrino scattering for neutrino wave-lengths less than the atomic spacing.The QM argument is essentiallydependent on the fact that the scatter-ing phases among the nuclei will berandom, leading to a scattering proba-bility proportional to N instead of N 2.

In later parts of the paper Weberdoes the relativistic QM scattering cal-culations to show that coherent scat-tering for all energies occurs—that is,neutrinos of all energies will suffercoherent scattering. Included in thesecalculations are terms involving thestiffness of the defect-free crystal, andso on. The conceptual idea is thatwhen the crystal as a whole recoils,like a Mössbauer Effect scattering,then one cannot determine (even inprinciple) where the nuclear scattering

of the neutrino took place. Hence theirresponses are in phase and offer equiv-alent alternative scattering paths. Onemust sum the amplitudes over all pos-sible paths—that is, all nuclei—toobtain the total amplitude for theneutrino scattering.

By QM rule 2, Ψ = ψ1 + ψ2 + ψ3 +. . . , and Ψ = N ψ1 with probability P = N2 |ψ1|

2, giving us coherent scat-tering proportional to N 2, where N isthe total number of nuclei in the bar,about 1023. One gains the enormousfactor of 1023 for neutrino scatteringover the noncoherent cross section!The only remaining contention iswhether all the phase relationships areproperly accounted for in this rela-tivistic calculation.

Weber (now deceased) actuallyconducted several experiments tocheck his relativistic calculations for along defect-free single crystal detector.He claims to have verified the turningon and the turning off of a nuclearreactor in blind tests, the leaking of tri-tium from a highly radioactive tritiumsource, and the twice-daily passing ofthe Sun though the long axis of hiscrystal detector. In 1995 he deter-mined that the total measured solarflux of neutrinos—all three types,because the detector did not distin-guish among them—was equal to thetotal neutrino flux expected by thestandard solar model. This predicted

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result agrees with the 2002 resultsreported by the heavy water detectorat the Sudbury Neutrino Observatory(SNO).

Ho, T. H. “Comments on the ‘Method forObservation of Neutrinos and Antineutri-nos.’” Physics Letters 168B (1986): 295.

Weber, J. “Method for Observation of Neu-trinos and Antineutrinos.” Physical ReviewC 31 (1985): 1468–1475.

166. Magnetic

Resonance Imaging

(MRI)

Nuclear magnetic resonance experi-ments began in the 1940s, and theycontinue to be very useful today. Theiralternative QM behavior is describedas a collection of spins acting together.Initially, the spin collection has a totalspin S in a collective quantum state Ψ= ψ1 + ψ2 + ψ3 + . . . and then thepulsed magnetic field rotates them allso slightly to S – α with respect to theoriginal direction—that is, they actcollectively and coherently. No onespin behavior is isolated from the oth-ers in the same microscopic atomicenvironment. All hydrogen nuclei inthe same environment respond thesame, while those in a different envi-ronment respond slightly differently.

The MRI instrument for magneticresonance imaging uses the differencesin the microscopic atomic environment

to allow different regions of the livingtissue to be “seen” separately. A com-puter algorithm analyzes the data fromnumerous RF detectors surroundingthe body and constructs an artificialimage on a display screen. A dynamicMRI instrument has a fast responsetime to show changes occurring in the microscopic environment in sec-onds or less, such as muscle action orheart contractions.

167. Heisenberg

Uncertainty

The uncertainty principle places nolimit to the accuracy of measuring theparticle’s position. The uncertaintyprinciple ∆px∆x ≥ h/4π forbids thesimultaneous measurement of bothposition and momentum in the samedirection to arbitrary accuracy, not anindividual measurement. Of course,practical design limitations exist thatprobably limit the measurement, butconceptually there is no limit. Thesame argument applies separately tothe momentum.

An application of the Heisenberguncertainty principle to the hydrogenatom is an insightful example. Thehydrogen atom is usually solved inspherical polar coordinates instead ofCartesian coordinates. In sphericalpolar coordinates, the uncertaintyrelations are a bit more complicated

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and the consequences can be some-what bizarre. For example, since thehydrogen wave function for the elec-tron about the z-axis—that is, in the φdirection—is known precisely for the1s atomic state, and hence the angularmomentum has no uncertainty, theuncertainty in φ is maximum. There-fore, in the φ direction, one finds anequal probability at all angles, produc-ing the smeared-out probability distri-bution in φ.

Many other uses for the uncer-tainty relation exist because it lies atthe very heart of quantum mechanics.However, one can see that any descrip-tion of a phenomenon using waves ofany kind will require an uncertaintyrelation. Engineers are familiar withthe fact that about a one-MHz band-width is required to reproduce a one-microsecond pulse: ∆f∆t ~ 1, forexample. Suppose there is a single-fre-quency wave defined by y = y1 sin k1x.This wave extends from – ∞ to + ∞,and the question “where is the wavelocated?” has no answer. By adding

together many single-frequency wavesof different frequencies with properlychosen amplitudes and phrases, wecan build up a lump in a narrowregion of space of approximate length∆x. The range of wavelengths ∆λneeded can be represented by the cor-responding range of wavenumbers ∆k.The approximate mathematical rela-tionship ∆ x ∆k ~ 1 can be establishedby considering several examples, asseen in the Krane reference below.

Bohr’s famous measurement dis-turbance argument is faulty. For half acentury physicists have regurgitatedthis argument of how the uncertaintyprinciple acts to defend quantum the-ory. In experiments that first refutedBohr’s argument, a beam of coldrubidium atoms is split to travel alongtwo different paths; call them A and B.The beams still overlap and combineat the end of their journeys to createan interference pattern. Now theresearchers looked to see which paththe atoms followed by tweaking thoseon path B into a higher energy state bya pulse of microwaves. These atoms in

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Two waves added

Twenty waves added

Collimated atoms

Collimator

µ waves Screen

Slit plate

A

B

Atom wave

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their internal states kept a record ofwhich path they had taken. Themicrowave pulse absorbed by an atom is insignificant by a factor ofabout 10,000 and can cause littlechange to the atom’s momentum, notenough to smear the interference pat-tern. Yes, QM worked still. With themicrowaves off, interference fringesappear. Turn them on so you can tellwhich path was taken, and the inter-ference pattern vanishes. The uncer-tainty principle is correct still, but theargument that “measurement disturbsthe system” to explain the double slitexperiment is wrong.

So what may be the deeper mecha-nism at work in the double slit experi-ment, for example? Perhaps quantumentanglement, in which every particleis linked to every other particle it hasinteracted with. Two-particle wavefunctions are linked in a six-dimen-sional configuration space with noone-to-one correspondence to physical3-D space, so the entanglement of Nparticles will be described by a wavefunction in 3N-dimensional configura-tion space with no one-to-one corre-spondence to 3-D physical space. Andnow the mathematics becomes messier!

Dürr, S., and G. Rempe. “Can Wave-ParticleDuality Be Based on the Uncertainty Rela-tion?” American Journal of Physics 68(2000): 1021–1024.

Englert, B.-G., M. O. Scully, and H. Walther.“The Duality in Matter and Light.” Scien-tific American 271, no. 6 (1994): 86–92.

Krane, K. Modern Physics, 2nd ed. NewYork: John Wiley & Sons, 1995, pp.93–106.

Styer, D. F. “Common MisconceptionsRegarding Quantum Mechanics.” Ameri-can Journal of Physics 64 (1996): 31–34.

Wick, D. The Infamous Boundary: SevenDecades of Heresy in Quantum Physics.New York: Copericus Books, 1996, pp.152–156.

168. Vacuum Energy?

There is always the zero-point energyin the vacuum. Whatever QM modelfor the vacuum is considered, all canbe reduced in a first approximation toa large number of harmonic oscilla-tors, which have a zero-point energyvalue that is non-zero. At present, QMcalculations of the energy density ofthe vacuum seem to be too large by atleast 30 orders of magnitude! The vac-uum energy density should be about10–11 J m–3 if this vacuum energy is thesource of the accelerated expansion ofthe universe determined by the Type1a Supernova measurements in 1998.

One can do an energy estimateusing the Heisenberg uncertainty prin-ciple. Or, if the vacuum has an effec-tive potential for a scalar field, theproduct of the visible matter densityand the potential will give the energydensity for an assumed radius of theuniverse. In either case, the assump-tions necessary to estimate this energydensity would take us too far astray.

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However, we can determinewhether an electrically neutral particleof mass ∆m popping into existence fora time interval ∆t can be detected by itsgravitational field. We use the uncer-tainty relation ∆E∆t ≥ h/4π in the formc2 ∆m∆t > h/4π. Suppose we have themost sensitive detector, a free particleof mass M initially at a distance Raway from ∆m; then in the Newtonianapproximation the detector willreceive a pulse P = F∆t. SubstitutingF = GM∆m/R2 into the uncertaintyrelation produces GM∆m∆t/R2 ≥GMh/(4πR2c2).The initial state of thedetector also obeys the uncertaintyrelation ∆P∆X ≥ h/4π, so that ∆m tobe noticeable requires the impulse P tobe greater than about 2∆P, or ∆X ≥4R (R/rg), where the Schwarzschildradius of the detector rg = GM/c2. Forobjects ranging in size from protons toplanets, rg lies within the object itself.So the momentum transferred by theimpulse will not be detected!

Haroche, S., and J.-M. Raimond. “CavityQuantum Electrodynamics.” ScientificAmerican 268, no. 4 (1993): 54–62.

Ostriker, J. P., and P. J. Steinhardt. “TheQuintessential Universe.” Scientific Ameri-can 284, no. 1 (2001): 46–53.

Stefanski, B. Jr., and D. Bedford. “VacuumGravity.” American Journal of Physics 62(1994): 638–639.

169. Casimir Effect

Although the classical vacuum is avoid, the quantum mechanical vacuum

is a soup of virtual particle-antiparticlepairs that interact with the real atomsin the metal plates, these pairs beingcreated and annihilated in extremelyshort time intervals in accordance withthe Heisenberg uncertainty principle.That is, the more the total energy ∆Ein the pair, the less time duration ∆t isits existence so that ∆E∆t ≥ h/4π. Thisvacuum pair “soup” pushes inward atboth plates when the plates are veryclose to each other because certainparticle-antiparticle pairs are practi-cally forbidden from momentarilyappearing between them. Essentially,if their deBroglie wavelength exceedsthe plate spacing, these pairs have amuch lower probability to be betweenthe plates. But these same pairs appearoutside the plates and provide theadditional forces, whence the netinward force. Known as the Casimireffect, it was first measured in 1958.

The Casimir force is too small to beobserved for plates that are not withinmicrons of each other. Two mirrorswith an area of 1 cm2 separated by adistance of about 1 µm have an attrac-tive Casimir force of about 10–7 N.Although this force seems very small, atdistances of less than a micrometer theCasimir force becomes the strongestforce between two neutral objects! Atseparations of 10 nanometer—roughly100 times the size of an atom—theCasimir effect produces a force that isthe equivalent of 1 atmosphere of pres-

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sure. The resurgence of interest in theCasimir force is because micromechan-ical devices on the scale of tens of nano-meters must accommodate its effects!

Haroche, S., and J.-M. Raimond. “CavityQuantum Electrodynamics.” ScientificAmerican 268, no. 4 (1993): 54–62.

Kleppner, D. “With Apologies to Casimir.”Physics Today 43, no. 10 (1990): 9–11.

170. Squeezing Light

Classically, a ray of light is an electro-magnetic wave having an amplitudeand a phase, both being expressed interms of the electric field componentsEx and Ey. Quantum mechanically, thenormal modes of the electromagneticfield are quantized and treated as anensemble of harmonic oscillators, oneharmonic oscillator per normal mode.The number of photons in each har-monic oscillator is the energy in thecorresponding oscillator. An harmonicoscillator obeys the Heisenberg uncer-tainty principle, so one expects the elec-tromagnetic field to behave likewise.

As the electric field in a light ray isreduced, even a ray from a lasersource, the fixed amount of intrinsicquantum noise in the light intensitybecomes more obvious. This quantumnoise in an electrical field is ever pres-ent. If you shine any light on a pho-todectector such as a photodiode,there will be fluctuations in the diodecurrent corresponding to the individ-ual photons being detected. One sees

that the photons are not spread outevenly in time nor in spatial extent.Heisenberg’s uncertainty relation dic-tates this behavior. The QM operatorsof phase- and amplitude-quadrature(i.e., for the perpendicular componentsof the E field) of the electromagneticfield do not commute, similar to posi-tion and momentum of a particle. Theproduct of phase- and amplitude-uncertainty has a fixed lower limit.The more precisely the phase of a lightwave is measured, the less determinedbecomes its amplitude and vice versa.States of the light with the smallestpossible amount of overall quantumnoise are minimum uncertainty states.

The reduction in quantum noise inone observable of the light (e.g., thephase) at the expense of enhancing itin the complementary observable (i.e.,the amplitude) can be done by para-metric amplification procedures. Theresulting states of the light are calledsqueezed states, since the quantumnoise got squeezed at a particularphase angle. Their wave packets oscil-late in time and get wider and nar-rower—that is, they breathe.

Alternately, the uncertainty in theamplitude of a laser beam can bereduced to a level below that normallyallowed by the Heisenberg uncertaintyprinciple, a level known as the zero-point quantum noise level. However,this increased knowledge comes at theexpense of greater uncertainty in the

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frequency of the light. Essentially, oneis using an uncertainty relation of theform ∆Ex∆Ey ≥ V, where V is a con-stant. Reducing the uncertainty in Ex

to gEx means that the uncertainty in Ey

becomes Ey /g to keep their productthe same.

Experiments with squeezed lightpromise to enhance our understandingsof quantum mechanics at the individualatom and photon levels. Recently, anew type of ultraprecise laser pointermade by “squeezing” a beam in twodirections was able to position thebeam with a precision of 1.6 Å, about1.5 times better than the theoreticallimit for a conventional laser.

Treps, N., et al. “A Quantum Laser Pointer.”Science 301 (2003): 940–943.

171. Electron Spin

Yes. Although the vacuum influenceon the electron spin is extremely small,the same effect of the vacuum on themuon’s spin has been measured atBrookhaven National Laboratory. Theinteraction magnitude is predicted bythe Standard Model (SM) of Leptonsand Quarks and their interactions. Allfundamental particle-antiparticle pairsmomentarily appear in the vacuumand disappear sporadically, so theelectron (and muon) see them all, ifonly for a fleeting moment. This vac-uum “soup” is slightly magnetic, so it

increases the magnetic moment of theelectron or muon to g = 2 (1 + a). Thesmall correction of about 0.12 percentis called the anomalous moment but isoften referred to as “g-2.” Its meas-urement with gradually increasingaccuracy presents spectacular agree-ment with calculation to better than24 parts per billion.

The muon is 206 times heavierthan the electron, so the muon’s mag-netic moment is 206 times smaller, butthe virtual particles in the quantumsoup can be more massive. As a result,the anomalous moment is 40,000times more sensitive to undiscoveredparticles and new physics at short dis-tances. There is agreement to 4 partsper million that must be regarded asthe best test of the theory, but there isalso a small discrepancy that needs tobe explained, a difference in mean val-ues of the experiments and the theoryby 2.6 standard deviations.

The muon g-2 result cannot atpresent be explained by the establishedSM. Recalculations of the predictedtheoretical value continue, and correc-tions have been done. Moreover, the g-2 calculation involves three of the four fundamental interactions—weak, electromagnetic, and color—sothere are many Feynman diagramsthat contribute.

Perhaps this unresolved g-2 differ-ence is the harbinger of new physics

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beyond the SM, such as new quarks,or supersymmetric particles, or a sur-prise in the vacuum.

Bennett, G. W., et. al. “Measurement of thePositive Muon Anomalous MagneticMoment to 0.7 ppm.” Physical ReviewLetters 89 (2002): 101804.

172. Superconductivity

The paired electrons in superconduc-tors that are in the superconductingstate show Bose-Einstein condensationto a single macrostate. There is somesmall energy width to this macrostatebecause the pairs are composed of spin1/2 particles, and they are showingremnant Fermi-Dirac behavior: notwo identical fermions can ever be inthe same state as defined by their four-momenta and spins no matter howthey behave collectively.

173. Superfluidity

The odd number of constituents in He-3 (two protons, one neutron, andtwo electrons) classifies it as a fermionthat obeys Fermi-Dirac statistics. So notwo He-3 atoms can share the samequantum state defined by the four-momentum (energy and three-momen-tum) and spin. The surprise in the early1970s was that He-3 can magneticallycouple with another He-3 to form aboson and become a superfluid liquidat the extremely low temperature of

2.7 millikelvins. The He-3 pairs formone momentum macrostate. Becausethe component He-3 atoms are notbosons, there should be some smallwidth to the macrostate momentum inaddition to the small width because theHe atoms are composed of fermions.

The pairs of atoms are magnetic,so the He-3 superfluid is more com-plex than its He-4 counterpart. In fact,superfluid He-3 exists in three differ-ent phases related to different mag-netic or temperature conditions. In theA phase, for example, the superfluid ishighly anisotropic—that is, directionallike a liquid crystal.

Scientific American. Special briefing on theNobel Prizes in Science, “A New Super-fluid.” Scientific American 276, no. 1(1997): 15–16.

174. Gap Jumping

This Josephson effect is really quan-tum mechanical tunneling across thephysical gap because the wave func-tion for the superconducting pairextends beyond the end of the materialinto the gap and to the other side. Ifthe superconducting material is actu-ally in the form of a ring, then match-ing of the wave function for the pairaround the ring must be made,restricting their angular momentumquantization to multiples of h/2π.

Clarke, J. “SQUIDS.” Scientific American271, no. 2 (1994): 46–53.

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175. Nuclear Decay

The wave function extends throughthe potential barrier to the outsideworld. Therefore the probability to beoutside the nucleus is not zero. So whydoes the wave function itself extendinto the barrier? All confinement prob-lems, classical and quantum, havesolutions with functions that extendinto the barrier, usually decreasingexponentially to almost zero within afew wavelengths. Atomic particleshave relatively long wavelengths com-pared to the barrier thickness. So whydoes the wave function itself not endin the barrier? Because the effectivebarrier height decreases with radialdistance.

The probability to tunnel throughthe barrier is proportional to Exp[–Ar√(U(r)–E)], where E is the energyof the incident particle, U(r) is the bar-rier potential as a function of distancer, and A is a constant that includesPlanck’s constant h. Some closelyrelated problems to be treated as tun-neling through a barrier are:

1. Bare copper wire is cut and the twoends are twisted together. In spite ofthe fact that the copper is coatedwith copper oxide, the twisted endsstill conduct electricity readily.

2. Tunnel diode operation.

3. Scanning tunneling microscope.

For a discussion about how a

nuclear decay rate may be influencedby its environment, see the Peres refer-ence below.

