General Relativity Reading Assignment

8/18/2019 General Relativity Reading Assignment

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302

N

o material object, particle or futurespaceship, can be accelerated to

the speed of light. Why this is

so has to do with momentum andenergy, which, in relativity theory,have new definitions. One of the

most celebrated outcomes of spe-cial relativity is the discovery that

mass and energy are one andthe same thing—as described by

E mc 2. Einstein’s general theoryof relativity , developed a decade

after his special theory of relativ-ity, offers another celebrated out-come, an alternative to Newton’s

theory of gravity. Both theories ofrelativity have changed the way we see

the universe.

How Can Space-Time be Modeled?

1. Stretch a plastic garbage bag tightly acrossthe top of a trash can. Tape the edges of thebag to the side of the can.

2. Place a pool ball or other heavy sphere in thecenter of the garbage bag. This should causethe bag to sag in the center.

3. Launch a marble by giving it a velocity tan-gent to the circumference of the can.

4. Try launching the marble with a variety of ini-tial velocities.

Analyze and Conclude

1. Observing Describe the motion of themarble. What effect does changing the initialspeed and direction of the marble have onthe shape of the orbit?

2. Predicting How might changing the mass ofthe heavy central sphere affect the motion ofthe marble?

3. Making Generalizations How closely doesthis model represent the motion of Earthsatellites?

discover!

RELATIVITY—MOMENTUM,

MASS, ENERGY, AND GRAVITY

According to special relativity, mass and energyare equivalent. According to general relativity,gravity causes space to become curved andtime to undergo changes.

THE BIG

IDEA . . . . . . . . . .

16

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RELATIVITY—MOMENTUM,MASS, ENERGY,AND GRAVITY

Objectives• Describe how an object’smomentum changes as itapproaches the speed of light.(16.1)

• Describe how mass and energyare related. (16.2)

• Describe how thecorrespondence principleapplies to special relativity.(16.3)

• State the principle ofequivalence. (16.4)

• Describe the relationshipbetween the presence of massand the curvature of space-time. (16.5)

• Describe Einstein’s predictionsbased on his theory of generalrelativity. (16.6)

discover!

MATERIALS plastic garbagebag, trash can, pool ball,marble

ANALYZE AND CONCLUDE

Slow moving marblesrapidly spiral into thecentral ball. Faster moving

marbles should “orbit”about the central ball.Depending on the directionof launch, orbits may becircular or elliptical.

A heavier ball will make asteeper well, simulating astronger gravitational field.

Done with care, the marblemay nicely simulate a planetorbiting the sun or a probe

orbiting Earth.

1.

2.

3.


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CHAPTER 16 RELATIVITY—MOMENTUM, MASS, ENERGY, AND GRAVITY 303

16.1 Momentum and Inertiain Relativity

If we push an object that is free to move, it will accelerate. If wemaintain a steady push, it will accelerate to higher and higher speeds.If we push with a greater and greater force, we expect the accelera-tion in turn to increase. It might seem that the speed should increasewithout limit, but there is a speed limit in the universe—the speedof light. In fact, we cannot accelerate any material object enough toreach the speed of light, let alone surpass it.

Newtonian and Relativistic Momentum We can under-stand this from Newton’s second law, which Newton originallyexpressed in terms of momentum: F mv/ t (which reduces tothe familiar F ma, or a F /m). The momentum form, interest-ingly, remains valid in relativity theory. Recall from Chapter 8 thatthe change of momentum of an object is equal to the impulse appliedto it. Apply more impulse and the object acquires more momentum.Double the impulse and the momentum doubles. Apply ten times asmuch impulse and the object gains ten times as much momentum.Does this mean that momentum can increase without any limit, eventhough speed cannot? Yes, it does.

We learned that momentum equals mass times velocity. In equa-tion form, p mv (we use p for momentum). To Newton, infinitemomentum would mean infinite speed. Not so in relativity. Einstein

showed that a new definition of momentum is required. It is

p

1mv

v 2

c 2

where v is the speed of an object and c is the speed of light. Thisis relativistic momentum, which is noticeable at speeds approach-ing the speed of light. Notice that the square root in the denominatorlooks just like the one in the formula for time dilation in Chapter 15.It tells us that the relativistic momentum of an object of mass m and

speed v is larger than mv by a factor of 1/1 (v 2/c 2) .As an object approaches the speed of light, its momen-

tum increases dramatically. As v approaches c, the denominatorof the equation approaches zero. This means that the momentumapproaches infinity! An object pushed to the speed of light wouldhave infinite momentum and would require an infinite impulse,which is clearly impossible. So nothing that has mass can be pushedto the speed of light, much less beyond it. Hence, we see that c is thespeed limit in the universe.

For:

Visit:

Web Code: –

Links on speed of light

www.SciLinks.org

csn 1601

At least one thingreaches the speed oflight—light itself! But aparticle of light has norest mass. A materialparticle can never be

brought to the speedof light. Light can neverbe brought to rest.

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16.1 Momentum andInertia in Relativity

Key Termsrelativistic momentum, rest mass

Common Misconception The momentum of an object isalways simply its mass 3 velocity.

FACT The relativistic momentumof an object of mass m and speedv is actually larger than mv .

Teaching Tip State that if youpush an object that is free tomove, it accelerates in accord withNewton’s second law, a 5 F/m. Themomentum version of Newton’ssecond law, F 5 D p / Dt , says that ifyou push an object that is free to

move, its momentum increases.Both the acceleration and thechange-of-momentum versionsof the second law give the sameresult. However, for very highspeeds, the momentum versionis more accurate. F 5 D p / Dt holdsfor all speeds, even those nearthe speed of light—as long asthe relativistic expression formomentum is used.


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What if v is much less than c? Then the denominator of the equa-tion is nearly equal to 1 and p is nearly equal to mv. Newton’s defini-tion of momentum is valid at low speed.

