6593.relativity

Concept of Modern Physics A. Beiser

Sunday, October 02, 2011 1Dr. Sushil Kumar, Chitkara University

Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 2

The concept of relativity had been well known since the time of Galileo.

It was used by Newton and Poincaredeveloped this idea.

Einstein said that he thought of the idea whilst riding his bicycle.

http://images.google.co.in/imgres?imgurl=http://upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Poincare.jpg/180px-Poincare.jpg&imgrefurl=http://en.wikibooks.org/wiki/Special_Relativity/Print_version&h=247&w=180&sz=7&hl=en&start=1&usg=__-_UC8BMxfS1xYcqSHjAPsStfTpM=&tbnid=2Zfu3OoQTnaSLM:&tbnh=110&tbnw=80&prev=/images?q=joules+poincare&gbv=2&hl=en

The theory of special relativity revolutionized the world of physics byconnecting space and time, matter and energy, electricity and magnetism …


Special Relativity

In 1905 the 26 year old Albert Einstein described in his theory of Special Relativity “how measurements of time and space are affected by the motion between the observer and what is being observed.”

Introduction of the chapterAfter completing the chapter you will be familiar with

the

1. Two basic postulates of the STR

2. Frame of reference, concept of ether and Michelson-Morley Interferometer

3. Galilean transformation

4. Lorentz transformation

5. Time dilation

6. Length contraction

7. Velocity transformation

8. Relativistic momentum and energy


Postulates of Special Relativity

Einstein built the special theory of relativity on two postulates:

1. The Relativity Principle: The laws of motion are the same in every inertial frame of reference.

2. Constancy of the speed of light: The speed of light in a vacuum is the same independent of the speed of the source or the observer.


Motion is always measured relative to a frame of

reference i.e. there is no absolute motion

Frame S Frame S’ V0 relative to frame S

V ’ = V - V0

Speed measured In frame S’

Speed measured In frame S

• When is it happening time t

• Where is it happening position (x,y,z)

• What reference frame

coordinate (t,x,y,z) measured with

respect to a particular observer at

(0,0,0,0) frame of reference

What is an event ?


Measuring an event

An event is something that happens, to whichan observer can assign three space coordinatesand one time coordinate

A given event may be recorded by any numberof observers, each in a different referenceframe

In general different observers will assigndifferent space-time coordinates for the sameevent.



Inertial Reference Frames

An Inertial Frame of Reference is one in which the basic laws of physics apply- e.g., a train moving at a constant velocity, in this, objects move “ normally”.

Objects obeying the Newton’s First Law.

v


Non-Inertial Frames

An accelerating or decelerating objects. If you are sitting / walking on that, then during this period, you are in a non- inertial frame.

For example: If you are in a Ferris Wheelyou are always accelerating inwards so non-inertial.


Relative Velocity:What is the black car’s velocity relative to your frame?

V= 70 km/hr V=50 km/hr

We know the answer intuitively (120 km/hr) ?


Same idea with velocity of light ?

C= 3 x 10^(8) m/s C= 3 x 10^(8) m/s

From the first example, we would expect the relative velocityto be 2c = 6 x 10^(8) m/s.

This is in fact WRONG !!!!!


The Luminiferous Ether

The Ether was the basis of understanding, and the term was used to describe a medium for thepropagation of light.

It was hypothesized that the Earth moves through this medium .

The Ether;Was transparentHad zero densityWas everywhereWas the substance which allowed light to propagate.

Ether


wind

Airplane

Observer fixed on earth

If the velocity of the wind is v1 relative to Earth, and v2 is the velocity of airplane relative to thewind, the speed of Airplane relative to the earth is (a) v1+v2 in the same direction, (b) v2-v1 in the

opposite direction and (c) in the Direction perpendicular to the wind.


v C c+ v

Determine the speed of light under these circumstances ?

v c c - v

c 22 vc

In our case the ether wind is blowing through our apparatus fixed to the Earth, determine the velocity of light ?


If the Sun is assumed to be at rest in the ether, then the velocityof the Ether wind would be equal to the orbital velocity of the eartharound the Sun. which has a magnitude of about 3 x 10^4 m/s compared to c= 3 X 10^8 m/s.

The change in the speed of light should be detectable !!!!