Halliday, D., and R. Resnick. Fundamentalsof Physics. New York: John Wiley & Sons,1988, pp. 1009–1010.

Peres, A. “Zeno Paradox in Quantum The-ory.” American Journal of Physics 48(1980): 931–932.

176. Total Internal

Reflection

Yes, the light goes a little beyond theinterface. One can treat this behavioreither classically or quantum mechan-ically. In QM the wave function forthe photon extends beyond the glass-air interface into the air.

You can see this behavior in thefollowing manner. Fill a drinking glasspartially full with water. Tilt the glassand look down into it at the side wallat such an angle that the light enteringyour eye has been totally internallyreflected from the wall. The wall willlook silvery when this condition holds.Then press your moistened thumbagainst the outside of the glass. Youwill see the ridges of your fingerprintbecause, at those points, you will haveinterfered with the total reflectionprocess. The valleys between theridges are still far enough away fromthe glass that the reflection hereremains total and you simply see a sil-very whorl.

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177. Annihilation

Fermi’s Golden Rule hints that weshould consider the phase space avail-able for the final particles, and thisphase space is related to the entropy ofthe final particles. If the entropy of thefinal state is greater than the entropyfor the initial state, the process occurs.In the simpler case, when an electronat rest and its antiparticle, the positronat rest, annihilate each other, two pho-tons are produced to conserve quan-tum numbers as well as energy andmomentum. The entropy of the prod-ucts is greater than the reactants.Why? Because there is much freedomin the direction of the photon polar-izations. The interacting particle andantiparticle begin with their spinsopposite but along a specific direction,thereby having a total spin of zero. Inthe final state with two identical pho-tons emerging in opposite directions inorder to conserve energy and linearmomentum, the photon spins areopposite—that is, both spin +1 or bothspin –1 with respect to their momen-tum directions—but the polarizationvectors can be in any direction in theplane perpendicular to the momentumdirections.

178. A Bouncing Ball

We present a simplified version of thecomplexities of this bouncing ball

action. The compression of the ball(and the concrete being struck) sendsphonons (quantum sound waves) run-ning around telling the material thatcompression is occurring and that theincreased energy density in parts of theball can be reduced by expanding backto its normal size. Of course, theexpansion overshoots and the ball“rings” as it leaves the concrete, eachextended state also increasing theenergy density in parts of the ball. Onecan model much of this behavior byassuming that the atoms and mole-cules are in a potential well somewhatsimilar to the parabolic well of theharmonic oscillator. However, insteadof a potential energy for an atom ver-sus the atomic separation distancebeing proportional to r2 only, theremust be additional terms proportionalto r3, etc., where r is the distance fromthe equilibrium location.

Eventually the phonons help theball get back into its normal shape, butthe atoms and molecules never quitemake their initial relative positionsagain, there being some residual dis-tortion. Even the concrete being struckby the ball never quite recovers. Wit-ness the eventual wear of a concretehighway by cars and trucks compress-ing the road, a more vigorous processbut conceptually the same.

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179. The EPR Paradox

There seems to be no classical thinkingthat would reproduce the data set. Apredetermined instruction set wouldbe akin to an algorithm for generatingrandom numbers—but no such sets ofnumbers are truly random. One mustaccept the conclusion that Nature isquantum mechanical and thereforeclassical physics is only an approxima-tion. The rules of QM agree with theresults, but the details are too compli-cated mathematically to presentherein. The references provide theextended discussion.

Even more surprising is the sugges-tion that locality is violated. That is,information from the first detectorpasses to the second detector withoutpassing through imaginary sphericalsurfaces surrounding each, as if moredimensions exist in our world! Some-one, someday, will determine a funda-mental reason for this behavior ofnature.

Eberly, J. H. “Bell Inequalities and QuantumMechanics.” American Journal of Physics70 (2002): 276–279.

Einstein, A., B. Podolsky, and N. Rosen.“Can Quantum-Mechanical Description ofPhysical Reality Be Considered Complete?”Physical Review 47 (1935): 777–780.

Mermin, H. D. “Is the Moon There WhenNobody Looks? Reality and the QuantumTheory.” Physics Today 38 (1985): 38–47.

Shimony, A. “The Reality of the QuantumWorld.” Scientific American 258, no. 1(1988): 46–53.

Silverman, M. P. A Universe of Atoms, anAtom in the Universe. New York: Springer-Verlag, 2002, pp. 92–102.

von Baeyer, H. C. Taming the Atom: TheEmergence of the Visible Microworld.Mineola, N.Y.: Dover, 1992, pp. 210–211.

180. Information and a

Black Hole

For certain, one should worry aboutquantum information loss, especiallyif quantum mechanics is to provide acomplete explanation for everything inthe world. Does the black hole infor-mation increase with the inclusion ofthe chair? Let’s see. A black hole hasmass, spin, and possibly electriccharge, weak charge, or color charge.That’s all! We cannot determine theinformation content of the black holefrom these quantities only. That is aproblem. The most likely solution thatwould prevent quantum informationloss is that the surrounding space just outside the event horizon of theblack hole takes care of the informa-tion equation to make everythingcorrect, emitting particles to compen-sate correctly.

The actual physics calculation ofinformation change in the gravita-tional field of a black hole is muchmore complex and difficult. Amongthe necessary concerns is the fact thatthe black hole has performed a non-unitary transformation on the state of

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system when it devoured the chair. Anon-unitary evolution is excluded in aquantum theory because it fails to pre-serve probability—that is, after a non-unitary evolution, the sum of theprobabilities of all possible outcomesof an experiment may be greater orless than 1. Quantum mechanics couldnot survive. Perhaps the QM of ablack hole will eventually be done andquantum gravity will save us from thiscatastrophe!

Bekenstein, J. D. “Information in the Holo-graphic Universe.” Scientific American 289,no. 2 (2003): 58–65.

Chapter 9Can This Be Real?

181. Carbon-14 Dating

The ratio of C-14 to C-12 in livingorganisms will depend on many fac-tors, including the local climate andthe amounts of C-14 in the atmos-phere, factors that can vary on timescales as short as tens of years. Theradiocarbon dating process assumes in its zeroeth order approximation no variation in these factors overhundreds and thousands of years. Butthe cosmic ray intensity reaching theatmosphere may vary considerably, so the amount of C-14 produced will

also vary. As the variations in thecosmic rays are determined by otherindependent methods, they can beincorporated into the C-14 dating asadjustments.

According to research literature,tree ring counts indicate that C-14 dat-ing has fluctuations of the C-14 con-centration in the atmosphere between1400 and 1700 B.C.E. Furthermore, acomparison of radiocarbon-deter-mined ages with ages of archaeologicalmaterials accurately established byother methods reveals that for theperiod from 100 B.C.E. to 1400, radio-carbon dating gives values that are toolarge, and that prior to 100 B.C.E. theradiocarbon values are too small.

At about 1600 B.C.E., the C-14date values are about 175 years (5percent) too small, increasing to about300 years (6 percent) at 3000 B.C.E.The discrepancy appears to be a resultof slight variations in Earth’s magneticfield over the years, which would alterthe cosmic ray intensities and hence C-14 production in the atmosphere.These corrections allow C-14 dates tobe corrected, and even for 100,000years ago the radiocarbon dates aregood to within 5 percent.

Staff of McGraw-Hill, eds. McGraw-HillEncyclopedia of Science & Technology. Vol.15. New York: McGraw-Hill, 2002, pp.136–144.

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182. Nuclear Energy

Levels

Even the shell model, often called theindependent particle model of thenucleus, fails to correctly predict manyof the energy level spacings unless thespin-orbit LS interactions are included.That is, the proton and neutron mag-netic moments interact with magneticfields produced by their orbitalmotions. These LS interactions addterms to the approximate constantpotential of the shell model to domi-nate the quantum state sequence insidethe nucleus. As a result, many energylevels change their relative positionson the energy scale, with levels fromdifferent principle quantum numbersbecoming interchanged! Once the LSinteraction was properly accountedfor, all its predictions were shown toagree with the empirical data.

This model of the nucleus alsoexplained why nuclei containing aneven number of protons and neutronsare more stable than others. Like theenergy levels for the electrons in quan-tum states outside the nucleus, theFermi exclusion principle allows twoidentical particles per quantum stateonly. The nuclear quantum states forthe protons are separate from thenuclear quantum states for the neu-trons, and any particular state is filledwhen there are two identical particleswith opposite spins. The proton levels

are higher in energy than the corre-sponding neutron levels because thereis the added Coulomb repulsion. Anyextra proton or neutron can be added,but this additional particle mustoccupy a higher energy state, usuallyleading to an unstable nucleus.

Jolie, J. “Uncovering Supersymmetry.”Scientific American 287, no. 1 (2002):70–77.

Serway, R. A. Physics for Scientists & Engineers with Modern Physics, 3rded. Philadelphia: Saunders, 1990, pp.1352–1354.

Tipler, P. A. Modern Physics. New York:Worth, 1978, pp. 427–432.

183. Nuclear Synthesis

Look at the binding energy curve forthe elements and you will see that atleast one isotope of Ni is well bound.Unfortunately, this isotope has a rapiddecay mode. In fact, all the Ni isotopesfrom Ni-49 to Ni-57 have half-lives ofonly milliseconds to at most 10 days.

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50 55 60 65

Mass number A

8.80

8.78

8.76

Cr-52

Cr-54Ni-60

Ni-62Fe-58

Fe-56

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Although the championship ofnuclear binding energy is often attrib-uted to Fe-56, this isotope actuallycomes in third. The most tightly boundof the nuclei is Ni-62. The bindingenergies are 8.790 MeV/nucleon for Fe-56 and 8.795 MeV/nucleon for Ni-62.The binding-energy curve shows thosenuclides that are close to the peak.

The most tightly bound nuclidesare all even-even nuclei. Fe-56 is abouta factor of ten more abundant in starsthan Ni-62. The Fewell referencebelow indicates that the reason lieswith the greater photodisintegrationrate for Ni-62 in stellar interiors. Oth-ers have suggested that the very lowrate of multistep production of Ni-62from Co-59 is the culprit.

Fewell, M. P. “The Atomic Nuclide with theHighest Mean Binding Energy.” AmericanJournal of Physics 63 (1995): 653–658.

Shurtleff, R., and E. Derringh. “The MostTightly Bound Nucleus.” American Journalof Physics 57 (1989): 552.

184. Heavy Element

Synthesis

The synthesis of the heavier elementsbeyond Fe is done during supernovaexplosions, in a few days or less, andthe atomic debris are spewed out intospace to later collect into new starsand planets and be there for incorpo-ration into life forms.

The fusion process for elements upto Fe in the periodic table yields energy,

and thus they occur in the normal stel-lar burning cycles. But since the “irongroup”—those elements with isotopemass number of about A = 60—is at the peak of the binding energycurve, the fusion of elements above Ferequires energy, with the exception ofthe most tightly bound isotope—Ni-62, for example.

The elements beyond Fe areexpected to be formed in the cata-clysmic explosions known as super-novae in which a large flux ofenergetic neutrons build up massapproximately one unit at a time toproduce the heavy nuclei. Followingneutron capture, some isotopes betadecay to change a neutron into a pro-ton plus an electron and an electronantineutrino, increasing the atomicnumber by one unit. Some samplesequences are:

Fe-56 + n → Fe-57 (stable)

Fe-57 + n → Fe-58 (stable)

Fe-58 + n → Fe-59 → Co-59 bybeta decay

Co-59 + n → Co-60 → Ni-60 bybeta decay

In principle this process could con-tinue indefinitely, but the elementsbeyond uranium (Z = 92) are allradioactive.

The layers of the star containingthe buildup of heavy elements may beblown off by the supernova explosionto provide the raw material of heavy

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elements within expanding hydrogenclouds that much later can condense toform new stars, planets, and the stuffof life.

Krane, K. S. Modern Physics, 2nd ed. NewYork: John Wiley & Sons, 1995, pp.290–291.

185. Neutron Decay

The failure of the neutron in a nucleusto decay is a quantum mechanicaleffect. According to quantum mechan-ics, the rate of decay is dictated byFermi’s Golden Rule, which states thatthe rate is proportional to the proba-bility of decay (i.e., the absolute valueof the square of the matrix elementconnecting the initial and final states)times the density of final states.Because the free neutron decays to aproton plus electron plus electron anti-neutrino, we know that the probabil-ity for this beta decay process is notzero and that there are available finalstates for the three product particles.Energy conservation dictates that thetotal final state energy equals the totalinitial energy of the free neutron.

Inside a nucleus, the decay of aneutron is a transition from an initialenergy state, the particular bound neu-tron state that the neutron occupies, toa final state consisting of a proton insome final proton energy state plus afree electron and a free electron anti-neutrino, the latter two particles

contributing to the energy of the finalstate. Therefore, energy conservationdictates that the proton will be in aproton energy state that is lower inenergy than the initial energy of theneutron. In many nuclei all availableproton states—that is, those that arenot occupied by protons—have higherenergies than the energy of the initialneutron state, so the decay cannotoccur.

The equivalent energy levels of theprotons in nuclei are higher than forthe neutrons because their energiesinclude the Coulomb repulsionbetween two protons and other prop-erties of the nuclear force, especiallythe spin dependence. Obviously thestable nuclei will include those forwhich neutron and proton decays donot occur!

Asimov, I. Understanding Physics. New York:Hippocrene, 1988, p. 245.

186. Finely Tuned

Carbon?

The crucial energy comparison tomake is not simply the radioactivestate energy of 7.65 MeV to the prac-tical limit value of 7.7 MeV, but onemust include the comparison of theradioactive state energy 7.65 MeV tothe energy 7.4 MeV of the reactants atrest. This energy of 0.25 MeV missesbeing too high for the production of

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carbon by the fractional amount of0.05 MeV/0.25 MeV, or 20 percent,which is not so critical after all.

Barrow, J. D., and F. J. Tipler. The AnthropicCosmological Principle. Oxford: OxfordUniversity Press, 1986, pp. 252–253.

Livio, M., D. Hollowell, A. Weiss, and J. W.Truran. “The Anthropic Significance of theExistence of an Excited State of C-12.”Nature 340, no. 6231 (1989): 281–284.

Weinberg, S. Facing Up: Science and ItsCultural Adversaries. Cambridge, Mass.:Harvard University Press, 2001, pp.235–237.

187. Proton-Proton

Cycle

The other stars are using the carboncycle for their fusion energy. The com-mon proton-proton cycle reaction isnot the source of fusion energy inmany stars burning hydrogen becausethe first reaction in this sequence hastwo protons combining to form adeuteron H-2, a very unlikely eventthat occurs slowly. A more likelysequence of reactions involves havingC-12 be a catalyst:

C-12 + p → N-13 + γ

N-13 → C-13 + e+ + ν

C-13 + p → N-14 + γ

N-14 + p → O-15 + γ

O-15 → N-15 + e+ + ν

N-15 + p → C-12 + He-4

Called the carbon cycle, this sequenceof reactions occurs much more rapidlythan the proton-proton cycle sequencebecause the C-12 acts as a catalyst,neither being produced nor consumedby the totality of reactions. The netprocess is still the same: 4 protons →He-4, and the net energy produced isthe same, but the rate of energy pro-duction is much higher.

The carbon cycle occurs at a highertemperature than the proton-protoncycle because the C and H Coulombrepulsion is greater than the H and Hrepulsion, so the Sun, with its internaltemperature of about 15 × 106 K, istoo cool to activate the carbon cycle,which requires about 20 × 106 K.

Krane, K. S. Modern Physics, 2nd ed. NewYork: John Wiley & Sons, 1995, pp.282–285.

188. Oklo Nuclear

Reactor

The reaction sequence shows how tobreed Pu from local U-238, which isthe most abundant naturally occurringuranium isotope. Initially, neutronscome from the fission of U-235. How-ever, the very high abundance of U-238 means that this isotope willabsorb some of the neutrons tobecome U-239, decay by beta decay toneptunium 239, and then decay to Pu-239. Some of the resulting Pu-239undergoes fission.

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U-238 + n → U-239 → Np-239 +e– + anti-ν

Np-239 → Pu-239 + e– + anti-ν

However, because the natural reactorsat Oklo probably operated for such along time, the Pu-239 had time todecay by alpha decay to U-235. Thusthe Oklo natural reactors were truebreeder reactors, fissioning more U-235 than originally existed in thereactors. The evidence for the breederprocess remains in the reactor as moreof the fission products than could pos-sibly be produced by the amount of U-235 that has been lost from each of thereactor sites.

A second piece of evidence for Pufission is the isotopic composition ofthe fission products in the mass range100 to 110. To breed Pu and addi-tional U-235, the reactors must haveoperated for periods significantlygreater than the half-life of Pu 239,about 24,360 years.

Cowan, G. A. “A Natural Fission Reactor.”Scientific American, no. 7 (1976): 36–47.

189. Human

Radioactivity

At one time in the history of radiationsafety, before extensive and long-termmeasurements, the recommended radi-ation limit was much less than thelimit today. During those times, in the mid-1900s, two people in close

proximity would have been emittingenough gamma radiation to exceed therecommended limit.

We can estimate the exposureamount and compare its value to therecommended limit today. There areapproximately 105 decays of K-40 iso-topes per second in your body, but thedecay chart tells us that only about 11percent yield a gamma ray, producingabout 1100 self-inflicted gamma raysper second, amounting to about 0.36mSv per year, well below the recom-mended limit today. Even a group of 10people closely packed would not pro-vide a radiation exposure more than 3.6mSv per year. So we are not radiationdangers to ourselves nor to our friends!

Cohen, B. L. “Catalogue of Risks Extendedand Updated.” Health Physics 61 (1991):317–335.

190. Nuclear Surprises?

Both are true statements.1. The only emissions from a

nuclear power plant are (a) watervapor from its cooling towers, (b)thermal energy in the external coolingwater, (c) any stray gamma rays notshielded (unlikely to be above normalbackground), (d) any radioactive iso-topes created in the external coolingwater (unlikely to be above normalbackground), and (e) electrical energy.

The emissions and safety proce-dures at a coal-burning power plant

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are not as strict and, because all coalnaturally contains radioactive materialwith many isotopes, some of theseradioactive isotopes escape into the airwhen the coal is piled in storage, whenthe coal is burned, and so on. Measure-ments at coal-burning plants verify thatradioactive atoms and molecules arereleased.

Scientific researchers in theMcBride et al. reference below haveconcluded from measurements “thatAmericans living near coal-fired powerplants are exposed to higher radiationdoses than those living near nuclearpower plants that meet governmentregulations. . . . The fact that coal-firedpower plants throughout the world arethe major sources of radioactive mate-rials released to the environment hasseveral implications. It suggests thatcoal combustion is more hazardous tohealth than nuclear power and that itadds to the background radiation bur-den even more than does nuclearpower. It also suggests that if radiationemissions from coal plants were regu-lated, their capital and operating costswould increase, making coal-firedpower less economically competitive.”