Trajectory of High-Speed Particles We often say that a par-ticle pushed close to the speed of light acts as if its mass were increas-ing, because its momentum—its “inertia in motion”—increases morethan its speed increases. The rest mass of an object, represented bym in the equation on the previous page, is a true constant, a propertyof the object no matter what speed it has.

Subatomic particles are routinely pushed to nearly the speedof light. The momenta of such particles may be thousands of timesmore than the Newton expression mv predicts. One way to look atthe momentum of a high-speed particle is in terms of the “stiffness”

of its trajectory. The more momentum it has, the harder it isto deflect it—the “stiffer” is its trajectory. If it has a lot of momen-tum, it more greatly resists changing course.

This can be seen when a beam of electrons is directed into amagnetic field, as shown in Figure 16.1. Charged particles movingin a magnetic field experience a force that deflects them from theirnormal paths. For a particle with a small momentum, the path curvessharply. For a particle with a large momentum, the path curves onlya little—its trajectory is “stiffer.” Even though one particle may bemoving only a little faster than another one—say 99.9% of the speedof light instead of 99% of the speed of light—its momentum willbe considerably greater and it will follow a straighter path in themagnetic field. Through such experiments, physicists working withsubatomic particles at atomic accelerators verify every day the cor-rectness of the relativistic definition of momentum and the speedlimit imposed by nature.

CONCEPT

CHECK . . . . . .How does an object’s momentum change as itapproaches the speed of light?

FIGURE 16.1

If the momentum of theelectrons were equal tothe Newtonian value mv,the beam would follow thedashed line. But becausethe relativistic momen-

tum, or inertia in motion,is greater, the beam fol-lows the “stiffer” trajectoryshown by the solid line.

At ordinary speeds,an object’s momentumis simply its classicalvalue, mv . For example,at 30 m/s (0.0000001c),the relativistic momen-tum differs from theclassical value by lessthan one trillionth of apercent. Newton’s defi-nition of momentum isvalid at low speeds.

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Teaching Tip Write theexpression for relativisticmomentum on the board.Point out that it differs fromthe classical expression formomentum by its denominator.A common interpretation isthat of a relativistic mass,

m 5

mo / √

12

v

2

/ c

2

, timesvelocity v . Because the increase inm with speed is directional (as islength contraction), and P ratherthan m is a vector, the conceptof momentum increase ratherthan mass increase is preferred inadvanced physics courses. Eithertreatment of relativistic mass orrelativistic momentum, however,leads to the same description ofrapidly moving objects in accord

with observations.

Teaching Tip Show thatfor small speeds the relativisticmomentum equation reducesto the familiar mv (just as forsmall speeds t 5 t o in timedilation). Then show that whenv approaches c , the denominatorof the equation approaches zero.This means that the momentumapproaches infinity! An object

pushed to the speed of lightwould have infinite momentumand would require an infiniteimpulse (force 3 time), whichis clearly impossible. Nothingmaterial can be pushed to thespeed of light. The speed of lightc is the upper speed limit in theuniverse.

As an objectapproaches the

speed of light, its momentumincreases dramatically.

T e a c h i n g R e s o u r c e s

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DVDs Special Relativity II

CONCEPT

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CONCEPT

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16.2 Equivalence of Mass and EnergyA remarkable insight of Einstein’s special theory of relativity is hisconclusion that mass is simply a form of energy. A piece of matter,even if at rest and even if not interacting with anything else, has an“energy of being” called its rest energy. Einstein concluded that it

takes energy to make mass and that energy is released when mass dis-appears. Rest mass is, in effect, a kind of potential energy. Mass storesenergy, just as a boulder rolled to the top of a hill stores energy. Whenthe mass of something decreases, as it can do in nuclear reactions,energy is released, just as the boulder rolling to the bottom of the hillreleases energy.

Conversion of Mass to Energy The amount of rest energy E isrelated to the mass m by the most celebrated equation of the twenti-eth century,

E mc 2

where c is again the speed of light. This equation gives the total energycontent of a piece of stationary matter of mass m. Mass and energyare equivalent—anything with mass also has energy.

In ordinary units of measurement, the speed of light c is a largequantity and its square is even larger. This means that a small amountof mass stores a large amount of energy. The quantity c 2 is a “conver-sion factor.” It converts the measurement of mass to the measurement

of equivalent energy. It is the ratio of rest energy to mass: c 2E/m

.Its appearance in either form of this equation has nothing to dowith light and nothing to do with motion. The magnitude of c 2 is90 quadrillion (9 1016) joules per kilogram. One kilogram of matterhas an “energy of being” equal to 90 quadrillion joules. Even a speck ofmatter with a mass of only 1 milligram has a rest energy of 90 billion

joules. (This is equivalent to the kinetic energy of a 3-ton truck mov-ing at over 20 times the speed of sound!)

Examples of Mass-Energy Conversions Rest energy, like any

form of energy, can be converted to other forms. When we strike amatch, for example, a chemical reaction occurs and heat is released.Phosphorus atoms in the match head rearrange themselves and com-bine with oxygen in the air to form new molecules. The resultingmolecules have very slightly less mass than the separate phosphorusand oxygen molecules. From a mass standpoint, the whole is slightlyless than the sum of its parts, but not by very much—by only aboutone part in a billion. For all chemical reactions that give off energy,there is a corresponding decrease in mass.

Can we look at the equa-tion E mc 2 in anotherway and say that mattertransforms into pureenergy when it is travel-ing at the speed of lightsquared?Answer: 16.2

think!

E = mc2 says that massis congealed energy.Mass and energy aretwo sides of the samecoin.

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16.2 Equivalence ofMass and Energy

Key Termrest energy

Common Misconception E 5 mc 2 means that energy ismass traveling at the speed oflight squared.

