ch 3 relativity

8/12/2019 Ch 3 Relativity

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3.1

Introductory ConceptsEinsteins Postulates

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The Special Theory of Relativity was published byEinstein in 1905 Involves drastic revisions of Newtonian physics Has important consequences in all areas of physics

Is confirmed by numerous experimental observations Many of the ideas seem strange different from oureveryday experience!


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This question is unclear! Do we mean relative to someone standing by the road?

Answer: (5 + 1) = 6 m/s forwards. relative to someone sitting on the bus?

Answer: 1 m/s forwards. relative to a car moving past the bus at 10 m/s? Answer: 10 6 = 4 m/s backwards

We cannot talk about velocity, only about velocityrelative to a particular observer .

We must define the f ram e of reference .

Ann is on a bus which is moving forward at 5 m/s.She walks up the aisle, towards the front of the bus,at 1 m/s. What is her velocity?

3.1.1 Frame of Reference


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A f rame o f reference can be thought of as a person (anobserver) with a ruler and a stopwatch. I.e. an observer is stationary in his/her own frame ofreference. (Hence for a moving particle we often talkabout the rest -frame of the particle.) One frame of reference may be moving with respect to

another frame of reference. y

xO

y

x O

vbus

Earth frame Bus frame

positionof Ann


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3.1.2 Inertial Frame of Reference

Newtons First Law: A body acted on by no net forcemoves with constant velocity.

An inertial frame is a frame in which this law is true.- Basically this means it is a frame which is notaccelerating.

Contrast : Imagine a ball lying on the floor of the bus.If the bus accelerates, the ball starts to move eventhough there is no net force on it.

- The earth can normally be treated as an inertial frame.

- Any two inertial frames have a constant relativevelocity.

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We often consider two inertial frames S and S asshown below. S has constant velocity u relative to S .O and O coincide at time t = t = 0

y

xO

y

x O

u

S S P

An event at P has space-time coordinates( x, y, z, t ) in S , ( x , y , z , t ) in S .

These equations are the Galilean coordinate transformation .

According to classical (Newtonian) physics,

x x ut

y y

z z

t t

Galilean Transformation


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Galilean Velocity TransformationNow suppose that at P we have a particle moving in

the x-direction.In S it will be observed to have velocity

In S it will be observed to have velocity x

dxv

dt

xdx

vdt

i.e. x xv v u

x xv v u

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x x ut dx dx dx

udt dt dt

t t soNow and


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Note: Measuring Space and Time

We have said that for any event, we can describe its

space-time coordinates in a certain frame, e.g. ( x, y, z, t )in S . But we need to think carefully about how we canmeasure these!

For convenience, we consider thatthroughout an inertial frame we have- an infinite 3D array of measuring rods- a corresponding 3D grid of tiny clocks,

all synchronized.

Then we can determine the space-time coordinates of anyevent (without worrying about the time taken for a signal to

travel from the event to an observer elsewhere in the frame.) 8


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3.1.3 Einsteins Postulates

Einsteins Special Theory of Relativity is based on just twobrief postulates:

These postulates may sound simple but they have far-

reaching consequences! 9

We say the speed of light in vacuum is c = 3 10 8 m/s

but in what frame of reference?

1) The laws of physics are the same in all inertialframes of reference.

2) The speed of light in a vacuum, c, is the same inall directions and in all inertial frames of reference.


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Einsteins First Postulate

The laws of physics are the same in all inertial frames.

Examples: 1) Throwing and catching a ball on a train

[ F r o m

Y o u n g

& F

r e e

d m a n

]

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2) E.m.f. induced by relative motion between an magnet anda coil.


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Einsteins Second Postulate

The speed of light in a vacuum, c, is the same in all

directions and in all inertial frames.

This accounts for the results of the Michelson-MorleyExperiment (see textbooks).

This means the speed of light is independent of themotion of the source.

If two observers in different inertial frames both

measure the speed of a beam of light, they get thesame answer!

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Classical mechanics works

Classical mechanics fails[From Young & Freedman]


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Einsteins second postulate also implies that: It is impossible for an inertial observer to travel at

speed c.

(Light must travel at speed c relative to any observer, sothe observer cannot be in the rest-frame of the light!)

The Ultimate Speed Limit

The Galilean Transformation Revisited

Consider again our two inertial frames, with a movingparticle P. If P is a beam of light, so has velocity c , theGalilean transformation predictsBut Einstein and experimental evidence says !

c c u

c c

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i.e. simultaneity is relative!

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Consider 2 events. Call them Red and Blue.Suppose that in frame S these events are simultaneous(i.e. occur at the same time).Suppose S moves at constant velocity u with respect to S .

Are the events simultaneous in S ?

Whether two events at different locations aresimultaneous or not depends on the state of motion ofthe observer.

Answer: No!

3.2 Simultaneity

The time interval between two events may be different

in different frames of reference.


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Simultaneity: A Thought Experiment

[ F r o m

H a

l l i d a y

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Sam concludes thatevents Red and Bluehappened simultaneously. Sally concludes that Redhappened before Blue.Both observers are correct

in their own frame!