G. J. Aubrecht, in the referencebelow, states that the radioactivitydanger from each coal-burning electri-cal plant is at least 100 times the dan-ger from each nuclear plant.

2. Background radiation levelscombining terrestrial (from K-40,

Th-232, Ra-226, etc.) and cosmicradiation (photons, muons, etc.) arefairly constant over the world in therange of 8–15 µrads per hour. Assum-ing maximum damage to human tis-sue, this present background radiationlevel corresponds to about 1.8 mSv peryear.

If one spreads all the human-pro-duced artificial radioactive materialsequally around the surface of Earth,the local increase in radioactivity isexpected to be minuscule compared tothis indigenous natural radioactivebackground. Suppose we had a millionmetric tonnes of human-made radioac-tive material to be dispersed overEarth of approximately 5 × 1014 m2.Each square meter would acquire anadditional 0.2 × 10–5 kg of radioactivematerial, compared to the naturalamount of radioactive material in thetop 10 centimeters of about 2 × 10–2

kg, producing an insignificant amountof local radiation unless the half-liveswere short, on the order of minutes todays. The additional amount addsonly 1 part in 10,000 when dispersedaround the globe.

Aubrecht, G. J. Energy, 2nd ed. Upper SaddleRiver, N.J.: Prentice Hall, 1994.

Cohen, B. L. “Catalogue of Risks Extendedand Updated.” Health Physics 61 (1991):317–335.

Eisenbud, M. Environmental Radioactivity:From Natural, Industrial, and MilitarySources, 4th ed. San Diego, Calif.: Acade-mic Press, 1987.

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McBride, J. P., R. E. Moore, J. P. Wither-spoon, and R. E. Blanco. “RadiologicalImpact of Airborne Effluents of Coal andNuclear Plants.” Science 202 (1978): 1045.

191. Cold Fusion

Cold fusion at room temperature is areal but unlikely possibility. The keyidea is the quantum mechanical over-lap of the wave functions of twonearby H-2 nuclei, for example. Theirwave functions always overlap, nomatter how far apart they are. How-ever, the bigger the value of the wavefunction overlap, the more probablewill be the possibility of the fusionprocess to make a He-4 nucleus.

Of course, there is a Coulomb bar-rier to be overcome. In the 1940s camethe proposal that muonic atoms—aproton with a muon replacing the elec-tron—might allow fusion because themuonic atom ground state puts themuon so close to the nucleus on aver-age that the muonic atom appears neu-tral to the approaching proton.However, calculations have shownthat the produced He isotope decaystoo quickly for this scheme to succeedin fusion energy production.

In gaseous form at room tempera-ture, two colliding H-2 nuclei do notget close enough for a large wave func-tion overlap, being strongly repelled byelectrical forces acting between twopositive nuclei. In a solid, however, at

room temperature, H nuclei in neigh-boring lattice sites experience enor-mous accelerations, as large as 1014

m/s2 in random directions. Sometimesthese accelerations are toward eachother, so the two protons can approachvery close and perhaps fuse into a Henucleus. However, the actual calcula-tion reveals the rarity of this event.

Despite the extreme improbabilityof deuteron fusion at room tempera-tures, so-called cold fusion, researchgroups worldwide continue its pursuit,as revealed in the references below.

Iwamura, Y., T. Itoh, M. Sakano, and S.Sakai. “Observations of Low-EnergyNuclear Reactions Induced by D2Gas Per-meation through Pd Complexes.” InfiniteEnergy 47 (January–February 2003):14–18.

Mallove, E. F. “The Triumph of Alchemy:Professor John Bockris and the Transmuta-tion Crisis at Texas A&M.” Infinite Energy32 (July–August 2000): 9–24.

Miles, M. H., B. F. Bush, and J. J. Lagowski.“Anomalous Effects Involving ExcessPower, Radiation, and Helium Productionduring D2O Electrolysis Using PalladiumCathodes.” Fusion Technology 25 (1994):478–486.

192. Fission of U-235

There are two major problems to beovercome in designing a fission device.The neutron distribution in a pure U-235 solid would decrease as theinverse distance squared from eachnuclear decay source, and the target

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nuclei would be moving away duringthe expansion, so one has a diffusionproblem complicated by moving tar-gets. The moving targets contribute atleast two difficulties: the density oftargets is rapidly decreasing, and theneutron-capture cross section is afunction of neutron kinetic energy asseen from the reference frame ridingwith each U-235 nucleus. Without theproper neutron capture rate by thereceding U-235 nuclei, the chain reac-tion fizzles out.

Of course, the nuclear device can-not be expected to be pure U-235because the isolation of enough quan-tities of U-235 from U-238 is too diffi-cult and too costly. Therefore, there ismostly U-238 in the expanding solidwith some U-235, so we have all thepreviously listed problems to solve butalso must account for the nuclearproperties of the U-238 as well as theU-235.

Apparently the Germans duringWorld War II did not solve these diffu-sion problems satisfactorily.

Bernstein, J. “Heisenberg and the CriticalMass.” American Journal of Physics 70, no.9 (2002): 887–976.

193. Minimal Nuclear

Device

Pure U-235 can be accumulated into acritical mass for a sustained nuclear

chain reaction, but pure Pu-239 can-not start itself because the loss rate ofneutrons exceeds the production rate.On the average, each U-235 fissionproduces 2.5 neutrons for every inci-dent neutron. At the critical mass offissile material the chain reaction willbe sustained. For U-235 this criticalmass is about 7 kilograms for idealbehavior, requiring a sphere about thediameter of a baseball of pure U-235.Surely this baseball would be too hotto handle!

Diffusion problems of an expand-ing material would require a neutron-reflecting strong tamper materialsurrounding the U-235 sphere to delaythe expansion for a few microsecondsto achieve additional fissions beforeexploding. At 100 percent efficiencythe explosion would be equivalent toabout 120 kilotonnes of TNT. How-ever, no nuclear device is that efficient.

Declassified records indicate thatabout 60 kilograms of highly enricheduranium was used in the nuclear devicethat was released over Hiroshima,Japan, in 1945. The explosive chargefor the device detonated over Nagasakithree days later was provided by about8 kilograms of plutonium-239 (>90percent Pu-239).

Bernstein, J. “Heisenberg and the CriticalMass.” American Journal of Physics 70, no.9 (2002): 887–976.

Pochin, E. Nuclear Radiation: Risks andBenefits. Oxford: Clarendon Press, 1983.

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194. Large Nuclei

Small nuclei that become excited anddeformed prefer to lose their energy bybreaking up into helium nuclei (alphaparticles) or C-12 nuclei wheneverpossible. In fact, researchers often talkabout “nuclear molecules” composedof these two entities.

The larger nuclei, with more than150 nucleons, usually spin faster whenenergy is added, and the result of ahigher angular momentum state is anucleus that is more deformed. As theyde-excite, up to about 40 gamma raysare emitted by descending an “excita-tion ladder,” producing a characteris-tic gamma ray emission spectrum.From this spectrum one can determinethe nuclear angular momentum statesand the nucleus’s deformation shape.Superdeformed nuclei were discoveredin this way.

Rotational motion of quantumobjects such as atoms and moleculeshas a long and distinguished physicshistory. Quantized rotational motionof molecules was first recognized fromthe absorption spectra of infrared lightin 1912. The occurrence of rotationalmotion of atomic nuclei first became atopic of interest in the late 1930s in aneffort to explain observed nuclearexcitation spectra by physicistsEdward Teller and John Wheeler inabout 1938.

Quantum mechanics dictates the

shapes. Upon excitation, the nucleusfirst deforms into a shape like a rugbyfootball, with a length-to-height ratioof about two to one. Mg-24 appears tobehave as if two C-12 nuclei are itsmajor components and seems to behaveas a superdeformed nucleus in thisrugby football shape. The next statewould have an elongated hyperde-formed shape as a result of perhaps sixalpha particles lined up along the longaxis. This nucleus is highly unstable,and this nuclear sausage would producean unmistakable debris pattern.

Recent detailed investigations ofseveral Pb isotopes have yielded sur-prises. The angular distribution andpolarization of the gamma rays showthat they were not electric quadrupole(E2) transitions but magnetic dipole(M1). Classically, M1 radiation is pic-tured as being emitted from a rotatingcurrent loop, with the field oscillatingat the same frequency as the frequencyof rotation. Similar gamma-ray emis-sion bands have recently been identi-fied in other nuclei in the mass regionaround 110, where the nuclei also arenearly spherical. These spectra have apattern that is typical of transitionsbetween rotation states, which posesan awkward problem: how can weexplain these regular patterns of M1gamma rays? Apparently there ismuch more to understand.

Clark, R. M., et al. “Evidence for ‘MagneticRotation’ in Nuclei: Lifetimes of States in

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the M1 Bands of Pb-198, Pb-199.” PhysicalReview Letters 78 (1997): 1868.

Macchiavelli, A. O., et al. “SemiclassicalDescription of the Shears Mechanism andthe Role of Effective Interactions.” PhysicalReview C. 57 (1998): R1073.

Nolan, P. J., and P. J. Twin. “SuperdeformedShapes at High Angular Momentum.”Annual Review of Nuclear and ParticleScience. 38 (1988): 533.

195. Human Hearing

The Mössbauer Effect has been usedto determine these actual displace-ments of an eardrum. The MössbauerEffect utilizes the recoil-less emissionof a 14.4 KeV gamma ray (photon)from an Fe-57 nucleus, say, and thisgamma ray is normally absorbed byan Fe-57 nucleus in another object inits path. When the emitters (Fe-57atoms placed on the eardrum) aremoving with the eardrum, the emittedgamma rays pass through the secondobject, a cooled thin film of Fe con-taining some Fe-57 atoms, to be cap-tured in a gamma-ray photon detector.

The important physical propertyhere is that the natural linewidth of theemitted gamma ray is very narrow,

about 10–8 eV, so that the Fe-57 recoilenergy of about 0.002 eV produces aDoppler shift so large that no absorp-tion in the cooled, thin film normallyoccurs. One can cancel this Dopplershift with a moving absorber or emit-ter of only 0.0002 m s–1. Therefore,when the eardrum moves forwardtoward the stationary cooled thin film,there will be some resonance absorp-tion of the gamma ray, so the detectorcount will decrease. When theeardrum moves opposite, there is noabsorption. Because the eardrumvibrates in a nonlinear fashion, thedetails are somewhat more compli-cated. From the geometries and thephysical properties of the emission andthe Mössbauer absorption, the dis-placement values of the eardrum canbe calculated. The sensitivity of thistechnique allows eardrum displace-ments that are only fractions of anuclear diameter to be detected.

196. 1908 Siberia

Meteorite

Willard Libby and Edward Tellerexplored this event with a reasonablehypothesis, since no rocky debris wasever found, and the amount of damagewas enormous. If the meteorite weremade of antimatter, then the ensuingmatter-antimatter annihilation in theatmosphere and during the ground

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collision would create plenty of ener-getic photons at 0.511 MeV, 935MeV, etc., due to electron-positronannihilations, proton-antiproton anni-hilations, etc. Many of these photonswould interact with nitrogen N-14 inthe atmosphere directly to make anexcited N nuclear state or indirectlyvia secondary neutron production inthe atmosphere. The excited state ofN-14 decays to C-14, which increasesthe atmospheric concentration of C-14in the carbon dioxide that is taken inby plants immediately after the event.

An increase in the C-14 to C-12ratio should appear in the radiocarbondating of living organisms such asplants, beginning in the year 1908 forlocal trees, and this increase in theratio should appear a few years laterfor trees in North America, caused byatmospheric mixing of the C-14. Oneof us (F. P.) was working for the sum-mer in Willard Libby’s laboratory andwas assigned to carefully separate thetree rings from an old oak tree, puttingpieces into vials, and then coding thevials so that only I knew which vialscontained which tree rings. The sam-ples were radiocarbon-dated, and thenthe results were plotted by the C-14 toC-12 ratio versus the calendar year.

W. Libby, E. Teller, R. Berger, L.Wood, and F. Potter did not publishtheir radiocarbon-dating results, whichdemonstrated a significant increase in

the 1911 C-14 to C-12 ratio detectedin the old oak tree from Wisconsin.Later, a research group of C. Cowan,C. R. Atluri, and W. Libby (1965) didpublish a similar result for the analysisof C-14 content in a 300-year-oldDouglas fir from Arizona showing anincrease in C-14 in 1911 with the sameinterpretation, supported by R. V.Gentry (1966). However, C-14 meas-urements of a tree by J. C. Lerman, W. G. Mook, and J. C. Vogel (1967)nearer the blast failed to show anincrease in the 1909 ratio.

Several other interpretations of the1908 meteorite event are possible.One of the main proposals is that anice-rock comet struck Earth, much likecomet Schumaker-Levy struck Jupiterin 1994. Also, one cannot rule out thepossibility that a massive rocky mete-orite just burned up completely—thatis, broke into small fragments thatburned up before striking the ground.

Chyba, C., P. Thomas, and K. Zahnle. “The1908 Tunguska Explosion: AtmosphericDisruption of a Stony Asteroid.” Nature361 (1993): 40–44.

Cowan, C., C. R. Alturi, and W. F. Libby.“Possible Anti-matter Content of the Tun-guska Meteor of 1908.” Nature 206(1965): 861–865.

Gentry, R. V. “Anti-matter Content of theTunguska Meteor.” Nature 211 (1966):1071–1072.

Lerman, J. C., W. G. Mook, and J. C. Vogel.“Effect of the Tunguska Meteor andSunspots on Radiocarbon in Tree Rings.”Nature 216 (1967): 990–991.

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197. The Standard

Model

As far as we know, no such definitiveargument for matching specific fami-lies exists in the Standard Model ofLeptons and Quarks and their interac-tions. As long as six leptons cancel outthe anomalies of the six quarks, forexample, all is well! Indeed, one canuse the second family of quarks to can-cel the anomaly contributions fromthe first family of leptons, the thirdfamily of quarks to cancel the secondfamily of leptons, and the first familyof quarks to cancel the third family ofleptons. In fact, any permutation ofthe traditional lineup of cancellationswould succeed.

This ambiguity in the cancellationsprobably indicates that the StandardModel as understood is incomplete.One would expect the traditionalscheme, but the conceptual under-standing provided by the StandardModel does not dictate uniqueness.

One of us (F. P.) has proposed aninteresting mathematical argument formatching lepton families to quarkfamilies based on correlations amongfinite rotational subgroups of the Stan-dard Model gauge group for the lep-tons and quarks. In this scheme, eachlepton family and each quark family isin a unique subgroup, and the one-to-one correlations are dictated by the

mathematics. Although the proposedscheme successfully predicted the massof the top quark, this geometrical basisfor the Standard Model awaits confir-mation of other specific predictions forcollisions, which are under way at Fer-milab and soon to be done using theLarge Hadron Collider.

Glashow, S. L. “Quarks with Color andFlavor.” Scientific American 233, no. 4(1975): 38–50.

Kane, G. “The Dawn of Physics beyond theStandard Model.” Scientific American 288,no. 6 (2003): 68–75.

Liss, T. M., and P. L. Tipton. “The Discoveryof the Top Quark.” Scientific American277, no. 3 (1997): 54–59.

Potter, F. “Geometrical Basis for the Stan-dard Model.” International Journal ofTheoretical Physics 33 (1994): 279–306.

198. Spontaneous

Symmetry Breaking

Yes. At least two other methods canachieve the same symmetry-breaking

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Electron neutrino

Electron

Muon neutrino

Muon

Tau neutrino

Tau

Up quark

Down quark

Charm quark

Strange quark

Top quark

Bottom quark

Top' quark

Bottom' quark

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result without requiring the Higgs par-ticle. The Standard Model is describedby its continuous gauge group SU(3)C

× SU(2)W × U(1)Y. The simplest way ofall is to spontaneously break this con-tinuous group to a discrete symmetrysubgroup of the continuous groupSU(2). That is, the lepton and quarkflavor eigenstates would be associatedwith finite rotational subgroups ofSU(2) instead of the continuous group.An analogy from geometry would beto begin with a sphere and then sym-metry-break to a regular tetrahedron,or a regular octahedron, or a regularicosahedron. Reconciling discretenesswith the continuous symmetry groupU(1) of quantum electrodynamics maybe a problem, however, where phasesare assumed to vary continuously.Another symmetry-breaking approachis the quark condensate method, whichalso does not require a Higgs particle.

At present, no Higgs particle hasbeen detected at the accelerators, eventhough its mass is expected to be below200 GeV/c2, within the energy range ofthe large accelerators. Of course, thedecay of such a Higgs particle is a fla-vor-changing neutral current weakdecay, which means that its decay rateis severely suppressed, so only a fewHiggs decays would have been detectedamong the particle debris so far. Whenthe Large Hadron Collider comesonline in 2005 or later there should be copious production of the Higgs

particle if this mechanism is truly thesource of symmetry breaking and theparticle masses. If the Higgs particledoes not show up, then spontaneoussymmetry breaking to a discrete groupremains an alternative possibility.

Coleman, S. Aspects of Symmetry. Cam-bridge, Eng.: Cambridge University Press,1985, pp. 113–130.

Icke, V. The Force of Symmetry. New York:Cambridge University Press, 1999, pp.232–248.

Potter, F. “Geometrical Basis for the Stan-dard Model.” International Journal of The-oretical Physics 33 (1994): 279–306.

’t Hooft, G. “Gauge Theories of the Forcesbetween Elementary Particles.” ScientificAmerican 242, no. 6 (1980): 104–140.

199. Proton Mass

Quantum chromodynamics describesthe interactions of the quarks. The up and down quark masses are listedas ~ 5 MeV/c2 each. However, these“current” quarks are not what ismeant by having them confined insidea proton by the color fields. Insteadone must use the effective mass—the“constituent” mass—which accountsfor this confinement and which can be estimated from the Heisenberguncertainty principle. Since δxδpx ≥h/4π, and each quark is confinedwithin the proton radius of about oneFermi, we estimate δpx ~ 100 MeV. In three dimensions, the total dp ~ ~ 170MeV/c2. So at least 510 MeV/c2 of the

( ) ( ) ( )δ δ δp p px y z2 2 2+ +

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proton mass is to be associated withthe “constituent” mass of the threequarks within the proton. The remain-der is the energy contributions of thegluons holding the proton together.