The Michelson-Morley Experiment

The Famous experiment designed to detect small changes in the speed of light with motion of an observer through the ether .

It was performed in 1887.

Albert A. Michelson

Edward W. Morley

The negative results of the experiment not onlymeant that the speed of light does not depend on the direction of light propagated but also contradicted the ether hypothesis.

Light is now understood to be a phenomenon that requires no medium for its propagation.


Actual Experimental set-up

Telescope

Mirror

Light source

Semi-silvered plateCompensating plate


The Michelson interferometer produces interference fringes by splitting a beam of monochromatic light so that one beam strikes a fixed mirror and the other a movable mirror. When the reflected beams are brought back together, an interference pattern results.


SourceA

B

C

Light velocity -c

V

Observer

L

L

Ether wind

A = Semi silvered plateB= Mirror totalC = Mirror total

M2

M 1

Arm one

Arm

tw

o


Precise distance measurements can be made with the Michelson interferometer by moving the mirror and counting the interference fringes which move by a reference point. The distance d associated with mfringes is

Michelson Interferometer


Michelson-Morley Interferometer has two arms of equal

length L.

First, the beam traveling parallel to the direction of the

ether wind

Velocity of light beam moves to the right, with respect to

the Earth is = c - v

Velocity of light beam moves to the left, with respect to the

Earth is = c + v

The total time of travel for the round –trip along the

horizontal path is

1

2

2

1 12

c

v

c

L

vc

L

vc

Lt


Now, the light beam traveling perpendicular to the wind, so in this case the speed of the beam relative to the

Earth is 2

122 vc

Total time of travel for the round-trip is21

2

2

21222 1

22

c

v

c

L

vc

Lt

Thus the time difference between the right beam traveling horizontally and the beam traveling vertically is

21

2

21

2

2

21 112

c

v

c

v

c

Lttt


3

2

21c

LvtttAfter simplification ,

Because

.12

2

c

v

The path difference corresponding to this time difference is

After rotating the interferometer through 90 degree

2

222

c

Lvtcd

The corresponding fringe shift is equal to this path differencedivided by the wavelength of light, lembda, because a change in path of 1 wavelength corresponds to a shift of 1 fringe,

2

22

c

LvShift

Change in path of the order of lambda = one fringe shift

If change is one then fringe shift is equal to 1 over lambda

If change is d in path then change in shift would be equal to d

into one upon lambda.


40.0100.5

102.2

102.2)/103(

)/103)(11(2

7

7

7

28

24

m

mdShift

msm

smd

The speed of the Earth about the Sun, gives a path difference of

Conclusion: 1. No detection of fringe shift in the pattern2. No motion of Earth with respect to Ether.3. The speed of light is same for all observers

“ Most famous negative result in the History of Physics”.


Questions/ doubts

1) How Michelson able detect change in the speed of light?

2) Concept of aether ?

3) Concept of frame of reference ?

4) How shifting of fringes decides whether the speed of light is same or not ?

5) In theory of relativity fourth coordinate Is of time , why we take it so?

because to define position of particles only distance from 3 co-ordinates

is needed?

6) How we come to know light is electromagnetic wave through

Moreley Experiment?

7) In Morley experiment is source of light is moveable or mirror is moveable?

8) The basic points how Morely had thought before doing experiment ?

9) Is theory of relativity and special theory of relativity same?

9) Conclusion about the persences of aether from M-M expt.?


11) Why there is change in velocity of light when shift = 1d

?


12) Why do hypothetical concept of ether was needed?

13) Mathematics of M-M Expt?

14) When Interferometer is rotated through 90 degree what happens then?

15) Derivation of M-M expt. ?

What Time Dilation means…………?

• A moving clock ticks more slowly than a clock

at rest

When two events are occurs at the same location in an inertial reference

frame , the time interval between them, measured in that frame, is called

the proper time interval or the “ proper time”

Measurements of the same time interval from any other inertial reference

frame are always greater.

The amount by which a measured time interval is greater than the

corresponding proper time interval is called time dilation.