Sally is in a spaceship travelling atconstant velocity u relative to Sam.

Two events (Red and Blue) leavemarks on both ships at positionsequidistant from Sally and Sam.


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3.3 The Relativity of Space and Time

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3.3.1 Relativity of Time

Consider another thought experiment. Again Sally in rest frame S is travelling past Sam in S . Two events happen at the same location in S but at

different times. Sally measures the time interval Dt = Dt 0.

Sam observes the events to happen at different points

in space, and with time interval Dt .

[From Y&F]

http://www.phys.unsw.edu.au/einsteinlight/
http://www.phys.unsw.edu.au/einsteinlight/http://www.phys.unsw.edu.au/einsteinlight/


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In time Dt , Sallys ship travels a distance u Dt .

So in S , the light travels distance Hence

In S the light travels further, so we expect Dt > Dt 0.

Sally measures 02d

t c

D

2 222 2 ( )u t l d

D

Squaring and solving for Dt gives 02 21

t t u c

DD

This is often written as where2 2

1

1 u c

0t t D D

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2 2 2 20( ) ( ) ( )2

22

22

2l t d

cc t u t u

ct

cD

DD D


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Note: so

u = 60 km/s = 0.0002 c gives = 1.00000002(non- relativistic)

u = 6.0 107 m/s = 0.2 c gives = 1.02 (relativistic)

u c 1

[From Y&F]

Time Dilation: Summary

Suppose that in a certain rest frame, two events occur atthe same location with time interval Dt 0 (the proper time

).

Then an observer moving at constant speed u relative tothe rest frame will measure the time interval to be

2 2

1

1 u c

0t t D D where


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Example 3.1

Muons have a mean lifetime of 2.20 10-6

s as measured intheir own rest frame. If a muon is moving at velocity 0.9 c relative to the earth, what will be the mean lifetime measuredby an observer on earth?

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Note: 1 so The proper time is the shortest time interval. The time

interval measured in any other frame is dilated (longer). This can also be summarized as moving clocks run slow

0t t D D


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Example 3.2Ed is on Earth, while Sheila flies past Earth at high speed in aspaceship. When Sheila passes Ed, they both start their

stopwatches. Choose the correct answers below:(i) As measured in Eds frame of reference, when his watchshows 10 s, Sheilas clock will show:

(a) less than 10 s, (b) 10 s, (c) more than 10 s.

(ii) As measured in Sheilas frame of reference, when herwatch shows 10 s, Eds clock will show:

(a) less than 10 s, (b) 10 s, (c) more than 10 s.

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Twins Paradox Consider a pair of identical twins. One stays on Earth.The other goes on a long, high-speed journey in space.When the travelling twin returns, he/she will be younger(i.e. have lived less long) than the twin who stayed at home.

Is the following true or false?


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3.3.2 Relativity of Length

To measure the length of an object, we must note thepositions of its two ends at the same time.

Suppose a rod has proper length L0 as measured in itsrest frame. An observer moving at speed c relative to thisrest frame, along the direction of the length of the rod, will

measure the rod to have length 0 . L L

Lengths perpendicular to the direction of relativemotion are unchanged.

0 L L1 so , so this is called l eng th con t rac t ion . It applies to all distances (not just lengths of objects)

21http://www.physicsclassroom.com/mmedia/specrel/lc.cfm
http://www.physicsclassroom.com/mmedia/specrel/lc.cfmhttp://www.physicsclassroom.com/mmedia/specrel/lc.cfm


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3.3.2 Length Contraction ProofSuppose Sam stands on a station platform. He measuresits (proper) length to be L0. Sally is on a train that movesthrough the platform at speed u.Consider Event 1: Sally passes the back of the platform

Event 2: Sally passes the front of the platform.

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Sally measures time interval Dt between these events.The platform is moving at speed u relative to her, so sheconcludes the the platform has length L = u Dt . Sam measures the time interval to be Dt = L

0 / u.

By time dilation we know Dt = Dt

0 , L Lu u

0 . L

L

So giving


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Example 3.4Sam is in a spaceship which flies past Anne at a constantspeed 0.210 c. Anne measures the time taken for the shipto pass by to be 3.57 ms. Find the proper length of the ship.

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Example 3.5Suppose a tiny spaceship flies past you at high speed. At acertain instant, you observe that the tip and tail of the shipalign exactly with the ends of a 1 m ruler that you are holding.Rank the following lengths in order from longest to shortest:

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a) the proper length of the 1 m rulerb) the proper length of the spaceshipc) the length of the spaceship as measured

in your frame of referenced) the length of the meter stick as measured

in the spaceships frame of reference


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Let us return to the general situation of an event

happening at a certain location at a certain time.If the event has space-time coordinates ( x, y, z, t ) in S ,then the coordinates in S are ( x , y , z , t ) where:

These are called theLorentz transformationequations .

What happens to these equations as u 0?

( ) x x ut y y

z z

2( )u

t t xc

3.4 The Lorentz Transformation


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Note1) Rearranging, we also have

(I.e. interchange primed and

unprimed quantities, and changethe sign of the relative velocity.)