Most of the properties of protons,except the spin, seem to be determinedby these three “valence quarks,” muchlike the valence electrons determinethe important chemical properties ofatoms. However, when the proton’sinnards are probed more energetically,more structure is found, up to four orfive more particles, called “virtualquarks.” In addition, up to 30 gluonscan be detected. The proton is reveal-ing its inner sanctum to investigators,and the view is becoming quite inter-esting. Quarks, antiquarks, and gluonscan be said to form a thick “soup”inside the proton, and theoretical andexperimental physicists are workingtogether to figure out the recipe.

Today we know that the threevalence quarks cannot alone accountfor the proton’s spin. The whole “sea”of quarks, antiquarks, and gluons eachpossess spin, so one must first deter-mine the contribution made by eachindividual member of this seethingmass. The results so far suggest that thesea of quarks makes a minimal contri-bution to the overall spin of a nucleon!

Abbott, D., et al. “Measurement of TensorPolarization in Elastic Electron-DeuteronScattering at Large Momentum Transfer.”Physical Review Letters 84 (2000): 5053.

Aniol, K. A., et al. “Measurement of the Neu-tral Weak Form Factors of the Proton.”Physical Review Letters 82 (1999): 1096.

200. Right- and Left-

handed Neutrinos?

No. The weak interaction is associatedwith the SU(2)-weak part of the Stan-dard Model gauge group that operatesin the unitary plane—a plane with twocomplex axes. That is, particle funda-mental lepton and quark states aredefined in this unitary plane. All rota-tions in the normal unitary planeinvolve only left-handed doublets andright-handed singlets, dictated solely bythe mathematics of the geometricaltransformation. Mathematicians callthese transformations right and leftscrew operations. So the physical prop-erty of left-handed doublet states forthe weak interaction is dictated by themathematical property of rotations inthe unitary plane. Nature simply“knows” the mathematics!

The antiparticle eigenstates are inthe conjugate unitary plane, which isgauge-equivalent (not equivalent) tothe normal unitary plane, so theenergy values of particles and antipar-ticles are the same, but all other prop-erties are opposites. In this conjugateunitary plane the mathematics dictatesright-handed doublets and left-handedsinglets. The existence of two gauge-equivalent but different 2-D complex

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spaces conjugate to one another dic-tates that the universe has both parti-cles and antiparticles. Why there existso many more particles than antiparti-cles in our present universe remains tobe resolved.

Altmann, S. L. Rotations, Quaternions, andDouble Groups. Oxford: Clarendon Press,1986, pp. 121–123.

201. Physics without

Equations

The best way to use cellular automata(CA) on computers is to incorporatethe fundamental interactions of theStandard Model of Leptons andQuarks plus the gravitational interac-tion of the general theory of relativity,or preferably its quantum gravita-tional version when available. Weknow that all these fundamental inter-actions in nature correspond mathe-matically to local phase changes, aprocess that can be simulated with CAwithout using equations by using aclever enactment of the path-integralapproach to doing all of physics in realtime. Not yet fully achieved except byvery crude approximation, the physicsof many-particle interactions will beaccomplished by large-scale grid com-putation methods or perhaps by theequivalent on a quantum computer.

The fundamental idea is to deter-mine the present behavior of a particle

by summing over all the phaseinformation from its local environ-ment. Of course, each particle alsoprovides phase information to its envi-ronment both near and far. The parti-cle’s new location is the region wherethe phases match best. The calculationgame requires a dynamic limit to howmany nearby cells are counted in orderto accumulate a good approximationof the phase information and to main-tain the local geometrical symmetry. A proof-of-concept calculation hasbeen done by one of us (F. P.) on adesktop computer using thousands ofnodes in a 3-D array, but a good cal-culation requires millions of cells orthe equivalent.

The marriage between physics andmathematics has been a happy andfruitful one over many centuries.Mathematical equations, from simplealgebraic ones to the more challengingdifferential equations, have allowed usto summarize an enormous amount ofphysical phenomena into a simple for-mat. The underlying fundamentalsymmetries of nature have been thetrue source of many of these equa-tions. However, formulating thesesymmetries as the Schrödinger equa-tion and Maxwell’s equations, forexample, and solving the equations arehuman processes. We cannot expectNature to do the same when the sim-pler process of looking locally for

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information is more direct. Thereforewe think that understanding the uni-verse by combining CA with path inte-grals will be the physics of futuregenerations.

Icke, V. The Force of Symmetry. New York:Cambridge University Press, 1999, pp.178–206.

Chapter 10Over My Head

202. Olbers’ Paradox

German astronomer Heinrich Olbers(1758–1840) was not the first scientistto ask “Why is the night sky dark?”but his name remains connected to thisparadox. The night sky is dark becausethe time required for the radiation fieldto reach thermodynamic equilibrium islarge compared to all other time scalesof interest—that is, the lifetime of starsis far too short for the sky to be asbright as the paradox suggests. In addi-tion, if all the matter in the universewere converted to radiation, the equi-librium temperature of the universewould be about 20 K, illustrating thatthere is insufficient energy to have abright sky. Edward R. Harrison in theearly 1970s determined this solutionand also determined that the usualexplanation, based on a cosmologicalredshift of the light from distant

sources, was not needed even thoughits argument would likewise produce adark night sky.

The critical quantity is the ratio ofthe average lifetime tave of a star to thetime T required for the universe toreach thermodynamic equilibrium.Starting with a uniform density ofstars, an observer can appreciate thatafter a clock time t = tave there will bean expanding sphere of burned-outstars beyond which lies a shell of lumi-nous stars. The radiation from thisshell has a maximum radiation densityequal to the surface radiation densityfrom the average star times the ratiotave/T as long as the clock time t << T.But tave is at most a little more than 10billion years, while T can be shown tobe tens of billions of years, so the nightsky remains dark. Harrison shows thatthis argument is true for all presentmodels of the universe and does notrequire a cosmological redshift. Heargues that Lord Kelvin (1901) wasthe first to give the correct answer,which Edgar Allan Poe had antici-pated in his qualitative cosmologicalspeculations. For the detailed calcula-tions, see the references below.

Harrison, E. R. “Why the Sky Is Dark atNight.” Physics Today 27, no. 2 (1974):30–36.

———. “The Dark Night-Sky Riddle.” Science 226 (1984): 941–945.

Pesic, P. “Brightness at Night.” AmericanJournal of Physics 66 (1998): 1013–1015.

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203. Headlight Effect

In the special theory of relativity (STR)a result called the headlight effectoccurs. One considers the Lorentz-Fitzgerald contraction of distances inthe direction parallel to the constantvelocity and no change in the perpen-dicular direction. If the primed frameis the vehicle frame, then the angle inthe two frames are related by cos φ =(cos φ′ + v/c)/(1 + v/c cos φ′ ). Substi-tuting the appropriate values tells usthat cos φ ~ 1, or φ ~ 0°! Therefore, allthe light is in a very small solid anglein the forward direction, and only anobserver directly along the line ofmotion will see the light. You will notsee the light from the relativistic vehi-cle passing nearby unless your eye iswithin the very narrow light cone.

In the rest frame of the source, thestar emits light in all directions, yet thecalculation reveals that for an observerof a very fast-approaching star, practi-cally all its light will be shining alongthe direction of motion! A fast-approaching star or galaxy will pos-sess a very narrow bright headlightbeam that could miss Earth. Mean-while, a fast-receding star or galaxymay not be seen at all because its lightis redshifted out of the visible rangeand practically all its light shines awayfrom us!

So in observing stars, there is thisSTR headlight effect to consider. There

also are the different clock rates forthe two frames of reference, so thenumber of photons emitted per secondon the star and received at Earth willdiffer. Moreover, the spectrum of lightwill be different as well.

Taylor, E. F., and J. A. Wheeler. SpacetimePhysics. San Francisco: W. H. Freeman,1966, p. 69.

204. Incommunicado?

No and yes! No, in the normal sensecase because the relative velocity cannever exceed the speed of light. Thesuccessive pulses may arrive less andless often, but you will never outrunthe light.

And yes, you would lose commu-nication contact if we allow the spaceitself to expand, analogous to theexpansion of the universe in presentcosmological models. The addition ofvelocities is the old classical physicsone, not the relativistic one. The pho-ton velocity is affected by the localenvironment; the local substratum(i.e., coordinate system) “drags” thephoton along. Imagine two localregions in rapid recession from eachother. If the person in one region firesa photon toward the other, the sub-stratum of the first region drags thephoton along, slowing the photon’sprogress toward its target. If theexpansion rate is high enough, the tworegions can be receding from each

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other at light speed or greater, pre-venting communication between youand your friend.

Higbie, J. “Radial Photon Paths in a CosmicModel: A Student Exercise.” AmericanJournal of Physics 51 (1983): 1102–1107.

205. Local

Accelerations

The presence of the massive body canbe determined by the trajectories of thetwo test masses upon their release. Inthe simple case in which the laboratoryis not moving with respect to the mas-sive body, when released equidistantfrom the object but separated fromeach other, the two test masses willmove toward each other faster thantheir mutual gravitational accelerationas they fall toward the body. In addi-tion, if they are separated vertically sothat one test mass begins closer to themassive body than the other, their ver-tical separation distance will change asthey fall. In a uniform gravitationalfield their separation distance wouldremain fixed in value in each test.

One can extend this problem toconsider a rotating massive body. Canobservers inside a spaceship determineby “local” measurements only—thatis, without looking outside—if they arein the field of a rotating central mass,or if they are just moving with velocityV on a Schwarzschild backgroundmetric? Yes they can; by using at least

four test particles inside their spaceshipand having the capability to measuretheir relative accelerations, they cansucceed in determining all the compo-nents of the Riemann tensor anddecide whether they are in the space-time curvature of a rotating centralmass. Note that gyroscopes do nothelp here because one would need tocheck their alignment with stars out-side, which is forbidden. The challengehere is to measure a new effect, calledintrinsic gravitomagnetism, introducedby the GTR, that the space-time geom-etry and the corresponding curvatureinvariants are affected and determinedby both mass-energy and mass-energycurrents relative to other mass—thatis, by mass-energy currents that cannotbe eliminated by a Lorentz transforma-tion. See the Ciufolini and Wheeler ref-erence below for the details.

Ciufolini, I., and J. A. Wheeler, Gravitationand Inertia. Princeton, N.J.: Princeton Uni-versity Press, 1995, pp. 358–360.

Kalotas, T. M., A. R. Lee., and R. B. Miller.“Einstein on Safari.” The Physics Teacher29 (1991): 122–124.

Martin, J. L. General Relativity: A Guide toIts Consequences for Gravity and Cosmol-ogy. Chichester, Eng.: John Wiley & Sons,Ellis Horwood, 1988, pp. 93–94.

206. Twin Paradox

Both special theory of relativity (STR)and general theory of relativity (GTR) explanations for the aging ofthe space-traveling twin should be

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considered. If by the STR we considerinertial reference frames only andignore any accelerations experiencedby the space traveler, a symmetrywould exist between the two frames,and the twins must both age at thesame rate. Therefore, the accelerationsexperienced by the space travelermake the difference in the aging.

One can handle these accelerationsin the STR or in the GTR. Some peo-ple argue that this twin paradox prob-lem requires only the STR becausethere is no curved space-time in theproblem—that is, both twins can beconsidered to be in a flat space-timebecause no gravitational accelerationsnear a mass are necessary. Then onewould handle the accelerations for thespace-traveler twin in terms of STRcalculations, perhaps via the velocityparameter technique. A true GTRproblem, by contrast, would requirethe physics of the curved metrics neara massive body.

The solution of the twin paradoxusing the GTR relies on clocks tickingslower in a gravitational potential neara mass. The clock at a far distancefrom the mass ticks at its fastest rateand, if brought closer to the mass,begins to tick slower and slower.Therefore a person closer to the mas-sive body, where the gravitationalacceleration is greater, ages slower.

In cases where the acceleration of a

spaceship can be approximated by anequivalent gravitational acceleration—that is, using the Equivalence Princi-ple—we can expect the traveling clockto tick slower during the acceleration.And that behavior is why the travelingtwin ages less. As seen by a thirdobserver at rest with respect to thestars and the stay-at-home twin, theclock on the spaceship is changing itsrate of ticking during the accelerations.

207. Twin Watches

A watch ticks at its fastest rate when atrest and when there is no gravitationalfield. So there are two effects to con-sider: (1) from the special theory ofrelativity (STR), the motion of thewatch with respect to the laboratoryframe affects the ticking rate; and (2)the change in gravitational potentialaccording to the general theory of rel-ativity (GTR) affects the clock tickingrate. For a watch in free fall, the twoeffects are exactly opposite and cancel!The two watches agree again when shetakes the second reading.

Now for the details. First, are thereany symmetry considerations thatwould simplify the calculation? Yes;the two parts of the journey for themoving watch—the upward and thedownward parts—are time reflectionsof each other, and these two partsrequire the same elapsed time in the

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laboratory frame and in the moving-watch frame.

Pick the laboratory frame of refer-ence. As the watch goes upward in thelab frame at its maximum velocity ini-tially, the STR makes the watch tickfaster as the velocity decreases, and theGTR makes the watch tick faster asgreater height is achieved. On thedownward journey, the watch ticksslower and slower by both STR andGTR effects. So we need only calculatethe changes in the tick rate when thewatch has gone upward by a smallamount—∆h, say.

From the STR, the time interval Tbetween ticks at velocity v is given by T= T ′/ , where T ′ is thetime interval between ticks of thewatch in its own reference frame. Attwo different heights—h1 and h2 = h1 +∆h—the time intervals between ticksare T1 = T ′/ and T2 =T ′/ ,respectively, becausethe velocities will be different at thetwo heights. Since v << c, and assum-ing a uniform acceleration approxima-tion for free fall, by the third goldenrule of kinematics, v2

2 = v12 – 2 g ∆h.

Substitute the velocities squared intothe watch’s time interval relations andexpand the square roots in the denom-inators by the Taylor series expansion1/ . . . . One calcu-lates T2 ~ T1 – T′ g ∆h/c2, a quantityproportional to the change in height.

From the GTR, the time interval Tbetween clock ticks at radial distanceR outside of a body of mass M is givenby T = T ′ . In thelimit of very large R, the clock ticks at its fastest rate. By definition, g =GM/R2 at the surface of Earth. Substi-tute the above heights for the two dis-tances from the massive body and takethe difference. One calculates that T2 ~T1 – T ′ g ∆h/c2, a quantity propor-tional to the change in height and aquantity from the GTR that changesas fast as the quantity from the STR.

So the total change in the tick rategoing upward is canceled by the totalchange in the tick rate coming down-ward, to make no net change whenthey are once again at the same height.If this argument has any flaws, do notblame either of my colleagues, RichardP. Feynman (deceased) or B. Winstein,for they know not what they hadwrought!

208. Global Positioning

Satellites

The general theory of relativity (GTR)plays an important role! Correctionsmust be made for clock rates in a grav-itational field in addition to the specialtheory of relativity (STR) correctionsto the clocks for the movement of thesatellite. Both relativistic effects foulup what should have been a pretty

( / ( ))1 2 2− GM Rc

( ) ~ /1 1 2− + +ε ε

( / )1 22 2− v c

( / )1 12 2− v c

( / )1 2 2− v c

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simple geometry calculation relatingdistance to time and velocity. Theclocks in the satellites tick at a slightlyfaster rate than identical clocks on theground because they are in a slightlyweaker gravitational field, being far-ther from the center of Earth. Theytick slower than the Earth-boundclocks because they are moving fasterwith respect to the stars.

We can estimate the sizes of theseeffects. From the STR, the time inter-val T between the ticks of a clockmoving at velocity v is given by T =T ′/ , where T ′ is the timeinterval between ticks of the clock inits own reference frame. At slowspeeds v << c, one expands the squareroot by the Taylor series expansion1/ . . . to obtain T~ T ′ (1+v2/(2c2)). For satellites orbitingEarth in about 720 minutes, theirspeed makes the time correction factorabout 1.1 × 10–10. Multiplied by thespeed of light, this time factor corre-sponds to a distance error of about 3.3centimeters for each second.

From the GTR, the time interval Tbetween clock ticks at radial distanceR outside of a body of mass M is givenby T = T ′ . Con-sider two different radii: r1 = 6.37 ×106 m and r2 = 2.02 × 107 m. Substi-tute the two radii for the two distancesfrom the massive body and take thedifference. One calculates a clock

correction factor of about 4.8 × 10–10

for this GTR effect, a little more thanfour times the STR effect, or about14.4 centimeters of error every second.So in 10 minutes even this small effectproduces an error of about 86 metersif not accounted for. Who would havethought that both STR and GTReffects are big enough to play impor-tant roles in such a useful practical sys-tem as GPS!

209. Solar Redshift

Even though there may be no relativeradial motion between the Sun and theobserver on Earth, there is still a grav-itational redshift dictated by the gen-eral theory of relativity (GTR). Recallthat the infinitesimal distance ds in aflat Euclidean space with coordinates(r, θ, φ) is defined by ds2 = c2 dt2 – dr2

– r2 dθ2 – r2 sin2 θ dφ2. In the gravita-tional field of mass M, this infinitesi-mal distance in the GTR becomes theSchwarzschild line element ds2 = (1 –rg/r) c2 dt2 – (1 – rg/r)–1 dr2 – r2 dθ2 – r2

sin2 θ dφ2, where rg = 2GM/c2 and G isthe gravitational constant.

We see that near a massive bodysuch as the Sun, the time coordinateincludes a factor , where ris the position of the light measuredfrom the center of the Sun. By evaluat-ing this factor at the Sun’s surface andat Earth’s distance, one finds that the

( / )1− rg r( / ( ))1 2 2− GM Rc

( ) ~ /1 1 2− + +ε ε

( / )1 2 2− v c

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clocks at the two distances are tickingat different rates, faster for bigger r.One approach is to assume that thephoton does not change its inherentphysical properties—for example, itmaintains its characteristic frequencyestablished during the emissionprocess at the surface of the Sun. Thenthe observer on Earth, who has thefaster-ticking reference clock withrespect to the stars, will measure alower photon frequency and see thelight as redshifted.

A second approach determinesthat the change in r for the photon canbe shown to correspond to a change ingravitational potential. The photonessentially begins in a gravitationalpotential energy valley and climbsupward to reach Earth. Its total energymust remain constant, so the increasein gravitational potential energy ismatched by the decrease in photonenergy—that is, a redshifted photon—because its energy E = hν.

Krane, K. S. Modern Physics. New York:John Wiley & Sons, 1983, pp. 438–442.

210. Orbiting Bodies

The general theory of relativity (GTR)in the Schwarzschild metric approxi-mation for the space-time metricabout the Sun predicts a precession ofthe planetary orbit. Mercury, forexample, accumulates a total GTR

precession of about 43 seconds of arcper Earth century. There are manyother precessional effects acting on theorbit, including effects from all theother planets orbiting the Sun, allthese perturbations amounting to awhopping 532 seconds of arc per cen-tury, all but the residual 43 secondsbeing explained with Newtonianmechanics.