FACT The equation gives the totalenergy content of a piece of

stationary matter of mass m.

Teaching Tip Write E 5 mc 2 on the board. State that this isthe most celebrated equation ofthe twentieth century. It relatesenergy and mass. Every materialobject is composed of energy—“energy of being.” This energy ofbeing is appropriately called restenergy.

Teaching Tip Stress that c

2

isa constant conversion factor andis NOT the speed of the mass.Also, stress that the equationE 5 mc 2 is NOT restricted tochemical and nuclear reactions.

Teaching Tip E 5 mc 2 saysthat mass is congealed energy.Mass and energy are two sides ofthe same coin.

The mass–energy equivalence is

important and usually generateshigh interest.


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In nuclear reactions, the decrease in rest mass is considerablymore than in chemical reactions—about one part in a thousand.This decrease of mass in the sun by the process of thermonuclearfusion bathes the solar system with radiant energy and nourisheslife. The sun is so massive that in a million years only one ten-millionth of the sun’s rest mass will have been converted to radi-

ant energy. The present stage of thermonuclear fusion in the sunhas been going on for the past 5 billion years, and there is sufficienthydrogen fuel for fusion to last another 5 billion years. It is nice tohave such a big sun! Nuclear power plants, such as the one shown inFigure 16.3, make use of the equivalence of mass and energy toproduce enormous amounts of energy.

The equation E mc 2 is not restricted to chemical and nuclearreactions. A change in energy of any object at rest is accompaniedby a change in its mass. The filament of a lightbulb has more masswhen it is energized with electricity than when it is turned off. Ahot cup of tea has more mass than the same cup of tea when cold.A wound-up spring clock has more mass than the same clock whenunwound. But these examples involve incredibly small changes in

mass—too small to be measured by conventional methods. No won-der the fundamental relationship between mass and energy was notdiscovered until the 1900s.

The equation E mc 2 is more than a formula for the conversionof rest mass into other kinds of energy, or vice versa. It states thatenergy and mass are the same thing. Mass is simply congealed energy.If you want to know how much energy is in a system, measure itsmass. For an object at rest, its energy is its mass. Shake a massiveobject back and forth; it is energy itself that is hard to shake.

CONCEPT

CHECK . . . .

. .

What is the relationship between mass and energy?

FIGURE 16.3

Saying that a power plantdelivers 90 million megajoulesof energy to its consumers isequivalent to saying that itdelivers 1 gram of energy toits consumers, because mass

and energy are equivalent.

FIGURE 16.2

In one second, 4.5 milliontons of rest mass are con-verted to radiant energy inthe sun.

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Teaching Tip State that themass of something is actually theinternal energy within it and thatthis energy can be converted toother forms of energy, such aslight.

Teaching Tip Mention thatthe 4.5 million tons of matter

that is converted to radiantenergy by the sun each secondis carried (as radiant energy)through space, so when we speakof matter being “converted” toenergy, we are merely convertingfrom one form to another—froma form with one set of units,perhaps, to another. Because ofthe mass– energy equivalence,in any reaction that takes intoaccount the whole system, thetotal amount of mass 1 energydoes not change.

Mass and energy are

equivalent—anythingwith mass also has energy.



• Problem-Solving Exercises inPhysics 9-2



CONCEPT

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16.3 The Correspondence PrincipleIf a new theory is to be valid, it must account for the verified resultsof the old theory. The correspondence principle states that newtheory and old must overlap and agree in the region where the resultsof the old theory have been fully verified. It was advanced as a prin-

ciple by the Danish physicist Niels Bohr earlier in this century whenNewtonian mechanics was being challenged by both quantum theoryand relativity. According to the correspondence principle, if theequations of special relativity (or any other new theory) are to be valid, they must correspond to those of Newtonian mechanics—classical mechanics—when speeds much less than the speed of lightare considered.

The relativity equations for time dilation, length contraction, andmomentum are

t

1t 0

p

1mv

1 v 2

c 2

v 2

c 2

v 2

c 2

L L0

We can see that each of these equations reduces to a Newtonian valuefor speeds that are very small compared with c. Then, the ratio (v /c )2

is very small, and for everyday speeds may be taken to be zero. Therelativity equations become

t 1 0

t 0

p

1 0

mv

1 0

t 0

L0

mv

L L0

So for everyday speeds, the time scales and length scales of movingobjects are essentially unchanged. Also, the Newtonian equationfor momentum holds true (and so does the Newtonian equationfor kinetic energy). But when the speed of light is approached,things change dramatically. Near the speed of light Newtonianmechanics change completely. The equations of special relativityhold for all speeds, although they are significant only for speedsnear the speed of light.

Equations remind usthat you can neverchange only one thing.Change a term on one

side of an equation andyou change somethingon the other side.

Much of nature is builton patterns, and look-ing for those patternsis the primary preoc-cupation of both artistsand scientists. We con-

nect things that werealways there but neverput together in ourthinking.

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16.3 TheCorrespondencePrinciple

Key Termscorrespondence principle,general theory of relativity

Teaching Tip The equationsin this section serve only to

illustrate the correspondenceprinciple. It is not necessary thatyour students memorize them.

Teaching Tip Show yourstudents that when small speedsare involved, the relativityformulas reduce to the everydayobservation that time, length,and the momenta of things do

not appear any different whenmoving. This is because thedifferences are too tiny to detect.

The correspondence principle isone of the neater principles ofphysics and is a guide to clearand rational thinking—not onlyabout the ideas of physics,but for all good theory—evenin areas as far removed from

science as government, religion,and ethics. Simply put, if a newidea is valid, then it ought to bein harmony with ideas that arevalid in the region it overlaps.