Time Dilation

One clock is at rest in a laboratory on the ground and

the other is in a spacecraft that moves at the speed v

relative to the ground. An observer in the laboratory

watches both clocks; Does he/she find that they tick

at the same rate ?

v



moving clocks run slow. This means that if two

events occur at the same place, such as the ticks of

a clock, a moving observer will measure the time

between the events to be longer. The relation

between a time measured by a stationary observer

t0 to the time t measured by an observer moving

with velocity v is:

The gamma factor is common in relativity, and we

will use it often. It is always greater than unity. If the

velocity were greater than c, it would be undefined.


Show Me The Derivation?

For our derivation, we will consider two

measurements. One taken by a rider with the

clock and the other measurement for the

clock will be made by a stationary observer

(referred to as the stationary for the mover).


Mete

r stick

mirror

mirror

Photosensitive surface

Light pulse

Recording device

Lo

A light pulse clock at rest

On the ground as seen by

An observer on the ground.

The light travels the total

Distance 2L at speed c, therefore

The time for entire trip is

c

Lt

2'


v

A

B

CD

Lo

0

t/2t

Clock is moving with v velocity in spacecraft. What observer notice who is in the

Rest with respect to spacecraft (seen from the ground). The time interval between

ticks is t.

2

ct

2

vt


to = 2L/c

Observer in same inertial frame and notice the events


Now, observer in another frame and seen events

from the ground …………


The light ray travels the path AB and mirror AD with velocity v.

The distance AB= ct/2 and AD=vt/2

22

0

22

0

2

2

0

2

11

/2

22

cv

t

cv

cLt

vtL

ct

To= time interval onclock at rest relative to an observer= proper time

T= time interval on clock in motion relative to an observer

v-= speed of relative motion

C= speed of light.


Example 1:

If you were to board a craft and travel at 0.2 c, how long would 1 hour be?


Example 2:

If you were to board a craft and travel at 0.8 c, how long would 1 hour be?


Example 3:

If you were to board a craft and travel c, how long would 1 hour be?


Example 4:

If you were to board a craft and travel at 300 m/s, how long would 1 hour be?


Question:

As we watch, a spaceship passes us in time t. The crew of the spaceship measures the passage time and finds it to be t'. Which of the following statements is true?

A) t is the proper time for the passage and it is smaller t'B) t is the proper time for the passage and it is greater than t'C) t' is the proper time for the passage and it is smaller than tD) t' is the proper time for the passage and it is greater than tE) None of the above statements are true.

Ans. C


QuestionSpaceship A, traveling past us at 0.7c, sends a messagecapsule to spaceship B, which is in front of A and istraveling in the same direction as A at 0.8c relative to us.The capsule travels at 0.95c relative to us. A clock thatmeasures the proper time between the sending andreceiving of the capsule travels:

A) in the same direction as the spaceships at 0.7c relative to usB) in the opposite direction from the spaceships at 0.7c relative to usC) in the same direction as the spaceships at 0.8c relative to usD) in the same direction as the spaceships at 0.95c relative to usE) in the opposite direction from the spaceships at 0.95c relative to us

Ans. D


Problem

You wish to make a round trip from Earth in a spaceship, traveling atconstant speed in a straight line for 6 months and then returning at thesame constant speed. You wish further, on your return, to find Earth as itwill be 1000 years in the future.

(a)How fast must you travel?

(b) Does it matter whether you travel in a straight line on your journey?If, for example, you traveled in a circle for 1 year, would it still find 1000years had elapsed by Earth clock when you returned?

Question

A millionairess was told in 1992 that she had exactly 15

years to live. However, if she travels away from the Earth

at 0.8 c and then returns at the same speed, the last New

Year's day the doctors expect her to celebrate is:

A) 2001

B) 2003

C) 2007

D) 2010

E) 2017

Ans. E


The Time Dilation equation for Relativity is:

What is t, to v, and c?