When the angular change aroundthe orbit is calculated with GTR in theφ-coordinate and then independentlyin the r-coordinate, there is disagree-ment, which is the conceptual sourceof the effect. Or one can assign anadditional equivalent mass distribu-tion for the energy in the gravitationalfield surrounding the Sun, creating ametric that does not correspond to a1/r potential, perhaps 1/r2 or 1/r3, orsome other function of r instead of the Newtonian inverse r potential. Allthese functions of r will exhibit a pre-cession of the orbit.

In addition, a body in orbit such as a planet orbiting the Sun actuallydoes not obey Kepler’s third law pre-cisely. That is, even when we ignorethe precession of the orbit by assumingits return to the same angle withrespect to the stars, the period of orbitneeds correction. This period of orbitcorrection is a so-called fourth inde-pendent general test of the GTR, inaddition to the gravitational redshift,

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the deflection of starlight, and the pre-cession of an orbit.

The correction to the period oforbit, as calculated by Preston andWeber in the reference below, beginsby putting the reference clock at thecenter of the orbit. The classical New-tonian period of orbit is given byKepler’s third law: T = 2π a3/2/√(GM) ,where a is the semimajor axis of theellipse and M is the mass value of thecentral body. For the elliptical orbit ofeccentricity ε, in the GTR one can cal-culate the period of orbit in the radialcoordinate Tr = T (1/α + 3/2 rg/r) andin the φ-coordinate Tφ = T (1/α – 3/2(rg/r) (ε

2/α)), where α = (1 – ε2)3/2. Fororbiting bodies near potential blackholes, this correction can get large asthe radial distance r approaches theSchwarzschild radius rg = 2GM/c2.

Krane, K. S. Modern Physics. New York:John Wiley & Sons, 1983, pp. 438–442.

Landau, L. D., and E. M. Lifschitz. The Clas-sical Theory of Fields, 4th ed. Sydney: But-terworth-Heinemann, 1987, pp. 328–330.

Preston, H. G., and J. Weber, “The Period ofOrbit as a Test of General Relativity.”Physics Essays 6 (1993): 465.

211. Gravitational

Lensing

The general theory of relativity (GTR)tells us that all forms of energy areaffected by a gravitational field,including the energy carried by pho-tons of light. In the Schwarzschild

metric surrounding a star, for exam-ple, the path of the light from a distantstar is diverted toward the Sun from astraight-line path when passing near a massive body such as our Sun. Halfof the deflection angle is caused by the Newtonian attraction of the Sun;the second half is caused by the geometrical modification, called cur-vature, of space by the Sun. Gravita-tional lensing ideas have been aroundfor about 200 years, but only in thepast decade or so has gravitationallensing played an important part inastronomical measurements.

If the intermediate massive body isa galaxy, then light from distancesources behind this galaxy will befocused somewhat by the two effects,

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Hubble space telescope

lensing image

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just like the light going through a glasslens is refracted to a focus. However,the geometry of the light focusing ismuch more complicated for gravita-tional lensing than for a simple sym-metrical convex lens, for severalreasons. The light may be focusedonto a line instead of a point, forexample, in an ideal case. Therefore,astronomers can use galaxies as lensesto gather more light from far objectsbetter. The focus may be poor, but thegreater intensity allows many spectro-scopic techniques to work better.

Usually the image resolution of the distant object is quite limited byinhomogeneities in the intermediategalaxy. However, these properties ofthe intermediate galaxy can be exam-ined quite well! In fact, if the classicalapplication of the GTR to calculate thefocusing effects in gravitational lensingis correct, then the total mass of theintermediate galaxy and its mass distri-bution can be determined. Galaxymasses determined with gravitationallensing have disagreed with the verysuccessful proposal for a modifiedNewtonian dynamics (MOND) withina galaxy region and have supported the“dark matter” models. However, thegame is not over for MOND becauselarge-scale gravitational quantization,either from a version of M-theory andsuperstrings or from some other quan-tization scheme, may come to the res-cue eventually.

Wambsganss, J. “Gravity’s Kaleidoscope.”Scientific American 285, no. 5 (2001):64–71.

212. Cosmological

Redshifts

There are three distinct causes for thespectral shift of light emitted (orabsorbed) by a galaxy: the kinematicalDoppler shift of the special theory ofrelativity (STR), the gravitational red-shift of the general theory of relativity(GTR), and the cosmological redshiftcaused by the expansion of the uni-verse. These three effects cannot bedistinguished from one another byobserving the spectrum of a singlegalaxy or other single light source.One can separate out the kinematicalDoppler shift for a cluster of galaxies,however, via statistical methods.

The standard explanation of thecosmological redshift says that thecoordinate system of the universe isexpanding while the galaxies remainat their local coordinate values. Onecan use an expanding balloon to“mimic” this type of behavior. Inflatea balloon enough to enable you todraw a coordinate system on its sur-face. Place some galaxies on the bal-loon surface. Now inflate the balloonfurther. The galaxies are farther apart,but they maintain the same coordinatevalues. Or one can use Earth. If Earthbegan to expand, Philadelphia and Los

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Angeles would move apart, yet eachwould retain its present longitude andlatitude. Distances are stretched, sowavelengths will become longer.

Perhaps another viewpoint will behelpful. One does not need to expandspace in this view. According to S.Weinberg in the Chown referencebelow, simply accept the fact that“every bit of the universe is rushingaway from every other bit”—that is,“the galaxies are exploding away fromeach other, as any cloud of particleswould do if they are set in motionaway from each other.” The matterinside the individual galaxies does nottake part in the general expansionbecause the local gravity holds thelocal matter together. The universe’sexpansion appears just beyond thefrontiers of the Local Group of galax-ies, about 4 million light-years fromthe Local Group’s center of mass.

M. L. Bedran in the reference listedbelow compares the Doppler redshiftto the cosmological redshift for agalaxy with z = 1, where 1 + z = exp[v/c], determining that a 2.4 percentdifference between the two redshiftsexists for this galaxy with an STRvalue of v/c = 0.6.

Bedran, M. L. “A Comparison between theDoppler and Cosmological Redshifts.”American Journal of Physics 70 (2002):406–408.

Chown, M. “All You Ever Wanted to Knowabout the Big Bang.” New Scientist (April17, 1993): 32–33.

213. Tired-Light

Hypothesis

The only two specific pieces of evi-dence are the time dilation arisingfrom the expansion of the universe,and the spectral shape of the cosmicmicrowave background. Astronomerssee that exploding stars in distantgalaxies brighten and fade moreslowly than those nearby. If the staremits a light pulse on January 1 and asecond pulse on February 1, these twopulses are separated by one light-month. As they travel toward Earth,their separation distance increases,perhaps doubling, so that they arereceived two months apart. The tired-light hypothesis cannot explain thisextended time interval. In fact, distantsupernovas are observed to wax andwane more slowly than nearby ones.

The observed spectrum of themicrowave background radiation is aperfect blackbody shape, easilyexplained by the expansion of the uni-verse from a thermodynamic equilib-rium condition. For the tired-lighthypothesis, an initial blackbody spec-trum does not remain a blackbodyspectrum as the light becomes red-shifted.

Croswell, K. The Universe at Midnight:Observations Illuminating the Cosmos.New York: Free Press, 2001, p. 76.

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214. Black Hole

Entropy

There should be radiation from theblack hole—that is, from the sur-rounding space, not from inside theblack hole, because nothing can getout. This Hawking radiation was firstcalculated in the 1970s and awaitsexperimental verification.

By taking quantum mechanics intoaccount, particles and antiparticles arebeing created continually via virtualpair creation in the vacuum. When thisprocess occurs near a black hole, oneparticle of the pair may be “eaten” bythe black hole and the other mayescape. In the thermal equilibriumstate, the amount of energy that theblack hole loses to Hawking radiationis exactly balanced by the energygained by swallowing other “thermalparticles” that happen to be runningaround in the “thermal bath” in whichthe black hole finds itself.

The temperature of a nonrotatingblack hole is given by T = hc3/(8πkGM),where h is Planck’s constant and k isBoltzmann’s constant. Note that thisexpression connects gravitation, ther-modynamics, and quantum mechanics.For black holes of a few solar masses,the temperature is only about 10–6 K!The smaller black holes with little masswill be at a much higher temperature,contrary to intuition.

Hawking, S. W. A Brief History of Time:From the Big Bang to Black Holes. NewYork: Bantam Books, 1988, pp. 104–110.

Penrose, R., The Emperor’s New Mind.Oxford: Oxford University Press, 1989,pp. 361–363.

215. Black Hole

Collision

Yes. The two black holes shouldcoalesce like two liquid drops. Weneed to ensure that the entropy after-ward in the coalesced final state isgreater than the entropy in the initialstate. We do that by adding up theentropy in the two states, separateblack holes versus one larger blackhole with gravitational waves carryingaway some energy and entropy in thefinal state.

The black hole entropy is propor-tional to the event horizon area, whichgrows as the mass to the fourth power.Suppose we take two black holes, oneof mass M1 and the other of mass M2.Their initial total entropy is propor-tional to M1

4 + M24. If they merge and

their final total mass is approximatelyM1 + M2, then their final total entropyis proportional to (M1 + M2)

4, whichyou can verify is greater than the orig-inal total entropy, so the reaction willgo. In cases where the final entropy isonly slightly greater than the initialentropy for the black hole parts only,one may need to add in the entropy in

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the gravitational waves to ensure abigger inequality

Thus a bigger mass means a biggersurface area, which means a biggerentropy than if the two smaller blackholes remained apart. Simulations in3-D of colliding black holes and theiremitted gravitational waves can beseen on the Internet.

Bekenstein, J. D. “Black-Hole Thermo-dynamics.” Physics Today 33, no. 1 (1980):24–31.

216. Centrifugal Force

Paradox

The conceptual resolution of this par-adox starts with the consideration oflight paths near the black hole. Thegeneral theory of relativity (GTR) pre-dicts that there should be light pathsaround the black hole that are circularat a radial distance of 1.5 times thegravitational radius rg = 2GM/c2.Around one of these circular lightpaths imagine a circular tube centeredexactly on the path of the circular ray

so that the axis of the tube and thepath of the ray coincide. Measure-ments with a straight ruler verify thatthe axis of the tube is circular, yetbecause of the bending of the lightrays, the tube is seen as absolutelystraight by an observer on the axis. Alamp held at the axis by a colleaguewill appear dimmer to you as he or shewalks away along the axis, but thelamp is never obscured, so you mustconclude that the tube is straight.Therefore, along this circular path onewould expect no centrifugal forceeffects.

Instead of the tube being aroundthe circular light path, consider thetube to be around a smaller circularpath centered on the black hole. Withrulers one can again verify that thetube curves to the left, with the blackhole on the left as one walks forward.The outward direction is to the right.Everyday experience tells us that thecentrifugal force pushes outward.Again your colleague walks away withthe lamp held along the axis of thetube. If somehow the light rays werenot bent by the gravitational field ofthe black hole, you would see the lampdisappear behind the left side of thetube, and you would conclude that the tube bends to the left. If the path isthe one discussed above, the lamp isalways in view. But the tube is so closeto the black hole that the light raysbend even more than circular rays. So

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Light rayBlack hole

Tube

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you actually see the lamp disappear tothe right. Therefore the outward direc-tion is to the left, and you must predictthat the centrifugal force would pushto the left!

Several related paradoxes for pathsnear a massive body are discussed inthe references, including the fact thatfor a rocket to maintain a constantspeed the boosters would need to fireperpendicularly to the path, but theirforce is not dependent on how fast therocket is moving.

Abramowicz, M. A. “Black Holes and theCentrifugal Force Paradox.” ScientificAmerican 268, no. 3 (1993): 74–81.

Abramowicz, M. A., and J. P. Lasota. “OnTraveling Round without Feeling It andUncurving Curves.” American Journal ofPhysics 54 (1986): 936–939.

217. Geodesics and

Light Rays

The two statements are not in conflict.One must always distinguish geodesicsin four-dimensional space-time fromgeodesics in three-dimensional space.Light rays always follow geodesics in4-D space-time, but these paths arenot necessarily geodesics in 3-D space.An analogy is helpful. Each great circleon a globe is a geodesic line on thetwo-dimensional surface but, being acircle, the great circle is not a geodesicline in the 3-D Euclidean space inwhich the globe sits.

In conventional geometry, thegeodesic is the shortest curve betweentwo points measured by counting howmany rulers fit along the curve. In aflat space—that is, in a space free fromgravitational fields—the geodesic is astraight line. In the GTR one candefine the distance between two pointsin space as half the time it takes forlight to travel from one point to theother and back, multiplied by thespeed of light. In flat space, the twodefinitions agree.

In 4-D space-time, light alwaysmoves along geodesics and traces thegeometry of space-time. In a 3-D spacewarped by a gravitational field, how-ever, the light rays are curved and donot coincide with geodesics in general,so the geometry of space is not tracedby light rays.

Abramowicz, M. A. “Black Holes and theCentrifugal Force Paradox.” ScientificAmerican 268, no. 3 (1993):74–81.

Abramowicz, M. A., and J. P. Lasota. “OnTraveling Round without Feeling It andUncurving Curves.” American Journal ofPhysics 54 (1986): 936–939.

Misner, C. W., K. S. Thorne, and J. A.Wheeler. Gravitation. San Francisco: W. H.Freeman, 1973, pp. 31–34.

218. Galaxy Rotation

To retain a Newtonian gravitationexplanation for the rotation propertiesof a galaxy, additional matter, called“dark matter,” in a halo surrounding

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the galaxy has been proposed, its massbeing about 10 times the mass of thevisible mass! Known particles such aselectrons, protons, neutrinos and soon cannot be its major constituents;otherwise the particles in the halowould have been detected already.Some exotic form of matter/energy isrequired in this galactic halo. How-ever, any type or types of possible“dark matter” in sufficient quantitieshas not yet been found.

An interesting proposal that doesnot require “dark matter” is calledMOND, an acronym for modifiedNewtonian dynamics, which applieswhenever the inward radial accelera-tion in the galaxy is below the value a0

= –1.1 × 10–10 m s–2, an incrediblysmall amount by Earth standards but avalue that occurs in most galaxies.Essentially, MOND replaces the New-tonian acceleration gN by g = √(gN a0) .All galaxies examined so far seem toobey the consequences of this ad hocrule by using the galactic visible matteronly, but its possible origin in terms offundamental physics principles isbeing investigated still. The majorproblem for MOND has been itsinability to accommodate the empiri-cal results on the focusing of distantstarlight by gravitational lensing.

An even more exotic solution hasbeen proposed to explain galaxy rota-tion without requiring “dark matter.”

Briefly, the large-scale structure andbehavior of the galaxy may result fromthe galaxy being in some quantizationstate. In modeling this type of theory,all the disk stars would be in the samequantization state independent of posi-tion radially and, by the virial theorem,must have the same tangential velocityV = GM2/(nJ), where M is the amountof visible mass, n is a small integer, andJ is the total angular momentum of thisvisible mass of the galaxy. Substitutingreasonable values for our Galaxy (theMilky Way), for example, one obtainsa value near to the measured value ofV = 220 km s–1. This theory predictsthat the next quantization state wouldhave exactly half the disk tangentialvelocity and, indeed, in 2003 a masscurrent of stars circulating the Galaxyjust beyond the “edge of the easily vis-ible disk” with a tangential velocity of110 km s–1 was determined serendipi-tously from data collected by the SloanDigital Sky Survey (SDSS)! Whetherthis proposed large-scale quantizationfaithfully represents gravitationalbehavior in galaxies and in the uni-verse remains to be fully examined.

Cline, D. B. “The Search for Dark Matter.”Scientific American 288, no. 3 (2003):50–59.

Milgrom, M. “Does Dark Matter ReallyExist?” Scientific American 287, no. 2(2002): 42–52.

Preston, H. G., and F. Potter. “ExploringLarge-Scale Gravitational Quantization

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without h-bar in Planetary Systems, Galax-ies, and the Universe.” E-print archive forphysics papers. http://lanl.arxiv.org/abs/gr-qc/0303112 (2003).

219. Cosmic

Background Radiation

Amazingly, cosmic background radia-tion (CBR) has a perfect blackbodydistribution! This CBR is uniform andisotropic in the universe to one part in100,000 and amazingly flat over largespatial regions—that is, large solidangles. One suspects that these largespatial regions, even those in oppositedirections in the sky, have always been in communication with eachother, to make them so uniform. Ofcourse, smaller regions have their indi-vidual characteristic galaxies, clustersof galaxies, and so on.

The most popular interpretation,the standard inflationary model of theuniverse, requires the universe to orig-inate with the far regions much closerto each other in thermal equilibriumfor the observed uniformity to develop,then for a very fast inflation to occurthat separated them out of communi-cation reach. Now we see these galax-ies, once close together, in oppositedirections in the universe and largeregions with the same large-scale char-acteristics in all directions. Their origi-nal collective blackbody spectrum at a

high temperature now exhibits a low-temperature blackbody spectrumbecause the expansion of the universehas “stretched the wavelengths.”

The stars we see are not in thermalequilibrium as a collective whole. Onecannot produce a perfect blackbodyspectrum at any temperature by sim-ply using billions of stars that are notin thermal equilibrium as a whole andadding up their radiation intensities inthe universe. One cannot obtain ablackbody spectrum from many otherhypotheses about the cosmologicalredshift of light from distant objects,such as the tired-light effect whereinthe light loses energy during tranversalof the universe.

One could, however, speculate thatthe galaxies have never changed theiraverage separations, that there hasbeen no coordinate expansion asstated in the standard inflationarymodel. The cosmological redshiftswould correspond to the redshift pro-duced by an “effective cosmologicalgravitational potential well”—forexample, in which the source sitslower in the well than the observer,true for all sources and all observers inthe universe. The galaxies would becloser together at all epochs, in com-munication with each other, and inthermal equilibrium, so the measureduniformity would be expected—thatis, all directions should look the same.

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One consequence would be that onecould never see galaxies beyond about12 billion light-years or so, the actualdistance value depending on the aver-age matter/energy density of the vacuum. The redshifts would be inter-preted as “effective recession veloci-ties” that reach light speed at this fardistance.

Hasinger, G., and R. Gilli. “The CosmicReality Check.” Scientific American 286,no. 3 (2002): 60–67.

Peebles, P. J. E. “Making Sense of ModernCosmology.” Scientific American 284, no. 1(2001): 54–55.