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So we see that advances in science take place not by discardingthe current ideas and techniques, but by extending them to revealnew implications. Einstein never claimed that accepted laws of phys-ics were wrong, but instead showed that the laws of physics impliedsomething that hadn’t before been appreciated.

The special theory of relativity is about motion observed in uni-

formly moving frames of reference, which is why it is called special.Einstein’s conviction that the laws of nature should be expressedin the same form in every frame of reference, accelerated as well asnon-accelerated, was the primary motivation that led him to developthe general theory of relativity —a new theory of gravitation, inwhich gravity causes space to become curved and time to slow down.

CONCEPT

CHECK . . . . . .How does the correspondence principle apply to

special relativity?

16.4 General RelativityEinstein was led to a new theory of gravity by thinking about observ-ers in accelerated motion. He imagined himself in a spaceship faraway from gravitational influences, as shown in Figure 16.4. In sucha spaceship at rest or in uniform motion relative to the distant stars,Einstein and everything within the ship would float freely; therewould be no “up” and no “down.” But if rocket motors were activatedto accelerate the ship, things would be different; phenomena similar

to gravity would be observed. The wall adjacent to the rocket motors(the “floor”) would push up against any occupants and give them thesensation of weight. If the acceleration of the spaceship were equalto g, the occupants could well be convinced the ship was not acceler-ating, but was at rest on the surface of Earth.

FIGURE 16.4

Imagine being on a spaceshipfar away from gravitational

influences. a. Everythinginside is weightless when thespaceship isn’t accelerating.b. When the spaceship accel-erates, an occupant insidefeels “gravity.”

Einstein actually imag-ined himself in eleva-tors, certainly morecommon at the timethan spaceships.

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According to thecorrespondence

principle, if the equations ofspecial relativity (or any othernew theory) are to be valid, theymust correspond to those ofNewtonian mechanics—classicalmechanics—when speeds muchless than the speed of light are

considered.



• Concept-DevelopmentPractice Book 16-1



16.4 GeneralRelativity

Key Termprinciple of equivalence

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The Principle of Equivalence Einstein concluded, in what isnow called the principle of equivalence, that gravity and acceleratedmotion through space-time are related. The principle of equiva-lence states that local observations made in an accelerated frameof reference cannot be distinguished from observations made ina Newtonian gravitational field. There is no way you can tell whether

you are being pulled by gravity or being accelerated. The effects ofgravity and the effects of acceleration are equivalent.

To examine this new “gravity” in the accelerating spaceship,Einstein considered the consequence of dropping two balls, say oneof wood and the other of lead. Figure 16.5 shows that when released,the balls would continue to move upward side by side with the veloc-ity that the ship had at the moment of release. If the ship were mov-ing at constant velocity (zero acceleration), the balls would appear toremain suspended in the same place since both the ship and the balls

move the same amount. But if the ship were accelerating, the floorwould move upward faster than the balls, which would soon be inter-cepted by the floor. Both balls, regardless of their masses, would meetthe floor at the same time. Occupants of the spaceship might attri-bute their observations to the force of gravity.

Both interpretations of the falling balls are equally valid. Einsteinincorporated this equivalence, or impossibility of distinguish-ing between gravitation and acceleration, in the foundation of hisgeneral theory of relativity. The principle of equivalence would beinteresting but not revolutionary if it applied only to mechanicalphenomena. But Einstein went further and stated that the principleholds for all natural phenomena, including optical, electromagnetic,

and mechanical phenomena.

FIGURE 16.5

To an observer insidethe accelerating ship, alead ball and a wood ballaccelerate downward

together when released, just as they would ifpulled by gravity.

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Teaching Tidbit Spaceelevator: Satellites in synchronousorbit can drop vertical cablesto the surface of Earth wherethey can be attached. Ratherthan rocketing material to thesatellite, material can be lifted inelevator fashion!


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Using his principle of equivalence, Einstein took another giantstep that led him to the general theory of relativity. He reasoned thatsince acceleration (a space-time effect) can mimic gravity (a force),perhaps gravity is not a separate force after all; perhaps it is nothingbut a manifestation of space-time. From this bold idea he derived themathematics of gravity as being a result of curved space-time.

According to Newton, tossed balls curve because of a force of grav-ity. According to Einstein, tossed balls and light don’t curve because ofany force, but because the space-time in which they travel is curved.

CONCEPT

CHECK . . . . . .

What does the principle of equivalence state?

16.5 Gravity, Space, and a NewGeometry

Space-time has four dimensions—three space dimensions (such aslength, width, and height) and one time dimension (past to future).Einstein perceived a gravitational field as a geometrical warping offour-dimensional space-time. Four-dimensional geometry is alto-gether different from the three-dimensional geometry introduced byEuclid centuries earlier. Euclidean geometry (the ordinary geometrytaught in school) is no longer valid when applied to objects in thepresence of strong gravitational fields.

Four-Dimensional Geometry The familiar rules of Euclideangeometry pertain to various figures that can be drawn on a flat sur-face. In Euclidean geometry, the ratio of the circumference of a circleto its diameter is equal to ; all the angles in a triangle add up to 180°;and the shortest distance between two points is a straight line. Therules of Euclidean geometry are valid in flat space, but if you drawcircles or triangles on a curved surface like a sphere or a saddle-shapedobject, as shown in Figure 16.9, the Euclidean rules no longer hold. If

you measure the sum of the angles for a triangle drawn on the outsideof a ball (positive curvature), the sum of the angles is greater than

180˚. For a triangle drawn on a “saddle” (negative curvature), thesum is less than 180˚.

FIGURE 16.8

The trajectory of a baseballtossed at nearly the speedof light closely follows thetrajectory of a light beam.

FIGURE 16.9

The sum of the angles of atriangle is not always 180°.a. On a flat surface, the sumis 180°. b. On a sphericalsurface, the sum is greaterthan 180°. c. On a saddle-shaped surface, the sum is

less than 180°.

