RELATIVITY

Einstein published two theories of relativity

In 1905

The Special Theory

For uniform motion 0a =

In 1916

The General Theory

For non-uniform motion 0a ≠ .

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First we will discuss The Special Theory

At the beginning of the century most believed

light was a wave.

Thus must have something that waves:

Sound has air

Water has water, etc.

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Physicists proposed that for waves of light

something must wave.

They called it the “ether” for light

This ether then must fill the universe.

The earth moves through the universe so:

How fast are we traveling through the ether?

To answer this we will start with a race

between two boats (that run at exactly the same

speed) in a river.

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The boat going downstream will have speed

c v+

The boat going upstream will have speed

c v−

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The calculated time for round trip is

// 2 2

2L L Lctc v c v c v

= + =+ − −

The speed for the boat going cross-stream will

have speed both ways

2 2c v−

Therefore

2 2

2cross

Ltv c

=−

Then the ratio of the two times will be

6

/ /2

2

1 11c r o s s

tt v

c

= >

−

Therefore we see that the boat that goes across

and back wins the race even though both boats

travel the same speed relative to the water.

The main point is the time to complete the

race is different for the two boats.

Note:

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We can measure ratio of times for boats and

calculate the speed of river if we know the

speed of boats.

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The Michelson-Morley Experiment

Diagram of Apparatus

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Drawing of actual apparatus

We know the speed of light.

We want the speed of the earth through the

ether.

10

When light arrives at the eye it has traveled two

paths to reach the observer.

There will be interference – either

constructive

or

destructive

The resulting image will be a series of lines.

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Spectrometer lines

Since the direction of the ether flow is not

known the apparatus must be rotated.

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First one than the other path will be parallel to

the flow of ether.

Therefore the interference lines should shift.

Michelson and Morley did the experiment very

carefully and did not find a shift.

The conclusion has to be that:

The Ether does not exist

or

the earth travels along with it.

13

Another experiment shows that we are

not moving with it.

Stellar Aberration is that experiment

While the light travels down the telescope the

telescope moves with the earth.

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The telescope has to be tilted to keep the image

in the center.

If the ether (the substance that waves to cause

the propagation of light) moves with the scope

there would be no need to tilt the it.

Therefore we conclude the ether does not

exist.

Classical Relativity

The transformation equations before Einstein

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'

'

'

'

x x v t

y y

z z

t t

= −

=

=

=

These are the Galilean Equations that allow

observers to compare observations in two

different frames moving relative to each other

with constant velocity.

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Observer on ground and observer on railroad

car moving in x-direction.

The observer on the ground observes the birds

separated by distance 2 1x x− .

The distances are equal.

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If an airplane flies over the railroad car

traveling in the x+ direction at a speed xu

measured by the observer on the ground what

will be the speed ( 'xu ) of the airplane measured

by the observer on the railroad car?

We can use the transformation equation for x

'x x vt= −

and the equation for t

't t=

Differentiate and divide to get

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'dx dx dvv t

dt dt dt= − −

' 0x xu u v= − −

if the velocity of the railroad car is constant.

If the observer on the ground measures the

velocity of the airplane as

xu

then the person on the railroad car will measure

'xu

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What if the person on the ground points a

flashlight in the x+ direction? What will be

the speed of light measured by the observer on

the railroad car?

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We get

'

'

x xu u vgivingc c v

= −

= −

We must keep this result in mind as we discuss

Einstein’s Theory.

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Einstein’s postulates for the Special Theory of

Relativity:

1. Fundamental laws of physics are identical

for any two observers in uniform relative

motion.

2. The speed of light is independent of the

motion of the light source or observer.

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These postulates cannot be satisfied using the

Galilean Equations, as we will see.

However Einstein found that the following

equations worked.

2

2

2

' ( )''

' ( )

1

1

x x vty yz z

vxt tc

where

vc

γ

γ

γ

= −==

= −

=

−

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These are the Lorentz Transformation

Equations.

Now consider the airplane flying over the

railroad car in the x-direction. What is the

speed of the airplane as measured by the

observer on the ground? What is the speed of

the airplane as measured by the observer on the

railroad car? We need to answer these

questions by using the Einstein-Lorenze

Equations.

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2

2

2

' ( )

' ( )

' ( )

' ( )

''

x x vtand

vxt tc

differentiatedx dx vdtand

vdxdt dtc

dividedx dx vdt

vdxdt dtc

γ

γ

γ

γ

= −

= −

= −

= −

−=

−

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divide by 'dt

2

'

2

''

1

1

xx

x

dx vdx dtdxdt vdt

cor

u vu vuc

−=

−

−=

−

Thus if an object (an airplane) flies over the

railroad car the observer on the ground will

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measure the speed in the x direction as xu . The

observer on the car will find 'xu .

What about the speed of light when a flashlight

is pointed in the x-direction?

The observer on the ground points a flashlight

in the +x direction. What will be the speed of

light measured by the observer on the car?

'

2( ) /1 1

xc v c v c vu cvc v c v c

c c

− − −= = = =

−− −

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Both observers, even though they are moving

relative to each other, measure the same value

for the speed of light.