The relativity of length and Length Contraction

The Relativity of Length

The length L0 of an object measured in the rest frame of the

object is its proper length or rest length. Measurements of the

length from any reference frame that is in relative motion

parallel to that length are always less that the proper length

2

2

0 1c

vLL


Speed of Spaceship Observed Length Observed Height

At rest 200 ft 40 ft

10 % the speed of light 199 ft 40 ft

86.5 % the speed of light 100 ft 40 ft

99 % the speed of light 28 ft 40 ft

99.99 % the speed of light 3 ft 40 ft


QuestionA measurement of the length of an object that is movingrelative to the laboratory consists of noting the coordinatesof the front and back:

A) at different times according to clocks at rest in thelaboratoryB) at the same time according to clocks that move withthe objectC) at the same time according to clocks at rest in thelaboratoryD) at the same time according to clocks at rest withrespect to the fixed starsE) none of the above

Ans. C


Question

A certain automobile is 6 m long if at rest. If it ismeasured to be 4/5 as long, its speed is:

A) 0.1cB) 0.3cC) 0.6cD) 0.8cE) > 0.95cAns. C


Problem

A cubical box is 0.50 m on a side.

(a) What are the dimensions of the box as measured by anobserver moving with a speed of 0.88c parallel to one of theedges of the box?

(b) What is the volume of the box as measured by thisobserver?


2

2

0 1c

vLL

L= 0.5 (1-(0.88)**2)1/2

=0.24m

The observed dimension are= 0.24*0.5*0.5=0.059m**3


An Important general question:

If we know the co-ordinate x, y, z and time t of an

event, as measured in a frame S, How can we find

the coordinates x’, y ’ , z ’ and t’ of the same event as

measured in a second frame S?


Galilean Transformation


To convert velocity components measured in the

S frame to their equivalents in the S’ frame according

to the Galilean Transformation, we simply differentiate

x’, y’, and z’ with respect to time:

zz

yy

xx

vdt

dzv

vdt

dyv

vvdt

dxv

'

''

'

''

'

''

Galilean transformation

violate both of the

postulates of special

relativity.

1). If we measure the speed

of light in the x-direction in

the S-system to be c,

however, in the S’ system it

will be

c’= c-v

Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 59Sunday, October 02, 2011 Dr. Sushil Kumar, Chitkara University 59

Lorentz’s Transformation

The primed frame moves with velocity v in the x direction with respect to the fixed reference frame. The reference frames coincide at t=t'=0. The point x' is moving with the primed frame.

The reverse transformation is:

Much of the literature of relativity uses the symbols β and γ as defined here to simplify the writing of relativistic relationships


Relativistic Velocity Transformation

No two objects can have a relative velocity greater than c! But what if I observe

a spacecraft traveling at 0.8c and it fires a projectile which it observes to be

moving at 0.7c with respect to it!? Velocities must transform according to the

Lorentz transformation, and that leads to a very non-intuitive result called

Einstein velocity addition.


Just taking the differentials of these quantities leads to the velocity

transformation. Taking the differentials of the Lorentz

transformation expressions for x' and t' above gives

Putting this in the notation introduced in the illustration above:


The reverse transformation is obtained by just solving for u

in the above expression. Doing that gives

Applying this transformation to the spacecraft traveling at 0.8c which fires a

projectile which it observes to be moving at 0.7c with respect to it, we obtain a

velocity of 1.5c/1.56 = 0.96c rather than the 1.5c which seems to be the

common sense answer.


When xu

and v are both much smaller than c (the non

relativistic case) , the denominator of equation

approaches unity and so u’x = ux –v. This

corresponds to the Galilean velocity

transformation, In the other Extreme, when ux =c

;

the equation becomes U’x = c ,

From this result, we see that an object moving with

a speed c relative to an observer in S also has a

speed c relative to an observer in S’- independent

of the relative motion of the S and S’.

Question:

Imagine a motorcycle rider moving with a speed of

0.800c past a stationary observer, as shown in figure

below, If the rider tosses a ball in the forward direction

with a speed of 0.700c with respect to himself, what is

the speed of the ball as seen by the stationary observer?

0.700c

0.800c


In this situation, the velocity of the motorcycle with respect to

the stationary observer is v=0.800c. The velocity of the ball

in the frame of reference of the motorcyclist is ux’=0.700c.

Therefore, the velocity, ux, of the ball relative to the stationary

observer is

c

ccc

cc

c

vu

vuu

x

xx

9615.01800.0700.01

800.0700.0

1

2

2

'

'


The length of any object in a moving frame will appear foreshortened in the direction of motion, or contracted. The amount of contraction can be calculated from the Lorentz transformation. The length is maximum in the frame in which the object is at rest.