220. Planetary

Spacings

The orbital radii of the planets onlyroughly follow the Titius-Bode law, sothis specific pattern is probably bogus.However, L. Nottale and his researchgroup have shown that the planetsobey a generalized Schrödinger-likewave equation (with one unknownparameter) that has solutions dictatinga regular pattern for where orbitingbodies reach an equilibrium radius.The planets of the Solar System occupyonly these radial positions and leavesome equilibrium radii unoccupied,perhaps a consequence of their historyof formation. However, even thoughNottale’s fits are extremely good, thereare several other sets of small integersthat statistically fit as well as the set

proposed by Nottale, including manysets with larger integers.

Extrasolar planetary systems withthree planets have been found, buttheir statistical fits allow several sets ofintegers also, so they are not yet thedefinitive test. Unfortunately, we mustwait for a definite extrasolar system ora precise laboratory test to resolve the issue of whether the patterns aresimply numerology or an exhibit ofpart of a new fundamental gravita-tional theory.

Lynch, P. “On the Significance of the Titius-Bode Law for the Distribution of thePlanets.” Monthly Notice of the RoyalAstronomical Society 341 (2003):1174–1178.

Nottale, L. “Scale-Relativity and Quantiza-tion of Extra-Solar Planetary Systems.”Astronomy & Astrophysics 315 (1996):L9–L12.

Nottale, L., G. Schumacher, and J. Gay.“Scale Relativity and Quantization of theSolar System.” Astronomy & Astrophysics322 (1997): 1018–1022.

221. Entropy in the

Big Bang

We quote from the reference listed.The “standard” answer attempting toexplain the paradox is:

True, the fireball was effectively in thermal equilibrium at thebeginning, but the universe at thattime was very tiny. The fireballrepresented the state of maximum

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entropy that could be permitted fora universe of that tiny size, but theentropy so permitted would havebeen minute by comparison withthat which is allowed for a universeof the size that we find it to betoday. As the universe expanded,the permitted maximum entropyincreased with the universe’s size,but the actual entropy in the uni-verse lagged well behind this per-mitted maximum. The second lawarises because the actual entropy isalways striving to catch up withthis permitted maximum.

This answer cannot be correct if theuniverse will eventually suffer a “bigcrunch,” for then the argument wouldapply again in the reverse direction!We are at an impasse.

Penrose, R. The Emperor’s New Mind.Oxford: Oxford University Press, 1989,pp. 317–330.

222. Gravitational

Wave Detectors

For all kinds of waves, for locationsbeyond several wavelengths, the solu-tions of the wave equation correspondto the radiation field transportingenergy and momentum from thesource into the surrounding space.When considering possible sources ofgravitational waves in the Galaxy andbeyond, the wavelengths are typically

several kilometers or more. One couldplace the rotating laboratory gravita-tional wave source several kilometersaway or more from the gravitationalwave detector, but the decrease in theradiation field intensity with distancesquared combined with the low sensi-tivity of the detectors make thisarrangement unlikely to work withpresent detectors. Therefore, as far aswe know, there has never been a truetest of the gravitational wave responseof any detector to gravitational radia-tion using laboratory sources of grav-ity waves.

There have been two fundamentaltypes of gravitational wave detectors:the Weber bar antenna, named afterpioneering physicist Joseph Weber,who began this research field in the1950s with his one meter diametersuspended aluminum bar; and theinterferometer type such as LIGO, firstanalyzed by the same Joseph Weberand his students. The classical calcula-tion of the resonant response of theWeber bar reveals just how limited isits sensitivity to gravitational wavesoriginating in our Solar System andGalaxy. However, if the Weber barantenna actually behaves differentlythan originally expected, as a collec-tive quantum oscillator respondingcoherently, say, then it will respondwell to all frequencies of the incidentgravitational waves. Hundreds to

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thousands of vibrational modes couldbe excited in a large range of frequen-cies with an increase in sensitivity ofmany powers of ten.

At this time there has been no sub-stantiated detection of gravitationalwaves by either type of detector.Weber reported a twice-daily responseof his two side-by-side almost-identi-cal bar antennas for orientationspointing toward the center of theGalaxy during a period of almost twodecades, but no other researcher hasverified this behavior with an inde-pendent detector. So we must wait forthe first detection of gravitationalwaves by LIGO or other detectors.Unfortunately, interferometer types ofdetectors such as LIGO and VIRGOcannot operate as a collective quan-tum oscillator.

223. Space Curvature

The proposed method for determiningthe curvature of space will work forboth continuous and discrete spaces. If we assume a uniform density ofstars, or galaxies if we choose to countgalaxies, the number N of this particu-lar kind of source within a sphere ofradius R in a Euclidean space (zerocurvature space) is given by N = ρ4πR3/3, where ρ is the uniform den-sity. When N is plotted against the dis-tance, N will fall short, match, or

exceed the cubic curve for the threetypes of spaces: positively curved, flat,or negatively curved, respectively.

On a “small” scale, when the totalnumber of sources is less than a fewhundred, there can be a relatively largeuncertainty in the general behavior ofthe plotted curve. But as more andmore sources are counted at fartherdistances, the asymptotic behaviorshould become apparent. However,adjustments must be made for thefinite velocity of light and for possibleevolutionary changes in the sources.Distant sources are sampled at an ear-lier time, possibly at a closer distance.

If the universe is actually represen-tative of a discrete space, one can showthat by counting many sources one candetermine the curvature in the limit asthe number of sources becomes large.Think of a lattice of points as one sim-ple example, such as a regular lattice of atoms in a solid. By counting the nearest neighbors only, then the nextnearest neighbors, and so on, one even-tually approaches asymptotically to aplotted line from which the curvaturecan be determined.

Of course, in a discrete space, onemust be careful not to count over andover the images of the same source.For example, imagine space dividedinto identical cubes next to each otherand filling all space. Standing insideone cube, we can look to our right to

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see ourselves inside the first cube onour right looking away to the nextcube, and so on. Each successive imagewill be dimmer and will be earlier intime because the light does not travelinfinitely fast. If our real space in theuniverse is discrete, the cube sizewould be enormous, certainly waybeyond the size of our Local Group ofgalaxies; otherwise we would havedetected this discreteness already byhaving seen multiple images of ourown Galaxy.

If space in the universe is curved,then cubes will not fill the space.Mathematicians point out that one ofthe dodecahedral spaces would be thesimplest space-filling for a negativelycurved discrete space, the most likelytype of space curvature for the uni-verse. However, the curvature of theuniverse is not known unambiguouslyyet, although a flat space with no cur-vature will nicely fit the present data inthe standard model of an inflationarybig bang universe.

Eckroth, C. A. “Counting Distant RadioSources to Determine the Overall Curva-ture of Space.” The Physics Teacher 30(1992): 92–93.

Gruber, R. P., A. D. Gruber, R. Hamilton,and S. M. Matthews. “Space Curvature andthe ‘Heavy Banana Paradox.’” The PhysicsTeacher 29 (1991): 147–149.

Levin, J. How the Universe Got Its Spots:Diary of a Finite Time in a Finite Space.Princeton, N.J.: Princeton University Press,2003, pp. 132–155.

Wolfram, S. A New Kind of Science. Cham-paign, Ill.: Wolfram Media, 2002, pp.433–540.

224. The Total Energy

Yes, there can be the creation of mat-ter out of nothing with no violation ofconservation laws! First proposed in1958 by H. Margenau and later recal-culated in more detail by N. Rosenand others in 1994, the gravitationalenergy cancels the mass energy in aclosed, homogeneous universe.

The simplest general approach wasdone by Margenau. Consider a finitespherical universe of radius R filledwith matter and radiation of equiva-lent total mass M. The gravitationalpotential energy is the negative quan-tity –kGM2/R, where G is the gravita-tional constant and k is a positivenumerical factor not greatly differentfrom 1. The total energy E in the uni-verse is then E = Mc2 – kGM2/R. Usingrepresentative values such as R = 1.3 ×1026 m and a mass density of 8 × 10–27

kg/m3, we estimate k ~ 2.4 when E =0. Nathan Rosen and others showedthat the gravitational energy cancelsout the mass energy without resortingto numerical estimates.

Cooperstock, F. I., and M. Israelit. “TheEnergy of the Universe.” Foundations ofPhysics 25 (1995): 631–635.

Jammer, M. Einstein and Religion: Physicsand Theology. Princeton, N.J.: PrincetonUniversity Press, 1999, pp. 201–203.

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Margenau, H. Thomas and the Physics of1958: A Confrontation. Aquinas Lecture23. Milwaukee: Marquette University Press,1958, p. 41.

Rosen, N. “The Energy of the Universe.”General Relativity and Gravitation 26(1994): 319–321.

225. Different

Universes?

If the lepton and quark masses are dic-tated by fundamental mathematicalquantities, we are behooved to con-sider that all fundamental quantities innature have an origin in fundamentalmathematics. There can be no alterna-tive universes, each supposedly havingdifferent fundamental constants, forthey must have the same mathematicsdictating the same physical constants.

In 1994 F. Potter, within the con-fines of the Standard Model of Lep-tons and Quarks, related the leptonand quark mass ratios to a mathemat-ical invariant called the elliptic modu-lar invariant J, which is invariantunder all linear transformations. Thecritical prediction is a fourth quarkfamily with a b′ quark mass of about80 GeV/c2 and a t′ quark mass ofabout 2,600 GeV/c2. Althoughsearches for a b′ quark have beenunder way at the Fermilab collider forseveral years, its existence cannot beruled out yet because the decay reac-tions have very low probability and

will be overwhelmed by many otherparticle decays into the same finalproducts. Perhaps when the LargeHadron Collider is turned on in a fewyears, with its very high rate of pro-duction of quarks, the statistics will beso much better that the b′ quarkshould be easy to find.

If the b′ quark is found, then weexpect that all other fundamentalphysical constants should also bederivable from mathematical invari-ants. If the proposed scheme is correct,then our universe is the only universepossible. Even exotic speculations suchas time travel could be eliminated ifthe direction of time is one of theinnate properties of the particle statedefinition. However, we must remem-ber always that Nature is more cleverthan we can hope to be, so we mustcontinue to test every reasonable pro-posal for the truth.

Linde, A. “The Self-Reproducing Inflation-ary Universe.” Scientific American 279, no.11 (1994): 48–55.

Potter, F. “Geometrical Basis for the Stan-dard Model.” International Journal of The-oretical Physics 33 (1994): 279–306.

Tegmark, M. “Parallel Universes.” ScientificAmerican 288, no. 5 (2003): 40–51.

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Chapter 11Crystal Blue Persuasion

226. Iodine Prophylaxis

The iodine tablets, usually potassiumiodine, work by “topping up” the thy-roid gland with stable iodine to reduceits accumulation of any radioactiveiodine that may have been releasedinto the environment by a nuclearaccident. Inhalation of the radioactiveiodine in the air is the major route ofentry into the body and the thyroid.For maximum benefit, the iodinetablets should be taken before theradioactivity fallout reaches your area;otherwise the tablets themselves willhave become radioactive.

227. Bicycle Tracks

In “The Adventure of the PriorySchool,” Sherlock Holmes not onlydraws a map of the neighborhood ofthe school but also examines severalsets of tyre tracks on the moor. Heneeded to determine the bicycle’sdirection of travel solely from hisinspection of the tracks. Holmes couldtell from the depth of the wheelimpression which track was made bythe rear tyre. You don’t have that

information, but a little mathematicsreveals the answer.

Note that the rear wheel of a bikealways points to the place where thefront wheel touches the ground.Therefore, the tangent to the rearwheel track will always cross the frontwheel track, while the front wheeltrack does not exhibit this geometricalproperty.

Once we identify the rear track, wecan pick two random points on it andextend the tangents to where theycross the front track in both direc-tions, measure the segments, anddetermine which direction yields seg-ments of the same length. Since a bicy-cle cannot change its length, we learnthe direction of travel.

228. Earth Warming

Yes, there can be fluctuations in thethermal energy conducted and con-vected to Earth’s surface from sourceswithin. There are several types ofthermal energy sources, includingradioactive nuclei emitting particles

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that transfer their kinetic energy tothermal energy, as well as frictionbetween internal rock flows, whichcould create local hot spots and/ortemporary changes in the flow proper-ties of liquid rock or the thermal con-ductivity of the rocks. Therefore, smallchanges in the rates of thermal energytransport to Earth’s surface are possi-ble and most likely happen continu-ally. Are these internal sources theculprits in the present slow averagetemperature increase? These fluctua-tions are expected to be too small, butno one knows for sure.

229. Frequency

Jamming

A spark gap usually is a wonderfulnoisy source of electromagnetic wavesat all frequencies simultaneously. Thegreater the current across the gap, themore intense will be the total radia-tion emitted at each frequency. Therewill be a distribution of intensity ver-sus frequency that can be “tuned” alittle by adjusting the gap spacing.

A simple spark gap would be asmall battery, such as a D cell, and two

wires in the process of making contact.Held near a radio, the small sparkacross the gap can be heard throughthe radio, indicating that many fre-quencies are being emitted. As anadditional demonstration, one caneven move a radio near a small electricmotor of the kind that has brushes tohear its rotation frequency because thebrushes make and break contact eachrevolution.

Of course, if one desires to have ahigher-current spark gap, a car batteryor a transformer can be used withproper safety precautions to provide ahealthy current that can be operated inan intermittent mode or in a continu-ous mode. Nearby radios, televisions,and so on will be affected by thislarger-current spark gap device. EvenGPS transmissions between 1,000MHz and 2,000 MHz may be affected,so some care must be taken not to vio-late federal transmission limitations.

230. Light Energy

A light source emitting light of fre-quency f approaching an observer atconstant velocity v will appear blue-shifted by an amount corresponding to the relativistic Doppler effect for-mula f ′ = fbecause the clock ticking rates will bedifferent for the source reference frameand the observer reference frame, andtheir separation distance is decreasing.

[( / )] / ( / )1 12 2− −v c v c

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When v << c, we can expand the for-mula in a Taylor series to obtain f ′ ~ f(1 + v/c – v2/2c2 + . . . ), so the leadingterm in powers of v/c is positive, cor-responding to the blueshift. Weassume that an acceleration itself doesnot produce an additional fundamen-tal frequency shift, although there willbe acceleration effects because thesource is changing instantaneouscomoving inertial frames.

For a photon, its energy is E = hfand its momentum is p = E/c, so bothenergy and momentum are different indifferent reference frames because theobserved frequencies are different.Notice that the recoil of the source onemission of light and of the observeron detection are not accounted for inthe discussion and that the energy andmomentum input necessary to keepthe relative velocity of the source andobserver fixed must be consideredalso. Of course, energy and momen-tum conservation laws are obeyed inthis example.

231. Acid Rain

Not so! The falling raindrops will notremain neutral at pH 7. Pure rainwa-ter falling through unpolluted air is anacid, with a pH of about 5.6, becauseas the drops form and fall they dis-solve carbon dioxide in the air andreact to produce carbonic acid,H2CO3. Officially, therefore, acid rain

is defined to have a pH of less than5.0, a condition that tends to occurmore often in industrialized areas ofthe world than in remote regions.

Human activities can increase theamount of CO2 in the air, but so domany natural resources, such as volca-noes, lightning strikes, cows, bacteria,and fires. When industrial and auto-mobile exhausts release sulfur com-pounds and nitrogen compounds,these molecules combine with oxygento form sulfuric acid and nitric acid,which can harm ecosystems, historicmonuments and buildings, and thehealth of people around the world.The reduction of sulfur and nitrogencompounds released into the air hasbecome a worldwide concern.

Trefil, J. The Nature of Science. Boston:Houghton Mifflin, 2003, pp. 6–7.

232. Electrical Current

The electrons in house wiring move ata snail’s pace, with an average driftvelocity of about a millimeter per sec-ond. These electrons, which are free tomove in the metal wires, are distrib-uted throughout, so when the switch isclosed to make a complete circuit, theymove en masse, sort of like water pass-ing through a continuous hose thatcloses on itself. The electron velocity islimited because its negative electriccharge interacts with the lattice of pos-itive ions during the movement.

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In addition to having a drift veloc-ity, the electrons experience a randomsequence of pinball-like collisions tochange their speeds and directions,essentially behaving as a free electrongas. Consequently, the host metalgains some thermal energy, and itstemperature rises. If a lamp is theincandescent type, the tungsten alloyfilament will gain enough energy bythis process to dramatically increase itstemperature to a new equilibrium tem-perature of about 2000 K to glow inthe visible and the infrared.

233. Earth’s Orbit

Although the general theory of relativ-ity dictates a precession of the perihe-lion for all planets, including Earth,this effect is very small compared toperturbations provided by gravitationalinfluences of all the planets. ApparentlyEarth’s elliptical orbit will pass througha repetitive cycling about every 93,000years from its present ellipse and orien-tation with respect to the stars, to a cir-cle, then back to an ellipse with anorientation perpendicular to the pres-ent orientation, to a circle again, backto an ellipse perpendicular again, etc.,until the present elliptical orientation isrecovered approximately. Of course, allthe planets are experiencing theseperturbation effects simultaneously, sothe detailed calculations become quiteinteresting!

234. Crystal Growth

The speed and precision of the crystalgrowth depend on many factors,including the temperature, concentra-tion, and purity of the solution.Assuming the ideal solution, eachadditional atom to be added from thesolution must first find a location onthe growing surface of the developingcrystal. But these atoms in solution arerandomly moving about, making ran-dom collisions with the crystal at ran-dom locations on the surface. Howcan they build a perfect single crystal?

Their little secret is that someatoms that have been added at mar-ginal locations, say, can escape fromthese surface locations to allow otheratoms from the solution to find a bet-ter location nearby, “better” heremeaning to be held electrostaticallytighter to the crystal. But these betterlocations do not occur in chronologi-cal order because they are determinedby the collective influence of numerousatoms already in the crystal, and thebest position one microsecond agomay not be the best position for anatom now. Therefore, the addition andsubtraction of atoms from the growingcrystal surface proceeds almost by trialand error! Consequently, one cannotwrite down an algorithm for placingatoms from the solution onto thegrowing crystal.

When the crystal grows slowly,

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there is plenty of time for the samplingprocess to proceed to fruition, and thecrystals tend to form with fewer dislo-cations and inclusions. When the crys-tal grows rapidly, errors in the crystalstructure become trapped, and thesecrystals tend to have many disloca-tions and inclusions.

235. Ruby, Sapphire,

and Emerald

How are ruby, sapphire, and emeraldcrystals related? Ruby and sapphireare color variations of the same min-eral, corundum. Rubies contain asmall amount of chromium. Purecorundum is a colorless, trigonal crys-tal that occurs in a wide variety of col-ors due to infiltrations of otherelements. All color variations ofcorundum, with the exception of ruby,are called sapphires.