This is in agreement with the Second Postulate.

LENGTH CONTRACTION

Read the section on Length Contraction in the

book. We will do it a little differently.

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The observer on the moving railroad car has a

rod moving with him. He measures the length

of the rod to be

' '2 1 0x x L− =

Use the Lorentz equations to get

'2 2 2

'1 1 1

( )

( )

x x vt

x x vt

γ

γ

= −

= −

Then putting these in the equation

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[ ]

0 2 2 1 1

0 2 1 2 1

( ) ( )

( ) ( )

L x vt x vt

L x x v t t

γ γ

γ

= − − −

= − − −

If the observer on the ground measures the far

end and near end of the rod at the same time

1 2t t=

Then

0 2 1( )L x x Lγ γ= − =

or

0LLγ

= and γ > 1

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So the observer on the ground with the rod

moving past in the x direction measures the rod

to be shorter than what is measured by the

observer at rest relative to the rod and on the

car.

Length Contraction is a prediction of the

Lorentz Equations.

TIME DILATION

Again we will find time dilation a different way

than the book. Then we will analyze the book’s

method.

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Observer on railroad car moving in x-direction

with firecrackers and another observer on

ground.

There is a firecracker on the top of the pole at

'1x and one on the top of the pole at

'2x . Let’s

say the firecracker on the top of the pole at '1x

explodes and then some time later the one on

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top of the pole at '2x explodes. We want to

consider the time between the two events.

The time interval between the two events as

measured by the observer on the ground will be

2 1t t t∆ = −

Using the transformation equations we get

'' 2

2 2 2

'' 1

1 1 2

( )

( )

v xt tc

v xt tc

γ

γ

= +

= +

So

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' '' '2 12 12 2

' ' '2 12

( ) ( )

( ( )

vx vxt t tc c

or

vt t x xc

γ γ

γ

∆ = + − +

⎡ ⎤∆ = ∆ + −⎢ ⎥⎣ ⎦

The time interval is different for the two

observers.

For simplicity consider two firecrackers on the

same post at

' ' '1 2x x x= =

then

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' ' ' '

2 12 ( )vt t x x tc

γ γ⎡ ⎤∆ = ∆ + − = ∆⎢ ⎥⎣ ⎦

and

1γ >

So the time interval for the observer on the

ground with the events moving relative to him

is longer.

MOVING CLOCKS RUN SLOWLY

If an observer is moving relative to a clock

the interval between ticks will be longer.

This is Time Dilation

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The book demonstration:

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For the girl on the railroad car, O’ frame, the

light travels up and back to the floor.

The distance traveled is 2d

The time to travel is ∆t’

The speed of light is c

So

' 2dtc

∆ =

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The observer on the ground sees the light travel

from the floor to the ceiling but during this time

the railroad car moves carrying the mirror with

it at a speed v.

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The observer on the ground sees the light travel

farther when it goes from floor to ceiling than

the observer on the railroad car.

For the girl distance traveled = d

For the boy distance traveled = 2c t∆

Both observers must measure the speed of light

as c.

Therefore, since the distance traveled is longer

for the boy than for the girl

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't t∆ > ∆

Consider triangle

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We see that

2 2

2

2 2c t v t d∆ ∆⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

Solve for t∆

2 2

2 2

4 4c vt d

⎛ ⎞∆ − =⎜ ⎟

⎝ ⎠

or

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2 2

4dtc v

∆ =−

and

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2 2 2

2

2 2

1

d dtc v vc

c

∆ = =−

−

Use

' 2dtc

∆ =

to get

'2d c t= ∆

and put in above equation to get

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' ''

2 2

2 21 1

c t tt tv vcc c

γ∆ ∆∆ = = = ∆

− −

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Einstein / Lorentz Transformation Equations

1. Satisfy Einstein’s Second Postulate

2. Predict Length contraction

3. Predict Time Dilation

But are these correct?

Must have an experiment to prove!!!

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Muon “lifetime” Experiment

Muons are particles created high in our

atmosphere.

They rain down continuously at high velocity at

approximately 82.994 10v x= m/s.

If at rest they only exist for approximately 62 10xτ −= seconds.

The experiment is to set up a detector at the top

of a mountain and stop the muons in the

detector and measure how long they exist.

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Then ask how far down would they have

traveled if they had not been stopped in the

detector.

distance = (speed) x (time they exist) 8 6(2.994 10 / ) (2 10 )d x m s x x s−=

= 600 m.

If the mountain is 2000m tall, at bottom should

find very few if any muons.

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Move the apparatus to the bottom of the

mountain and measure the number of the same

type muons that reach sea level.

The experiment showed that as many reached

sea level as passed through the atmosphere at

the level of the top of the mountain.

Why?

When moving relative to us the observer they

exist not for τ = 2 x 10-6 s but for

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6 6

82

8

16 16(2 10 ) 32 102.994 101 ( )

3 10

tt t x s x sx

x

τ γ − −= = = = =

−

Thus distance traveled will be 8 6(2.994 10 / )(32 10 ) 10,000d x m s x s m−= =

well below sea level.

Or viewed from the muon

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Experiment also confirms length

contraction.