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The principle ofequivalence states

that local observations made inan accelerated frame ofreference cannot bedistinguished from observationsmade in a Newtoniangravitational field.





16.5 Gravity, Space,

and a New GeometryKey Termsgeodesic, gravitational wave

CONCEPT

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Similarly, the geometry of Earth’s two-dimensional curved sur-face differs from the Euclidean geometry of a flat plane. As shown inFigure 16.10a, the sum of the angles for an equilateral triangle (theone here has the sides equal

14

Earth’s circumference) is greater than180°. Earth’s circumference is only twice its diameter, as illustrated inFigure 16.10b, instead of 3.14 times its diameter.

Of course, the lines forming the triangles in Figures 16.9 and

16.10 are not “straight” from the three-dimensional view, but arethe “straightest” or shortest distances between two points if we areconfined to the curved surface. These lines of shortest distance arecalled geodesics.

The path of a light beam follows a geodesic. Suppose three exper-imenters on planets Earth, Venus, and Mars measure the angles ofa triangle formed by light beams traveling between them. The lightbeams bend when passing the sun, resulting in the sum of the threeangles being larger than 180°, as illustrated in Figure 16.11. So thethree-dimensional space around the sun is positively curved. The

planets that orbit the sun travel along four -dimensional geodesics inthis positively curved space-time. Freely falling objects, satellites, andlight rays all travel along geodesics in four-dimensional space-time.

The Shape of the Universe Although space-time is curved“locally” (within a solar system or within a galaxy), recent evidenceshows that the universe as a whole is “flat.” This is a striking knife-edge condition. There are an infinite number of possible positivecurvatures to space-time, and an infinite number of possible negativecurvatures, but only one condition of zero curvature. A universe of

zero or negative curvature is open-ended and extends without limit.

Look at an airplane’sflight path drawn ona flat map and you’llsee that the line is

curved. The same linedrawn on the surfaceof a globe would be ageodesic—a “straight”(shortest-distance-between-two-points)line on Earth’s curvedsurface.

FIGURE 16.10

The geometry of Earth’stwo-dimensional curvedsurface differs from theEuclidean geometry of a

flat plane.

FIGURE 16.11

The light rays joining thethree planets form a tri-angle. Since the sun’s grav-ity bends the light rays, thesum of the angles of theresulting triangle is greater

than 180°.

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Teaching Tip One importantpoint to make is that relativitydoesn’t mean that everythingis relative, but rather that nomatter how you view a situation,the physical outcome is the same.There is a general misconceptionabout this. Point out that inspecial and general relativity the

fundamental truths of naturelook the same from every pointof view—not different fromdifferent points of view!


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If the universe had positive curvature, it would close in on itself, just as the surface of Earth closes in on itself. If you march straightahead on Earth, never turning, you will eventually return to yourstarting point. And if you shine a flashlight into a space of positivecurvature, the light will eventually illuminate the back of your head(if you wait long enough!). No one knows why the universe is actu-

ally flat or nearly flat. The leading theory is that this is the result ofan incredibly large and near-instantaneous inflation that took placeas part of the Big Bang some 13.7 billion years ago.

General relativity, then, calls for a new geometry: a geometrynot only of curved space but of curved time as well—a geometry ofcurved four-dimensional space-time.16.5.1 Even if the universe at largehas no average curvature, there’s very much curvature near massivebodies. The presence of mass produces a curvature or warpingof space-time; conversely, a curvature of space-time reveals thepresence of mass. Instead of visualizing gravitational forces betweenmasses, we abandon altogether the idea of force and think of massesresponding in their motion to the curvature or warping of the space-time they inhabit. General relativity tells us that the bumps, depres-sions, and warpings of geometrical space-time are gravity.16.5.2

We cannot visualize the four-dimensional bumps and depressionsin space-time because we are three-dimensional beings. We can geta glimpse of this warping by considering a simplified analogy in twodimensions: a heavy ball resting on the middle of a waterbed, whichis illustrated in Figure 16.12. The more massive the ball, the more it

dents or warps the two-dimensional surface. A marble rolled acrosssuch a surface may trace an oval curve and orbit the ball. The planetsthat orbit the sun similarly travel along four-dimensional geodesics inthe warped space-time about the sun.

Gravitational Waves Every object has mass, and thereforemakes a bump or depression in the surrounding space-time. Whenan object moves, the surrounding warp of space and time moves toreadjust to the new position. These readjustments produce ripples inthe overall geometry of space-time, similar to moving a ball that restson the surface of a waterbed. A disturbance ripples across the water-bed surface in waves; if we move a more massive ball, then we get agreater disturbance and the production of even stronger waves. Theripples that travel outward from the gravitational sources at the speed

of light are gravitational waves.

Whoa! We learnedpreviously that the pull ofgravity is an interactionbetween masses. And welearned that light has no

mass. Now we say thatlight can be bentby gravity. Isn’t this acontradiction?Answer: 16.5

think!

FIGURE 16.12

Space-time near a star iscurved in a way similar tothe surface of a waterbedwhen a heavy ball restson it.

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Teaching Tidbit Gravitybends around lumps of matterlike light bends in lenses. Lightemitted from a source travelsalong multiple geodesic paths toan observer who sees multipledistorted images of the sourceprojected onto the sky. So likea light lens, a gravitational lens

can produce multiple imagesthe way a fun-house mirrordoes with light. The amount ofgravitational bending dependson the mass.

Teaching Tidbit Planetsin our solar system don’t crashinto the sun only because theirtangential velocities are sufficientfor orbit. Likewise for stars ingalaxies: Stars with sufficient

tangential velocities orbit aboutthe galactic center. But slowerstars are pulled into and gobbledup by the galactic nucleus, which,if massive enough, is usually ablack hole.