Rubies are red variations of themineral corundum, a crystalline formof aluminum oxide and one of themost durable minerals that exists.Only diamonds are harder. Rubies’rich, red colors arise from the substi-tution of a small number of aluminumatoms by chromium atoms. Whenexposed to high temperatures, rubiesturn green, but they regain their origi-nal color after cooling. Some rubiesphosphoresce with a vivid red glowwhen illuminated by ultraviolet light.

Sapphire is aluminum oxide withtrace impurities of iron and titaniumatoms, which are responsible for thedeep blue color shades most peopleassociate with sapphire. Several othercolors of corundum, such as yellow,reddish-orange, and violet, also areclassified as sapphire. Synthetic sap-phires have been produced commer-cially since 1902 and are used forscratch-resistant watch crystals, opti-cal scanners, and in applications wherephysical strength and transparency toultraviolet irradiation are important.

Emeralds are quite different fromrubies and sapphires in that emeraldsare the green form of beryl, colored bythe presence of chromium or vana-dium. The crystal structure of berylemeralds is hexagonal (six-sided), witha hardness slightly higher than quartzbut considerably less than diamond.Emeralds are notorious for containingflaws, and flawless stones are rare andgreatly valued.

The colors in these gemstones areproduced by characteristic downwardatomic transitions involving F-centers(from the German word farbe, mean-ing color) at the chromium or otheratoms. In a simple view of an F-center,the ambient light excites one electronin the chromium atom, for example,so the atom can be treated analogousto the hydrogen atom, with a largeaverage radius for the location of theelectron away from the chromium

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nucleus. The electron transition backto a lower energy state causes theemission of a photon in the visible.

Nassau, K. “The Causes of Color.” ScientificAmerican 243, no. 4 (1980): 124–154.

Perkowitz, S. “True Colors: Why ThingsLook the Way They Do.” The Sciences(May–June 1991): 22–28.

236. Kordylewski

Clouds

Joseph L. Lagrange in the late 1700scalculated via Newton’s laws thatthere are five special positions forobjects bound by any two-bodysystem. Now called Lagrange points,positions L1, L2, and L3 are unstable,while L4 and L5 are stable. Severalspacecraft have been placed at or near these Lagrange points, and there have been proposals for building space colonies at the L4 or L5positions.

Applying Kepler’s third law for aparticle of mass µ between Earth’s massm and the Sun’s mass M orbiting withEarth’s period T, one obtains after sev-eral steps GMµ/(r–R)3 – Gmµ/(R2(r–R))= GMµ/r3 where r is the Earth–Sun dis-tance and R is the Earth–particle dis-tance. The particle at L1 will be about0.01 times the distance to the Sun. TheL3 point on the night side of Earth canbe calculated in the same way, replac-ing r–R with r+R. However, the otherthree points must be calculated withthe gravitational attractions of theother planets included.

Similar calculations have beendone for the five Lagrange points forthe Earth-Moon system. Polishastronomer K. Kordylewski in 1961reported the observation of dust cloudsat the L5 point, but some observershave not seen them. Particles here maynot remain long before being ejected,according to calculations.

Kordylewski, K. “Photographische Unter-suchungen des Librationspunktes Lim5System Erde-Mond.” Acta Astronomica 11(1961): 165–169.

O’Neill, G. K. “The Colonization of Space.”Physics Today 27, no. 9 (1974): 32.

237. Twist Scooter

If the plane of the V arms of the twistbike remained horizontal at all times,there would be no forward motionexcept by pushing with a foot on theground. By tilting the vertical shaft of

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Moon

L2 L3

L4

L1

Earth

L5

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the handlebar sideways about 10degrees or so, the front of the scooteris lowered a bit, and a slight unbal-anced outward push of the arms of theV by the rider’s legs provides a for-ward net force similar to the resultingnet force exercised by an ice skater.The initial forward movement fromrest may be difficult on an upslopeexceeding a particular angle deter-mined by the size of the wheels and bythe possible tip angle of the handlebar.

238. Unruh Radiation

The Equivalence Principle tells us thata particle accelerating in the vacuum isequivalent to a particle at rest in auniform gravitational field. If there isradiation in one case, there must beradiation in the other equivalent case.The former is called Bekenstein radia-tion, the latter Unruh radiation,named after physicists who studied theproperties of the radiation theoreti-cally. No one has ever measured thisradiation because its intensity is manypowers of 10 too faint to be detected.

239. Star Diameters

One can determine the interferencediameter of a distant star even whenoptical parallax resolution of its diam-eter is impossible by utilizing quantuminterference between the photons fromthe left side of the star arriving out of

phase with the photons from the rightside. In other words, the photons arenot expected to be in phase. Theirphase difference depends on threeparameters: their initial phase differ-ence, the distance to the star, and thediameter of the star. By slowly chang-ing the separation of the two photode-tectors on the arms of an intensityinterferometer, one can sweep across arange of phase differences to determinethe diameter of the source. One labo-ratory analogy might be consideringhow one would determine the spacingbetween the slits of a two-slit interfer-ence experiment with a similar appara-tus. Ultimately, the amplitudes and notreally the intensities interfere. How-ever, the phase correlations depend onthe product of intensities, in contrastto the two-slit interference example.

The original experiment is knownas the Brown-Twiss experiment,named after the two researchers whofirst succeeded in using the techniqueto determine a star diameter back in1957. Interference associated with thesuperposition of separate light intensi-ties was viewed with considerableskepticism. Apparently, as the storygoes, one of the original researcherswas giving a physics talk at Caltechsoon after their first measurements. Inthose days, several Nobel physicistswould sit in the front row along withRichard Feynman and other prominentphysicists. About 10 minutes into the

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talk, Feynman walked out, much to thedismay of the speaker. About 40 min-utes later, just near the end of the talk,Feynman walked back in and sat downin his seat again. The speaker thenasked why he walked out and thenreturned. Feynman responded that hehad walked out because he did notbelieve that the physics was correct. Heexplained that he had gone back to hisoffice and worked out the problem,only to discover that the physics hadbeen done correctly. He then returnedto acknowledge the cleverness of thespeaker and his colleague. Now thespeaker was in dismay again, amazedthat someone could have worked outthe many details in so short a time!

Brown, R. H., and Twiss, R. Q. “A NewType of Interferometer for Use in Radioas-tronomy.” Philosophical Magazine 45(1954): 663.

Silverman, M. P. A Universe of Atoms: AnAtom in the Universe. New York: Springer-Verlag, 2002, pp. 102–126.

240. Glauber Effect

Yes, a standard incandescent lightbulbdoes emit single photons, and some-times there are photon pairs, andtriplets, and so on. In the ideal chaoticphoton source—a hot, incandescentwire that has physical dimensionssmaller than a wavelength of theemitted light, for example—the firstspontaneously emitted photon can

stimulate the emission of a secondphoton from a nearby atom, and thetwo can stimulate the emission of athird photon, and so on. In principle,the photons arriving at the receptorcan be single, double, triple, and so on,the actual photon state depending onhow many stimulated photons werepicked up before escaping the lightsource. The receptor receives a differ-ent energy burst with each absorption.Since the probability for stimulatedemission into the same final state isproportional to the number of pho-tons in that state already, thesemultiple photon processes occur quitereadily.

Real light sources such as incan-descent bulbs have huge physicaldimensions compared to the wave-length of light. There will be numerousideal chaotic sources along the fila-ment wire simultaneously and ran-domly emitting photons toward thedetector. These photons tend to arrivein bunches, with the photons withinany one bunch coming from severalplaces in the source. Very seldom doesone find a steady stream of photonswith nearly equal time spacing arriv-ing from the lightbulb when one lookson the nanosecond time scale.

Loudon, R. The Quantum Theory of Light,3rd ed. Oxford: Oxford University Press,2000, chap. 6.

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241. Bird Sounds

Some birds can emit just a fundamen-tal frequency with no harmonics. Justhow the bird eliminates the harmonicsoriginally generated within is beinginvestigated. The present conjecture isthat a cavity resonance amplifies justthe fundamental before the sound isemitted. If the fundamental frequencychanges, then the cavity must changeto accommodate the new fundamentalin “live time.”

242. Spouting

Alligator

To eject water droplets upward, thealligator head vibrations must providethe initial energy to create nearlystanding waves in the shallow wateron the back of its head. As wave crestsbecome larger, droplets of water breakoff and are projected high above thesurface. One can simulate this effectby sliding a styrofoam cup filled withwater across a finished wooden sur-face at about 10 centimeters per sec-ond. Water droplets will shoot upwardto about 20 centimeters.

Jargodzki, C., and F. Potter. “Spouting WaterDroplets.” In Mad about Physics: Brain-twisters, Paradoxes, and Curiosities. NewYork: John Wiley & Sons, 2001, p. 39.

243. Hair-Raiser

Function

For the HRF of a non-integer, oneneeds to write down a few more exam-ples of the given integer description.Then take the logarithm of each exam-ple to discover that they all can beexpressed as log N = nn–1 log n. By tak-ing the exponential of both sides withthe proper grouping, the final expres-sion becomes N = (n)^(nn–1)—that is, nto the power (nn–1). With HRF(x) =(x)^(xx–1), the HRF of non-integer val-ues for x becomes an easy calculationwith the appropriate calculator, onecapable of many decimal places. Whatis the limit as n approaches zero?Complex numbers can be used, as wellas irrationals such as π.

A plot of the HRF using integersshows a remarkably steep rise for evensmall integers; hence its name! Youmight want to compare its rise to anexponential function. And if all youdesire is an approximate value for theinverse or for the HRF of a non-integer,the plot provides a visual image and ameans to satisfy your curiosity.

However, as far as we know, theinverse HRF is awkward, and no easycalculation algorithm is known. Wedon’t even know whether the inversecan be expressed as the limit of aseries! One can determine the inverseby successive approximation to any

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number of decimal places with theappropriate calculator.

Of what use is the hair-raiser func-tion? The question reminds us of twoclassic quotes from Michael Faradaywhen he was attempting to explain adiscovery to the visiting prime minis-ter. He was asked: “But, after all, whatuse is it?” To which Faraday replied,“Why, sir, there is the probability thatyou will soon be able to tax it.” Andwhen the prime minister asked of anew discovery, “What good is it?,”Faraday replied, “What good is a new-born baby?”

244. Space Crawler

The U.S. Patent Office awarded patent5966986 in 1999 to this propulsiondevice. We quote the patent abstract:

A propulsion system which isdesigned to be used on a payloadplatform such as a spacecraft, satel-lite, aircraft, or an ocean vessel. Tooperate the system electrical power is

required. However, during operationthe system does not require fuel orother mass be expelled into the envi-ronment to move in space. The sys-tem is designed to operate in twooperational modes: in Mode I thesystem incrementally moves the pay-load platform forward with eachoperational cycle. In this first mode,the velocity imparted to the payloadplatform is not additive. In Mode IIthe payload platform accelerates for-ward a discrete increment of velocityduring each operational cycle. In thissecond mode the increments ofvelocity are additive.

There is no problem with energyconservation because the onboard bat-tery supplies the energy. The inventorVirgil Laul claims that this propulsiondevice when attached to spacecraftwill be able to propel spacecraft out inspace. We leave this problem as a finalchallenge. What is the physics here?Are any conservation laws violated?Will the device work in space as wellas it does on the air table?

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Indexabsorption

atmospheric, 216 cancellation, 212

academicdisputes, 44 types, 118

accelerationangular, 144car driver, 146collision time, 165free fall, 30local, 107movie of, 17pseudo-force, 168rocket ship, 107in sandglass, 144skiers, record, 167time reversal of, 16twin paradox, 108uniform, 44

acid rain, 119action-at-a-distance

Boscovich, R. J., 66Riemann, G. B., 111

airboundary layer, 197can, air pressure, 7density, 169engine, 50pressure, 125, 126, 132soup can, 129in straw, 128viscosity, 144

algebra, 103alien beings, 31alligator, spouting, 122alpha particle, 96amino acid triplet, 149ammonium maser, 68angular momentum

conservation of, 55electron in atom, 67of flywheels, 54quantization, 67quantum, total, 67

annihilation, 87, 97,171Anthropic Principle, 96antiparticles, 87, 255Apollonius, 21archaeology, 53Archimedes

gravestone, 22principle, 3

Aristotle, 50arm contortions, 25asteroid, 36astronauts, 123astronomers, 30atmosphere

absorption by, 216carbon-14 in, 93cosmic rays, 124expansion of, 35greenhouse, 70, 71ozone layer, 70refraction by, 140, 152UV, 103wings in, 31

atomabsorption by, 135Bohr, N., 68Bose-Einstein, 74carbon, 69, 70clusters, 55, 59emission by, 66, 67quantum dots, 74X-ray laser, 73, 222

atomic modelBohr, N., 67Nagaoka, H., 73

Rutherford, E., 73axis of rotation, 24, 31

b’ quark, 276Babylonians, 13, 20ball, bouncing, 87baseball bat, 53BBs, 37Be-8 synthesis, 95beans, parboiling, 6beets, peeling, 7Bekenstein radiation, 283Bell inequalities, 87benzene

carbon atoms, 69energy levels, 69

beta decay, 244, 245bicycle tracks, 118big bang

entropy, 114Hoyle, F., 114inflationary, 41, 112microwaves, 99

billiards table, 20bird sounds, 122birth dates, 15black hole

collisions, 110entropy, 110forces, radial, 111and information, 89time symmetry, 150

blackbody spectrum, cosmos, 112, 266, 271 plasma, 218

blueberry muffins, 4body cushion, 30Bohr, N.

atom, 68completeness, 83

287

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Bohr, N. (continued)double-slit, 232habits, 70quantization, 67religion, 65

boiling point, 134boiling water

altitude effects, 5beets in, 7in ice, 8salt added to, 7watched pot, 8

Boltzmann, L., 44book rotation, 25Bose-Einstein

condensate, 74, 208Cooper pairs, 84He-4, 85statistics, 177

bosons, 85, 177Bragg scattering, 74, 79brain

connections, 23magnetic field, 85power needs, 54

bread kneading, 3Brownian motor, 50bubble collapse, 218bullet

fireworks, 34impact, 30

Bushmen, 172butter, measuring, 3

cadmium selenide, 59caffeine, 7, 79calendar

Gregorian, 142Julian, 142lunar, 14, 143Mayan, 52rice planting, 14

caloriesfrom fat, 5 human needs, 4

campfire, igniting, 134can, pressure in, 7car driver, 15, 42

carboncycle, in stars, 245 nuclear levels, 95synthesis in stars, 95

carbon-14 dating, 93carbon dioxide

air amount, 279in bread, 126 greenhouse gas, 70, 71moderator, 98plants, 93in water, 9

carbonic acid, 279Carnot cycle

ferrofluid, 52photon, 61quantum, 61

cartoons, 30, 35Casimir effect, 84cellular automata, 104centrifugal force, 111CFC, 216chaotic systems

competition, 57hot wire, 284identical, 57

Chernobyl, 179Chinese cooking, 6chlorophyll, 135circadian rhythm, 149classical mechanics, 51clocks,

atomic, 15eternal, 15identical, 43light, 16molecular, 17

coal burning, 98coffee, 7, 79coherent

light scattering, 227scattering, 80, 81, 176X-rays, 227

coin tossesrandom walk, 50randomness, 50

cold fusion, 98

collisionasteroid, 36body cushion, 30bullet, 30molecules, 167spaceships, 46wall, in cartoons, 30

colorcadmium, 59F-centers, 281nanophase, 59ruby, 120

communicationblack hole, 106delays, 175jamming, 118spaceships, 43, 106

computeratomic, 73DNA, 221java quantum, 221nuclear spins, 221quantum, 79

concrete, 239conductivity

electrical, 200, 220thermal, 130, 131, 132

conics, 21consciousness, 86convection, 132Cooper pairs, 84copper

cladding, 175nanophase, 55oxide tunneling, 238X-ray laser, 73

Coriolis, G., 55cosmic rays, 93, 124, 221Coulomb

barrier, 244, 245, 248 blockade, 56

Crab Nebula, 182creative thinkers, 85crystal structure

Bragg scattering, 79diamond, 70graphite, 70growth, 120

288 Index

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lattice, 120, 248quantum dots, 74

crystalline, 55, 198cubes

array, 21space-filling, 22symmetry, 24, 160

cup, rotating, 27cuprates, 55Curie, M., 93, 95, 102 Curie, P., 93, 95, 102Curie temperature, 102, 199cylinder, 22

dark matter, 269dates, 9day length

Egyptians, 12hours in, 12sidereal, 37solar, 37tidal effects, 57winter, 61

de Broglie λ, 67, 222, 234decay

competitive, 57neutron, 95nuclear, 86

decoherence, 224defrosting tray, 5demonstrations, 34density, cream, 127Descartes, R., 57detector

calibration, 114gravitational, 80, 114neutrino, 81

determinism, 88deterministic

behavior, 57chaos, 57

detonation, 37deuteron, 96diamond, 70, 220digital timer, 15dimensions

defined, 159fractal, 159

fractional, 24space, 21, 22, 24space-time, 27

dimples in bat, 54dipole-dipole, 135dispersion, 59distance, 20DNA

clock, 17computer, 221

dodecahedron, 22, 24, 153Doppler shift, 65double-slit, 83, 213, 232driving, switching, 15

Earthatmosphere, 35climate, 140magnetic field, 241Moon, seen from, 77orbit change, 119rotations in year, 13speed in orbit, 12temperature, 178, 216warming, 118

ecliptic, 14eddy

current, 199turbulent, 197

efficiencylight bulb, 209photon engine, 61thermal, 52, 61

egg and bottle, 2Egypt

day length, 12pyramids, 54

Einstein, A.Beiblätter, 44Bible study, 107completeness, 83Curie letter, 104deductions by, 45determinism, 88dice quote, 100emission, 208EPR paradox, 87God, 100, 110

height, 69letter to Mileva, 43light bending, 106mass-energy, 41paper in 1905, 46photon emission, 68reality, 114relativity 1905, 27space-time, 89

electricalcircuit, 58conductor, 200current, house, 119diode, pickle, 137

electrical chargefractional, 200on pinhead, 56transferred, 200tunnel junction, 56

electronin a box, finding, 78Cooper pairs, in, 84density, metals, 130house current, 119single, tunneling, 200spin, vacuum effect, 84 Standard Model, 101

elementspun, 122 synthesis of, 94, 95, 123

ellipse, 21, 119elliptic functions, 116, 276emerald, 120energy

Earth warming, 118field, charge, 47kinetic, 51, 55mass-energy, 40, 42rotational, 99in universe, 115

energy levelsbenzene, 69C-12 nucleus, 95hydrogen, 67nuclear, 93quantum dot, 223

engineair-driven, 50

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engine (continued)Brownian, 50magnetocaloric, 52photon, 61

entropyasymmetry in, 18big bang, 114black hole, 110disorder, 58final state, 239heat flow, 58liquids, 196

eponymy, 24EPR paradox, 87equations, time in, 18equinoxes, 12, 14eternal clock, 15evaporation, 9explosions in space 33exponential growth, 57

faces, resolution of, 80fame, 59Faraday, M.

flux law, 202quotes, 286

farmergoose chase, 23rice, planting, 14

faster than cspotlight, 41quasar, 41wave function, 225

Fe-56 stability, 94Fermi, E.

energy, 220exclusion, 242Golden Rule, 87

Fermi-Dirac, statistics, 237

fermions, 177, 237ferromagnetism, 54, 102Feynman, R. P.