( ) x x ut

2( )ut t xc

2) For a pair of events, say 1 and 2, we measure

and in Sand in S

whereor

2

( )

( )

x x u t

ut t xc

D D D

D D D

2 1 x x x D

2 1 x x xD 2 1t t t D 2 1t t t

D

Dt = 0 gives (time dilation)

Dt = 0, D x = L0, D x = L gives (length contraction)0 L L 27

2

( )

( )

x x u t

ut t xc

D D D

D D D

t t D D


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Example 3.6

Two events in a frame S have space-time coordinates:

Is there a frame S in which these events aresimultaneous? If so, find the velocity of S relative to S and the time at which the events occur in S .

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0 01 1 0 2 02, ) ( , ), , ) (2( ( , ).2 x x

x xt x t xc c


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The Lorentz Velocity Transformation

Now consider a particle moving in the x-direction. How isits velocity in S related to its velocity in S ?

2 2 2 2

( )( ) 1 1

dx

dt x x u u u dx u xdt c c c c

uv udx dx u dt dx u dt v

dt dt dx dt dx v

So

xdx

vdt

xdx

vdt

Hence we have the Lorentz velocity transformation :

21 x

x x

v uv

uv c

What happens to these equations as u 0? 29

2( )u

cdt dt dx ( )dx dx u dt

From the Lorentz transformation, taking differentials weget and .

21 x

x x

v uv

uv c


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Example 3.7

A spaceship moves away from Earth at speed 0.900 c.The ship fires a probe in the same direction as its motionwith speed 0.700 c relative to the ship. Find the velocityof the probe relative to Earth.

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One of the important conservation laws in physics is thePrinciple of Conservation of M o m e n tu m .We need this to be true in all inertial frames of reference. But defining momentum as , it is not conserved!

3.5 Momentum and Energy

mp v

Relativistically, to preserve the principle of conservationof momentum we need a new definition:

m : measured in rest-frame of particleD x : measured by a stationary observerDt 0 : measured in rest-frame of particle

Consider a particle of mass m moving with constant speedv in the x direction. Classically, .

x p mv m

t

D

D

0.

x p m

t

D

D

D D


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If a particle has mass m as measured in its rest frame,then at velocity v , its relativistic momentum iswhere

m p v 12 21

.v c

With this new definition ofmomentum, Newtons secondlaw is still true in the form

but !d

dt

p F mF a

[From Y&F]

0

m x t p

t t

D D

D D 0t t D D p mv We have and so

Relativistic Momentum

What happens to this equation as v 0? If v doubles, does p more or less than double?


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3.5 Relativistic Kinetic Energy

Classically, kinetic energy is given by1 22

K mv

Relativistically it can be shown that the kinetic energy ofa particle of rest mass m moving with velocity v relative tothe observer is

where .12 21 v c

2( 1) K mc

What happens as v 0?

If v doubles, what

happens to K ? [From Halliday]

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A mass 1 kg moves at speed 2 10 8 m/s .Find its momentum and kinetic energyaccording to (a) classical physics, (b) relativity .

Example 3.8

E l 3 9


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Example 3.9

(a) How much work must be done on a mass m to accelerate it(i) from rest to speed 0.90 c ? (ii) from speed 0.90 c to 0.99 c ?

(b) At what speed is the momentum of a particle twice as great

as the result obtained from the non-relativistic expression mv ?


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For a particle of mass m,

E total energy

Einsteins theory of relativity indicates an equivalencebetween mass and energy!

Total Energy and Rest Energy

2 2 E K mc mc

For K = 0 , .2 E mc

Name some scientific processes and technologieswhich are based on E = mc 2.

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mc 2 rest energy

K kinetic energy

+=


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It can be shown (by eliminating velocity) that total energy

and momentum are related by

Energy and Momentum

If p = 0 then again 2 E mc

2 2 2 2 2( ) . E mc p c

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E pc

Massless particles do exist! For example the photon.Such particles always travel at speed c.

If m = 0 then


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Electron Volts

Particle energies are often expressed in electronvolts . When a charge

q moves through a potential difference

V its energy changes by qV . The electron volt is the energy of an electronaccelerated through a p.d. of 1 V. So 1 eV = 1.6 10 -19 J.

Particle masses may be expressed in eV/ c2.Particle momenta may be expressed in eV/ c.

NoteIf a physicist talks about, for example, a 2 MeV particle,the figure 2 MeV refers to the particles kinetic energy , not its total energy.


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(a) Find the rest energy of an electron, in joules and in eV .

Now consider a 1.00 MeV electron.(b) Find its total energy (in MeV ).(c) Find its momentum (in MeV/ c).

Example 3.10

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[Electron mass: me = 9.11 10 -31 kg ]

l


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(a) A neutral pion is said to have mass 135 MeV/ c2. Find itsmass in kilograms.

Example 3.11

(b) The pion is unstable and can decay into electromagneticradiation. If it is at rest before it decays, find the total energy ofthe radiation produced .

ch 3 relativity

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