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Any accelerating object produces a gravitational wave. In gen-eral, the more massive the object and the greater its acceleration, thestronger the resulting gravitational wave. But even the strongest wavesproduced by ordinary astronomical events are the weakest known innature. For example, the gravitational waves emitted by a vibratingelectric charge are a trillion-trillion-trillion times weaker than the

electromagnetic waves emitted by the same charge. Detecting gravita-tional waves is enormously difficult, but physicists think they may beable to do it, and searches are under way at present.

CONCEPT

CHECK . . . . . .What is the relationship between the presence

of mass and the curvature of space-time?

16.6 Tests of General RelativityUpon developing the general theory of relativity, Einstein pre-

dicted that the elliptical orbits of the planets precess about thesun, starlight passing close to the sun is deflected, and gravitationcauses time to slow down. Later, his predictions were successfullytested and confirmed.

Precession of the Planetary Orbits Using four-dimensionalfield equations, Einstein recalculated the orbits of the planets aboutthe sun. Planets and comets travel along curved paths because of thecurvature of space-time. With only one minor exception, his theorygave almost exactly the same results as Newton’s law of gravity. The

exception was that Einstein’s theory predicted that the ellipticalorbits of the planets should precess independent of the Newtonianinfluence of other planets, as shown in Figure 16.13. This precessionwould be very slight for distant planets and more pronounced closeto the sun. Mercury is the only planet close enough to the sun for thecurvature of space to produce an effect big enough to measure.

Precession in the orbits of planets caused by perturbations ofother planets was well known. Since the early 1800s astronomersmeasured a precession of Mercury’s orbit—about 574 seconds of

arc per century. Perturbations by the other planets were found toaccount for the precession—except for 43 seconds of arc per century.Even after all known corrections due to possible perturbations byother planets had been applied, the calculations of scientists failed toaccount for the extra 43 seconds of arc. Either Venus was extra mas-sive or a never-discovered other planet (called Vulcan) was pulling onMercury. And then came the explanation of Einstein, whose generalrelativity equations applied to Mercury’s orbit predict the extra 43seconds of arc per century!

FIGURE 16.13

Einstein’s theory predictedthat elliptical orbits of theplanets should precess.

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The presence of mass produces a curvatureor warping of space-time;conversely, a curvature ofspace-time reveals the presenceof mass.





16.6 Tests ofGeneral Relativity

Key Termgravitational red shift

CONCEPT

CHECK . . . . . .

CONCEPT

CHECK . . . . . .


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Light traveling “against gravity” is observed to have a slightlylower frequency due to an effect called the gravitational red shift.Because red light is at the low-frequency end of the visible spectrum,a lowering of frequency shifts the color of the emitted light towardthe red. Although this effect is weak in the gravitational field of thesun, it is stronger in more compact stars with greater surface grav-ity. An experiment confirming Einstein’s prediction was performedin 1960 with high-frequency gamma rays sent between the top andbottom floors of a laboratory building at Harvard University.16.6

Incredibly precise measurements confirmed the gravitational slowingof time.

So measurements of time depend not only on relative motion,as we learned in special relativity, but also on gravity. In special rela-tivity, time dilation depends on the speed of one frame of referencerelative to another one. In general relativity, the gravitational red shiftdepends on the location of one point in a gravitational field rela-

tive to another one. It is important to note the relativistic nature oftime in both special relativity and general relativity. In both theories,however, there is no way that you can extend the duration of yourown experience. Others moving at different speeds or in differentgravitational fields may see you aging slowly, but your aging is seenfrom their frame of reference—never your own. As mentioned earlier,changes in time and other relativistic effects are always attributed to“the other guy.”

CONCEPT

CHECK .

. . . . .What three predictions did Einstein make based on

his general theory of relativity?

Why do we not noticethe bending of light bygravity in our everydayenvironment?Answer: 16.6

think!

Newton’s and Einstein’s Gravity Compared From Newton’s law,one can calculate the orbits of comets and asteroids and even predictthe existence of undiscovered planets. Even today, when computing

the trajectories of space probes throughout the solar system andbeyond, only ordinary Newtonian theory is used. This is becausethe gravitational fields of these bodies are very weak, and from theviewpoint of general relativity, the surrounding space-time is essentiallyflat. But for regions of more intense gravitation, where space-time ismore appreciably curved, Newtonian theory cannot adequately accountfor various phenomena—like the precession of Mercury’s orbit close tothe sun and, in the case of stronger fields, the gravitational red shift andother apparent distortions of space and time. These distortions reachtheir limit in the case of a star that collapses to a black hole, wherespace-time completely folds over on itself. Only Einsteinian gravitationreaches into this domain.

Link to SPACE SCIENCE

The medieval philoso-pher William of Occamsaid that when decid-

ing between two com-peting theories, choosethe simpler explana-tion—don’t make moreassumptions thanare necessary whendescribing phenomena.

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Teaching Tidbit A double-pulsar system of two radio pulsarsthat orbit one another quicklyand with high accelerationhas recently been identifiedfor tests of general relativity.Research reports precisiontiming observations for a 3-yearperiod. With mass measurements

possible, four independent testsconfirm the validity of generalrelativity at the 0.05% level inthe strong-field regime.

Upon developing thegeneral theory of

relativity, Einstein predicted thatthe elliptical orbits of the planetsprecess about the sun, starlightpassing close to the sun is

deflected, and gravitationcauses time to slow down.





• Next-Time Questions 16-1,16-2

CONCEPT

CHECK . . . . . .

CONCEPT

CHECK . . . . . .


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Concept Summary ••••••

• As an object approaches the speed of light,its momentum increases dramatically.

• Mass and energy are equivalent—anythingwith mass also has energy.

• According to the correspondence principle,if the equations of special relativity (or any

other new theory) are to be valid, they mustcorrespond to those of Newtonian mechan-ics—when speeds much less than the speedof light are considered.