Brown-Twiss, 283ratchet, 50twin watches, 108

field theoryBoscovich, J., 66

nuclear shell, 93fission

Pu-239, 246U-235, 99, 245U-238, 246

Fitzgerald, G. F., 96fluid

binary, 52ferrofluid, 52inmiscible, 52magnetorheological, 52

fluorescencelights, 66quantum dots, 74

flywheels, coupled, 54FM radio frequency, 69force

aperiodic, 150applied, 44, 51, 55buoyant, 170centrifugal, 111field, in movies, 36fluctuating, 51geometry, 111torque, 144van der Waals, 135

forensics, paint, 654-D, 24, 27four-momentum, 40free energy, 196, 202free fall

acceleration, 30body cushion, 30cartoons, in, 30Galileo, 59Philoponus, J., 50twin watches, 108

friction, static, 128 Friday 13th, 79fullerene, 215

g-2, muon, 236Galaxy center, 274galaxies, 110, 111, 258, 265Galileo, G., 59, 142galvanometer, 58gamma rays, 250, 251Gamow, G., 109

geodesics, 111glass

absorption in, 135old window, 54

Glauber effect, 122gluons, 255glycemic index, 9gold density, 128GPS signals, 108grain

boundaries, 200 sand, atoms in, 65size, 55, 65

graphite, 70grasshoppers, 57gravitational

bound system, 113clock rate, 60, 108, 260detectors, 80, 114lensing, 109tidal effects, 205twins, 60waves, 80

gravitomagnetism, 259gravity

artificial, 31in galaxies, 112geodesics, 111geometry, 111

greenhouse effectgases, 70, 71, 216water vapor, 216

Groundhog Day, 14group velocity, 213growth, 57, 120gunfight, 30

hair-raiser function, 123Hall effect, quantized, 72halogen lamp, 138hamburgers, 131hardness, metals, 55Hawking radiation, 267He-4

burning in stars, 95superfluid, 85synthesis in stars, 96

headlight effect, 106

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heat bath, 61heat engine

air, 50Brownian, 50photon, 61

Heisenberg, W.matrix model, 84on reality, 78uncertainty principle, 83

helium burning, 95helium liquid, 72, 85Hubble telescope, 264human

aging, 45, 47, 60brains, 23, 54, 85collision damage, 166DNA, 17energy needs, 4faces, 35, 80health in UV, 67hearing, 100height, 69, 123memory, 156radioactivity, 97SAD, 17shrunken size, 32space debris, 32space journey, 45, 46UV need, 210walking, 59

Huxley, A., 119Huygens, C., 51hydrogen

atom, 67, 164bonds, 126, 196burning, star, 97MRI, 82nuclear device, 37in space, 174stability of atom, 27

icebond angles, 8floating, 6, 127fluorocarbons, 216in microwave, 8polar melt, 33in ponds, 133

ice cream, 5

icosahedron, 22, 25, 160ignition

oxygen, 133spontaneous, 133

inertia, 56information

black hole, and, 89phase, 214, 256quantum, 89speed of, 182storage, 73

infraredimages, 35, 80resolution, 176

insulation, fat, 10integrated circuits, 73interference

double-slit, 83intensity, 121laser, 213quantum, 75star light, 121

Internet, game play, 34invariants, 40, 41, 51iodine, 118, 179iron filings, 52

Josephson effect, 85

Kepler’s 3rd law, 264Kerr effect, 59kinetic energy

discovery, 51, 55vis viva, 55

Kordylewski cloud, 120

Lagrange points, 282Lamb shift, 84Landau levels, 220Larmor, J., 41laser

beams in space, 34kinetic, 68noninversion, 68security system, 34strobe, 146Weber, J., 68X-ray, 73, 212

LCD, 71

leap years, 13LED, 71lens, 66Lenz’s law, 199Lewis, G., 71, 83Libby, W., 93, 100life, 120, 122, 133light

ambient, 97approaching, 106bending of, 106, 109bulb, 122, 209and car driver, 42ceramic response, 59clock, 16cosmic background, 112efficiency, sources, 209faster than, 32, 41, 182flash, 15, 33, 34, 71, 173forensics, paint, 65geodesics, 111infrared, 80moving source, 119oven, 10sabers, 36scattering, 59, 80, 174, 177signals, 43squeezing, 84therapy, 17tired-light, 110tweezer, 66UV, 80

light tweezer, 66lightning, 33LIGO, 229Lincoln, A., 32lipstick color, 59liquid helium siphon, 72locality violation, 240Lorentz, H.,

contraction, 42 relativity, 41

magneticdipole, 198

magneticfield, 52, 72, 85induction stove, 10

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magnetic (continued)resonance, 82

Marsdebris on, 116moons, 30

Martians, 33maser, 68mass quadrupole, 80mass-energy

confusion, 42creation of, 115definition, 40, 41electron, 47particles, system of, 42Poincaré, H., 45, 47proton, 103

materialceramic, 59dispersive, 59empty space in, 68glass flow, 197hardness, 55magnetic, 54nanophase Cu, 55smart, 203stretching of, 35

mathematicians, 118, 121Matthew effect, 59Maxwell’s equations

failure of, 208 invariance of, 41symmetry of, 102

measurementBohr, 232Moon distance, 20quantum, 78, 83simultaneous, 191uncertainty principle, 83

meatChinese style, 6cooking roast, 5preserving, 5

Meissner effect, 55memory, 17men, 4, 10metals

density of, 128thermal conductivity, 130

meteor, 120meteorite in 1908, 100methane, 70, 71metronomes, two, 18Michelson, A., 42microtubules, 156microwaves

absorption, 135background, 99, 266heating water, 3ice absorption, 8metals in, 6oven, 6, 10

milk and cream, 3Milky Way, 112, 113, 274Miller, J. S., 2Minkowski, H., 27minute, origin, 13mirror types, 67, 211miscible liquids, 52modular function J, 276molecular

clock, 17displacement, 176, 177scattering IR, 177

momentum, 57, 164, 170MOND, 265, 270Moon

calendar, 14in daylight, 142distance, 20, 206Easter, 143full, 13size of, 25tidal effects, 57

Mössbauer Effecteardrum, 251gamma rays, 208Weber bar, 230

moviesbattle sounds, 33body cushion, 30gunfight, 30sounds, 33spaceships, 32warp speed, 32western, 165

MRI, 82

MSG, 2muon, 101, 236, 248

Nagaoka, H., 73nanophase cluster

cadmium, 59copper, 55energy levels, 203

neurons, 23, 95neutrino

detectors, 81, 231handedness, 103mass, 45, 103proton cycle, 97scattering, coherent, 81solar, 81supernova 1987A, 173Standard Model, 101

neutroncapture, 246, 249 decay of, 95

Newton, I.birth date of, 142comets, 110firstborns, 122height, 69 mass-energy, 42Principia, 23quip about, 54

Newton’s Laws2nd law, 125, 128, 1323rd law, 170torques, 144

Ni isotopes, 94night sky, 106nitrogen, liquid, 5NMR, 226noise

jamming with, 118robotic, helpful, 60shot, electrical, 84

north, finding, 18nuclear

beta decay, 244, 245binding energy, 242decay, 86device, 36, 37, 94, 180emergency, 118

292 Index

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energy levels, 93, 95fission, 99gamma rays, 250reactor, 97, 98, 179spin flip. 226spin-orbit, 242submarine, 36synthesis, 94, 95waste, 98

nucleuslarge, 99models, history of, 73neutron in, 95rotation of, 250shell model, 93stable, 94, 99

octahedron, 22, 24, 160Oklo nuclear reactor, 97Olbers’ paradox, 106optical

cavity, 61color, lipstick, 59solitons, 59

orbits, 27, 109, 119oscillators, 150oscilloscope, 182osmosis, 129ozone layer, 70

paintings, 65paramagnetism, 54parboil, 6Pauli, W., 80Pb bullets, 175Penrose, R., 114, 151, 155Penrose tiles, 153perceptions, 119perihelion, 12perpetuum mobile, 51, 124pH, 119phase

conjugate, 67diagram, 52fixed, 177, 226information in, 256local change in, 256locking, 18

random, 82, 227space, 239uncertainty, 235velocity, 213

philosophical, 74phosphorous, 175photon

absorption, 65bunching, 122emission, 65, 67engine, 61scattering, 78, 177UV, 66, 210

photosensitivity, 143photostrictive effect, 203physicist, 18, 20, 121, 165physics

equationless, 103mathematics, 114, 116

pi, 22, 23, 26pickle, electric, 9pinhead, charge on, 56Planck, M., 115planet orbits

Aryabhata, 99extrasolar, 272Kordylewski, 120precession, 109spacings, 113stability, 64, 108, 168

plasmabubble, 218display, 71ions, 66

Plato, 24, 116Platonic solids

defined, 24importance of, 160

playing card, 77plutonium, 36, 123Poincaré, H., 45, 47polar molecules, 8polytopes, 160pond freezing, 133population

growth, 57inversion, 68, 212noninversion, 68

positronium, 87pot, watched, 8potassium-40, 98potato, 4, 9Potter, F., 160, 253, 256, 276power plant, coal, 98precession, 109, 119, 139preserving food, 5prism, 87probability

amplitude, 75quantum, 75, 76reaction rate, 87

procrastinating, 13projectile impact, 30protein, 148proton, 103, 161, 255psychology, 165Pu-239, 99, 180pyramids, 54Pythagoras, 20Pythagorean

theorem, 24 triplets, 20

quadrupoleelectric, 207gravitational, 80

quantumCarnot cycle, 61coherence, 79, 205computer, 79dots, fluorescence of, 74engine, 205entanglement, 233information, 89interference, 75, 232noise, 235state, collective, 231, 273vacuum, 84

quantum mechanicsamplitude ψ, 75angular momentum, 67bosons, 85, 177bouncing ball, 87decoherence, 224development of, 82Einstein & Bohr, 100

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quantum mechanics (continued)EPR paradox, 87Fermi exclusion, 242fermions, 177, 237interpretation of, 83matrix origin, 84measurement, 83relativistic, 230rules of, 75, 224scattering, 80, 81, 214stationary states, 67superposition, 76time asymmetry, 151time operator, 181tunneling, 85, 237, 238wave function, 75, 78

quarkb’ quark, 276masses, 103, 116, 160families, 101, 276proton mass, 103Standard Model, 101

quasars, 41, 171quaternions, 163

Rabi, I. I., 72, 81radiation

cosmic, 112, 124infrared, 8Unruh, 121

radioactivitybackground, 98, 247C-14 dating, 93, 100K-40 in human, 98safety, 36

Ramanujan, S., 23randomness

Brownian motor, 50coin tosses, 50crystal growth, 280molecular, 51, 136phases, 227stability, for, 59thermal, 51in walk steps, 50in water, 136

ratchet, 50recoilless absorption, 65

rectangular array, 21redshift

cosmological, 41, 110Doppler, 65solar, 109tired-light, 110

reflection, total internal, 86refractive index, 59, 152refrigerator, 10, 24relativity, general

centrifugal force, 111clock rate, 190, 191, 260curvature, 115, 120equivalence, 107, 121, 190geodesics, 111GPS accuracy, 109light bending, 109redshift, 109, 262test masses, 107twin paradox, 46, 60, 108twin watches, 108twistors, 150

relativity, specialaccelerations, 44charge, 47clock rate, 261, 262Doppler formula, 278Einstein, A., 41energy defined, 40headlight effect, 106invariants, 40Larmor, J., 41Lewis, G., light, 71light energy, 119mass-energy, 41, 42Planck, M., 115Sagnac effect, 186Terrell effect, 162, 183twin paradox, 46, 60, 108twin watches, 108

rice, 9, 14, 143Riemann, G. B., 111robot, 23, 59Roman numeral V, 25rotation

book, 26cup, 27double, 25

of galaxy, 111nuclear, 99space station, 31

ruby, 120Rutherford, E., 73, 100

Sagnac effect, 43salt

in water, 7preservative, 5sea salt, 134

sand grain, 65sandglass, 14satellite

debris, 32, 37GPS, 108

scatteringcoherent, 80, 81, 176, 227constructive, 80neutrino, 81photon, 78

Schrödinger equationbenzene, 69de Broglie waves, 101nuclear, 93symmetry lack, 102

Schrödinger’s cat, 77Schwarzschild, K.

metric, 262, 263 radius, 234

science fiction, 35scientific method, 97scientist, 16, 23, 36, 59scissors closing, 182scooter, twist, 120Seaborg, G. T., 123SET diode, 200sexagesimals, 13Sierpinski triangle, 158silicon, 81sodium D lines, 138solar heating, 195Solar System, 113solitons, 59solstices, 12, 13, 14sonoluminescence, 71sound

archaeology, 53

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birdsong, 122existence of, 164explosions, 33in friction, 129into light flash, 71in space, 33, 36speed of, 35, 173, 182in Weber bar, 229

soup, can of, 4space

configuration, 23, 78, 233crawler, 124curvature, 115dimensions of, 27discrete, 115expansion, 41, 171, 258,

265filling, 21, 22, 153, 2754-D, 24, 269imaginary, 27

spaceshipaccelerating, 44, 45black hole, 269collisions, 46communication from, 43energy need, 170in movies, 32, 34nuclear, 32propulsion, 171, 286relativistic, 41, 43, 44temperature of, 179warp speed, 32

space-timeEinstein, A., 89coordinates, 27curvature, 120distorting, 171fabric, tear of, 37interval, 40imaginary space, 27Minkowski, 27quaternions, 163

spark gap, 278spectral fingerprint, 206spheres

intersecting, 25reactions on, 137three-sphere, 25

spider and fly, 20spin flip, 226spin one-half, 161, 177spotlight, 41square root, 27SQUIDS, 85St. Augustine, 11stability, 27, 94, 108, 164Standard Model

b’ quark, 276discrete subgroups, 254families, 101, 276gauge group, 254Higgs particle, 102, 254mass ratios, 160weak interaction, 147

starC-12 synthesis, 95diameters, 121element synthesis, 95Fe-56 in, 94helium, 95hydrogen, 97lifetime, 257proton-proton cycle, 96shapes, 113

stereoscopic view, 184Stevin, S., 59Stigler’s law, 24stimulated emission

laser, 68noninversion, 68

strain gauge, 42straw in potato, 4sugar

igniting, 7preserving with, 5sucrose in blood, 9in water, 3

summer in January, 12sun

proton cycle, 96redshift, 109

sun rhyme, 34sundial, 14sunlight spectrum, 210sunrise, sunset, 12, 13superconductor

Cooper pairs, 84Meissner effect, 55quantum effects, 84SQUIDS, 85suspension, 55

superfluidity, 72, 85supernova

element synthesis, 243intensity curve, 2661987A, 173Type 1a, 233

superpositionbenzene, 214computer states, 79electron in box, 78rule, 75two states, 77, 214

super-radiance, 74Swift, J., on Mars, 30syllogism, 22, 97symmetry

asymmetry, 50breaking, 101discrete, 162, 254ferromagnet, 102group theory, 160polyhedral, 160potential, 50rotational, 24, 214spherical, 180SU(2), 162tetrahedral, 160, 215time, 18

tasting food, on, 2tau, 101tea water, heating, 3teakettle, water vapor, 8Teller, E., 100, 250temperature

boiling, 5Earth, 178, 216infrared, 8nanoKelvin, 223radiation, 8

Terrell effect, 183, 184Tesla, N., 58, 61tesselation, 153

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tetrahedron, 22, 24, 160theory and truth, 24thermal energy

burning in air, 133equilibrium, 178flow direction, 58mantle, as source, 217random motion, 51

thermal stress, ice 133Thomson, J. J., 47thinking too much, 119thunder, 33tidal effects, 57time

coordinate, 27direction, 276elapsed, 46, 108goes by, 12gravity, 60interval, 42, 43, 190reversal, 16, 210St. Augustine, 11symmetry, 18travel, 34twin watches, 108

tomato, peeling, 7tritium, 230tunnel junction, 56tunneling

light, 86nuclear, 86SQUIDS, 85

twin paradox, 46, 60, 108two-photon

absorption, 207 H emission, 207

U-235, 36, 94, 97, 99, 180uncertainty

in amplitude, 235digital, 145hydrogen, 231in phase, 235principle, 83, 233, 254relation, 232simplified, 89

universecurvature, 275

different ones, 115, 124dimensions of, 153expansion, 106, 110, 265inflationary, 41, 271potential well, 271saying about, 52total energy in, 115

Unruh radiation, 121

vacuumelectron spin, 84Casimir effect, 84Higgs particle, 102Lamb shift, 84muon g-2, 236quantum, energy in, 84particle “soup”, 84, 234Unruh radiation, 121

van der Waals force, 135vapor pressure, 135velocity

limit c, 41parameter, 188, 189redshift, 41relativistic, 41, 45

Venus, orbit period, 68Verne, Jules

memorial, 35Moon trip, 31weightlessness, 31

video games, 21, 35volume

cooking meat, 6to surface area, 133, 135

watchfinding north, 18free falling, 108 mountains, rate in, 14

waterboiling, altitude, 5boiling, pot, 8carbon dioxide in, 9cloud weight, 33cup of, spouting, 285density, max, 133evaporation, 9expansion, 172

heating for tea, 8helium in, 217ice in, 6salt in, 7spouting, 122sugar in, 3in universe, 3

water molecule, 8, 9water vapor

absorption, 216condensing, 135greenhouse gas, 70, 71teapot, 8

wave functioncollapse, 78configuration space, 78Cooper pair, 237coordinate space, 78defined, 75two-particle, 224

weak interactionbosons, 103doublet states, 255left-handedness, 147symmetry, 102Z boson, 103

Weber, J.bar detector, 81coherence, 81, 82gravitation, 81laser and maser, 68neutrinos, 81

weightlessness, 31Weinberg, S., 266Wells, H. G., 33, 37 Wheeler, J. 250, 259wings, beating, 31women, 4, 10wrist watch, 87

X-rayBragg scattering, 79laser, 73, 212refractive index, 69

year length, 13, 15

zero-point energy, 235

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