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/ltrans.html

Questions/ Problems/ Doubts:

1). I know the derivation but I have problem in basic concept, I did not

understand the application of Galilean and Lorentz transformation?

2.) I know the concept, but I face problem when I try to solve the numerical

problem?

3.) No proper notes/material of subject?

4.) Problem in length contraction, when observer is moving or object is

moving?

5.) Why Galilean transformation do not obey laws of physics?

6.) Transformation equation?

7.) When we move away from a building it becomes smaller and smaller, is it

the case of length contraction or some other physical process?

8.) If object is in moving frame of reference, is there any change of

dimension?

9.) Why there is need of transformation, what we get from it?

10.) How a large building contract, when seen in the glass/mirror of moving

vehicle?

Questions/doubts:

11. WHY THERE IS CHANGE IN VELOCITY OF LIGHT WHEN FRINGE SHIFT = 1 ?

12. When an observer and object of length L is moving with velocity v1 and v2 respectively, what will be the length contraction, v2> v1 ?

13. Not able to understand why the length seems less while moving……..Length contraction?

14. How length contraction takes place in MUON’S DECAY ?


Questions/ doubts……..417

We consider earth an inertial frame of reference, and there are hardly any motions comparable to speed of light. Why do we refer to the relativity then ?

does time, length and velocity all relativistic

quantities?


Questions/ doubts

How does time change in S’ as per Lorentz transformation & Physically why is this change of time?

Explain the use of telescope and compensating plate in the setup of experiment?

How does the height of the object changes when observer is along x-axis and object is moving along y-axis ?


Questions/ doubts/ problems

How to implement these formulas in typical questions?

Galilean & Lorentz transformation with numericals?

Tried to understand the topic ‘ Length contraction’ using text book but was unable to understand, also tried numerical problems on the topic ‘ time dilation’ and was unable to solve them too…......!!!!!

Explain length contraction of a thin road, when observer moves perpendicular to the rod ?


How much Do I know?

What Do I need to learn?

What I have Learned ?



Relativistic mass

Mass m0 of an object measured when it was at rest and mass m measured when it was in motion with velocity v,

2

2

0

1c

v

mm

Relativistic Mass Increase

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1

Speed ( c = 1)

Mass

Q. The total energy of a proton is three times its rest energy.(a) Find the proton’s rest energy in electron volts.


MeV

JeVJ

smkg

cmp

938

)106.1/1)(1050.1(

)/100.3)(1067.1(

1910

2827

2

Q. With what speed is the proton moving?


smcu

or

c

u

givesuforsolving

cu

c

u

cmcmE

mcE

p

p

/1083.23

8

9

11

1

13

1

3

8

2

2

22

2

2

2

2

2

Relativistic Energy How Does the Total Energy of a Particle Depend on

Speed?

We have a formula for the total energy E = K.E. + rest energy,


22 0

2 21 /

m cE mc

v c

so we can see how total energy varies with speed


How Does the Total Energy of a Particle Depend on Momentum?

It turns out to be useful to have a formula for E in terms of p.

Now2 4

2 2 4 0

2 21 /

m cE m c

v c

2 4 2 2 2 4

0

2 4 2 2 2 2 4

0

2 4 2 2 4 2 2 2

0

(1 / )m c v c m c

m c m v c m c

m c E m c m c v

hence using p = mv we find 2 4 2 2

0E m c c p

If p is very small, this gives 2

2

0

02

pE m c

mthe usual classical formula. If p is very large, so c2p2 >> m0

2c4, the approximate formula is E = cp


The High Kinetic Energy Limit: Rest Mass Becomes

Unimportant!Notice that this high energy limit is just the energy-momentum relationship Maxwell found to be true for light, for all p. This could only be true for all p if m0

2c4 = 0, that is, m0 = 0. Light is in fact composed of “photons”—particles having zero “rest mass”, as we shall discuss later. The “rest mass” of a photon is meaningless, since they’re never at rest—the energy of a photon

22 0

2 21 /

m cE mc

v c

is of the form 0/0, since m0 = 0 and v = c, so “m” can still be nonzero. That is to say, the mass of a photon is really all K.E. mass. For very fast electrons, such as those produced in high energy accelerators, the additional K.E. mass can be thousands of times the rest mass. For these particles, we can neglect the rest mass and take E = cp.

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