• The principle of equivalence states that localobservations made in an accelerated frameof reference cannot be distinguished fromobservations made in a Newtonian gravita-tional field.

• The presence of mass produces a curvatureor warping of space-time; conversely, a cur-vature of space-time reveals the presence ofmass.

• Upon developing the general theory of rela-tivity, Einstein predicted that the ellipticalorbits of the planets precess about the sun,starlight passing close to the sun is deflected,and gravitation causes time to slow down.

relativisticmomentum (p. 303)

rest mass (p. 304)

rest energy (p. 305)

correspondenceprinciple (p. 307)

general theory ofrelativity (p. 308)

principle ofequivalence (p. 309)

geodesic (p. 312)

gravitational wave (p. 313)

gravitational red

shift (p. 316)

16.2 No, no, no! Matter cannot be made to move

at the speed of light, let alone the speed of

light squared (which is not a speed!). The

equation E mc 2 simply means that energy

and mass are “two sides of the same coin.”16.5 There is no contradiction when the mass-

energy equivalence is understood. It’s true

that light is massless, but it is not “energy-

less.” The fact that gravity deflects light is

evidence that gravity pulls on the energy of

light. Energy indeed is equivalent to mass!

16.6 Earth’s gravity is too weak to produce a

measurable bending. Even the sun produces

only a tiny deflection. It takes a whole gal-

axy to bend light appreciably.

think! Answers

Key Terms ••••••

For:

Visit:

Web Code: –

Self-Assessment

PHSchool.com

csa 160016

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16 ASSESSCheck Concepts ••••••

Section 16.1

1. What would be the momentum of anobject if it were pushed to the speed oflight?

2. What is meant by rest mass?

3. What relativistic effect is evident when abeam of high-speed charged particles bendsin a magnetic field?

Section 16.2

4. What is meant by the equivalence ofmass and energy? That is, what does theequation E mc 2 mean?

5. What is the numerical quantity of theratio rest energy/rest mass?

6. Does the equation E mc 2 apply only toreactions that involve the atomic nucleus?Explain.

7. What evidence is there for the equiva-lence of mass and energy?

8. When the mass of something decreases,does it emit or absorb energy?

9. Compare the relative amounts of masslost in nuclear reactions and in chemicalreactions.

Section 16.3

10. What is the correspondence principle?

11. What results when low everyday speedsare used in the relativistic equations for timeand length?

12. Do the equations of Newton and Ein-stein overlap, or is there a sharp break

between them?Section 16.4

13. State the principle of equivalence.

14. Compare the bending of the paths of base-balls and of photons by a gravitational field.

Section 16.5

15. What is a geodesic ?

16. According to general relativity, in whatpaths do planets travel as they orbit the sun?

Section 16.6

17. What is the evidence for light bending near

the sun?18. Which runs faster, a clock at

the top of the Sears Towerin Chicago or a clock on theshore of Lake Michigan?

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ASSESS

Check Concepts

1. Infinite

2. The mass of an object orparticle at rest

3. It doesn’t bend as much. It hasa “stiffer” trajectory.

4. Mass and energy are twosides of the same coin.

5. c 2, or 9 3 1016 J/kg

6. No; it is universal.

7. Solar, chemical, and nuclearpower (Check students’ workfor other examples.)

8. It emits energy.

9. For nuclear reactions, aboutone part per thousand; forchemical reactions, about onepart per billion

10. Old and new laws agree inthe region of overlap.

11. The same results as with thesimpler classical formulas

12. Overlap smoothly

13. Local observations madein an accelerated frame

of reference cannotbe distinguished fromobservations made in aNewtonian gravitational field.

14. Both are attracted by gravity.Baseballs are noticeablydeflected only because theytravel with less speed thanphotons.

15. A line of shortest distancebetween two points

16. Planets travel along4-dimensional geodesicsin warped space-time aboutthe sun.

17. Displacement of stars whoselight grazes the sun during asolar eclipse

18. The higher clock; the one atthe top of the skyscraper runsfaster.


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CHAPTER 16 RELATIVITY—MOMENTUM, MASS, ENERGY, AND GRAVITY 319CHAPTER 16 RELATIVITY—MOMENTUM, MASS, ENERGY, AND GRAVITY 319

19. Moving “downhill” in a gravitational fieldhas what effect on the frequency of light?

20. Does Einstein’s theory of gravitationinvalidate Newton’s theory of gravitation?Explain.

Think and Rank ••••••

Rank each of the following sets of scenarios inorder of the quantity or property involved. Listthem from left to right. If scenarios have equalrankings, then separate them with an equal sign.(e.g., A B)

21. Electrons are fired at different speedsthrough a magnetic field and are bent fromtheir straight-line paths to hit the detector

at the points shown. Rank the speeds of theelectrons from highest to lowest.

22. To an Earth observer, metersticks on threespaceships are seen to have these lengths.Rank the speeds of the spaceships relative toEarth from highest to lowest.

Think and Explain ••••••

23. What happens to the momentum of a mas-sive object as its speed gets closer and closerto the speed of light?

24. When a charged particle moves through amagnetic field, what is the evidence that itsmomentum is greater than the value mv ?

25. According to E mc 2, how does the amountof energy in a kilogram of feathers compare

with the amount of energy in a kilogram ofiron?

26. Does a fully charged flashlight battery weighmore than the same battery when dead?Defend your answer.

27. Two safety pins, identical except that one islatched and one is unlatched, are placed inidentical acid baths. After the pins are dis-solved, what, if anything, is different aboutthe two acid baths?

28. A friend says that the equation E mc 2

has relevance to nuclear power plants, butnot to fossil-fuel power plants. Anotherfriend looks to see if you agree. What do

you say?

319

19. Frequency is reduced by aphenomenon known as thegravitational red shift.

20. No; Einstein showed thathis equation reduced to theNewtonian equation forgravitation in the weak-fieldlimit.

Think and Rank

21. C, B, A

22. C, B, A

Think and Explain

23. Its momentum gets larger andlarger, approaching infinity.

24. It is deflected less than itwould be if its momentumwere the Newtonian mv.

25. The energy is the same.Energy depends only on mass.

26. The charged battery weighsmore because it has moreenergy.

27. The bath with the latchedpin would be slightly warmerbecause it has the energy ofthe pin.

28. E 5 mc 2 refers to all changesin energy, whether in a fossilfuel or a nuclear power plant,or even in the process ofstriking a match.

29 N i i i l f


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16 ASSESSConcept Summary ••••••

ASSESS (continued)

320

16 29. Is this label on a consumer product

cause for alarm? CAUTION: The mass ofthis product contains the energy equivalentof 3 million tons of TNT per gram.

30. An astronaut awakes in her closed capsule,which actually sits on the moon. Can she tellwhether her weight is the result of gravita-tion or of accelerated motion? Explain.

31. An astronaut is provided “gravity” whenthe ship’s engines are activated to acceler-ate the ship. This requires the use of fuel. Isthere a way to accelerate and provide “grav-ity” without the sustained use of fuel? (Hint:Recall simulated gravity in Chapter 12.)

32. What happens to the separation distance be-tween two people if they both walk north at

the same rate from two locations on Earth’sequator?

33. Your friend whimsically says that at theNorth Pole, a step in any direction is a stepsouth. Do you agree?

34. We readily note the bending of light byreflection and refraction, but why are we notaware of the bending of light by gravity?

35.

Light does bend in a gravitational field. Whyis this bending not taken into considerationby surveyors who use laser beams as straightlines?

36. Your friend says that light passing the sunis bent whether or not Earth experiences asolar eclipse. Do you agree or disagree, andwhy?

37. In 2004 when Mercury passed between thesun and Earth, light was not appreciablybent as it passed Mercury. Why?

38. During the first second of its flight, a bulletfired horizontally drops a vertical distanceof 4.9 m from its otherwise straight-linepath in a gravitational field of 1 g . By whatdistance would a beam of light drop from itsotherwise straight-line path if it traveled ina uniform field of 1 g for 1 s? For 2 s?

39. A photon changes its energy when it “falls”

in a gravitational field. This change in en-ergy is not evidenced by a change in speed,however. What is the evidence for thischange in energy?

40. Do you age faster at the top of a mountainor at sea level?

320

29. No, it is simply a statement ofmass–energy equivalence.

30. Without other clues, shecannot tell the difference.

31. A person in a rotatinghabitat is provided gravity bycentripetal force.

32. Their separation distance

decreases. At the pole, itbecomes zero.

33. Yes; at the North Pole everystep is in the southerndirection.

34. Mainly because Earth’s gravityis too weak and the speed oflight is too fast

35. Bending is negligible for shortdistances.

36. Agree; the eclipse simply

offers a good way to see thebending.

37. Mercury’s mass is too small fornoticeable deflection.

38. It is the same drop for both!In 2 s, both a bullet and abeam of light would drop19.6 m (with g 5 9.8 m/s2).

39. A change in frequency,wavelength, and momentum

40. You age very slightly faster atthe top of a mountain.

41 They should live on the


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CHAPTER 16 RELATIVITY—MOMENTUM, MASS, ENERGY, AND GRAVITY 321CHAPTER 16 RELATIVITY—MOMENTUM, MASS, ENERGY, AND GRAVITY 321

41. Should a person who worries about growingold live at the top or at the bottom of a tallapartment building?

42. Is light emitted from the surface of amassive star red-shifted or blue-shiftedby gravity?

43. From our frame of reference on Earth,

objects slow to a stop as they approach blackholes in space because time gets infinitelystretched by the strong gravity near theblack hole. If astronauts accidentally fall-ing into a black hole tried to signal back toEarth by flashing a light, what kind of wave-lengths of light would best be looked for inEarth-based telescopes?

44. Gravitational waves are difficult to detect. Is

this due to having long wavelengths or shortones? High energy or low energy?

Think and Solve •••••• 45. A 100-watt light bulb consumes

100 joules of energy every second. Howlong could you burn that light bulb fromthe energy in one penny, which has a massof 0.003 kg? (Assume all the penny’s mass isconverted to energy.)

46. The fractional change of mass to energy in afission reactor is about 0.1 percent, or 1 partin a thousand.a. For each kilogram of uranium that under-

goes fission, how much energy is released?b. If energy costs three cents per megajoule,

how much is this energy worth in dollars?

Activity ••••••

47. Write a letter to your grandparents explain-ing how Einstein’s theories of relativityconcern the fast and the big—that relativityis not only “out there,” but affects this world.Tell them how these ideas stimulate yourquest for more knowledge.

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41. They should live on theground floor to age very veryslightly more slowly.

42. Red-shifted; gravity takesenergy away from light.

43. Far infrared

44. Long wavelengths and lowenergy

Think and Solve

45. E 5 mc 2 5 (0.003 kg) 3 (33 108 m/s)2 5 2.7 3 1014 J;(2.7 3 1014 J) / (100 J/s) 5 2.73 1012 s (almost86,000 years!)

46. a. E 5 mc 2 5 (0.001 kg) 3 (33 108 m/s)2 5 93 1013 J, or93 107 MJ.b. Multiply by $0.03 per MJand the amount of energy in1 gram is worth $2.7 million!

Activity

47. Letters will vary, but the mainidea is to say how relativityis an everyday phenomenon,even though its effects arenormally too small to besensed. Relativity, sensed or

not, does affect the everydayworld. If it intrigues thestudent and helps to focuson a wider view of things,wonderful.


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