Special Relativity

An Introduction To ‘High Speed’ Physics

What Is Light?

Newton – light is corpuscular

There were two contradicting theories as to the nature of light:

Huygens – light is a wave

20th Centurywave-particle duality

19th Centurydiffraction/interference

18th CenturyNewton must be right!?

If light is a wave, through what does it propagate?

The Aether

Space is permeated by an invisible lumineferous aether(light-bearing medium)

Medium through which light can propagate

The Earth must be moving relative to the aether

So light will travel faster or slower, depending on the orientation

Differences can be determined by experiment

Test for the existence of the aether

The Michelson-Morley Experiment

The Michelson-Morley ExperimentTest for the presence of an aether using an interferometer

v ms-1 relative to the aether

speed c – v

speed c + v

incoming light

A is a half-silvered mirrorB/C are mirrorsO is a detector

1

2

2

12

c

v

c

d

vc

d

vc

dtACA

d

There is a phase difference between the two beams

Light at O should be phase-shifted, but no phase shift was observed

21

2

2

12

c

v

c

dtABA ABAACA tt

A

B

C

O

Maxwell’s Predictions

Electric and magnetic fields interact

00

1

c

E-Field B-Field

Accelerating charges produce EM waves

changing E-Field

changing B-Field

Maxwell’s equations predict that these waves propagate through a vacuum at a constant speed

u ms-1

What Has Gone Wrong?

v ms-1

The resultant velocity of the person is u + v

The resultant velocity of the light is c not c + v

Is Maxwell wrong? Are Michelson and Morley’s results wrong?

Or is Galileo wrong?

Galilean velocity transformation

Consider a train:

Einstein’s Postulates

We can now state two postulates:

1) The laws of physics are the same in all inertial frames ofreference

2) The speed of light in a vacuum is the same in all inertialframes of reference

But what is an inertial frame of reference?

Frames Of Reference

A frame of reference is the coordinate system of an observer

x

y

z

stationary frame

x

y

z

v ms-1

frame moving at constant velocity v

accelerating frame

x

y

z

a ms-2

inertial frames indistinguishable from a gravitational field

The Galilean Transformation

Consider two inertial frames, S and S’

x

y

z

S S’

x’

y’

z’

v ms-1

S’ is moving at velocity v away from S, and vy = vz = 0 ms-1

object in frame S’

The object has the same y- and z-coordinates in both frames

The time measured at any instant is the same in both frames

The x-coordinate is constant in S’, but changes in S

The Galilean Transformation

At a time t, the x-axis of S’is a distance vt from the x-axis of S

tt

zz

yy

vtxx

'

'

'

'

'

'

'

'

tt

zz

yy

vtxx

The transformation from S’to S

The transformation from S to S’

Galilean Transformation

So the x-coordinate in S is the x-coordinate in S’+ vt

All other coordinates are unchanged

This transformation between frames can be written as:

A New Transformation…

The Galilean transformation contradicts Einstein’s second postulate

We need to derive a new transformation, with the properties:

The speed of light must be a constant

It must tend to the Galilean transformation for low velocities

We are not assuming that time is absolute

So we need to be careful when referring to time

An event has both space and time coordinates

This can be written as a four-vector (x, y, z, t)

A Thought Experiment

Alice is on a long train journey, and is rather bored

She decides to build a clock using her mirror and a torch

d

What is the time interval between a pulse leaving and returning to the torch?

c

dt

2

A Thought Experiment

Bob is standing on platform 9¾ and watches Alice in the train

d v ms-1

9 ¾What is the time interval t’ in Bob’s frame of reference?

l l

22

2

'

tv

dl2

2

2

'22'

tv

dcc

lt

22

2

'

2

2'

2

tvtc

ct

tcd 2

2

1

'

c

v

tt

vt’

The Lorentz Factor

The factor is the Lorentz Factor

2

2

1

1)(

c

vv

Speed / ms-1c0

1

For v << c, (v) = 1

As v tends to c, (v) tends to infinity

Time Dilation

We have this relationship, but what does it mean?

tt '2

2

1

1

c

v

If a body is travelling slowly w.r.t. an observer ( = 1) time intervals are the same for the body and the observer

If a body is travelling fast w.r.t. an observer ( >> 1) time intervals appear longer to the observer

If you are in a spacecraft travelling close to c, time will pass normally for you, but will speed up around you

Notice, therefore, that photons do not age

The Twin ParadoxOnce upon a time there were two twins…

Bill Ben

Ben goes on a journey into space, but Bill stays on Earth

When Ben returns, which of the twins is oldest?

Bill thinks he will be older, as Ben travelled very fast away from him

Ben thinks he will be older, as Bill travelled very fast away from him

Who is right?

Bill is older, because he stayed in the same inertial frame, but Ben had to accelerate in the rocket

The viewpoints are not identical

Length Contraction

How do lengths appear in a different frame?

01LL 2

2

1

1

c

v

Similar derivation as for time dilation

There is no change in the directions perpendicular to travel

In the direction of travel, we can show that:

So, for a body travelling with v close to c, relative to an observer, the body will appear shorter to the observer, in the direction of v

How do you measure your velocity?

It is meaningless to have an absolute velocity

Velocity can only be measured relative to another body

But what about length contraction and time dilation?

If lengths are shortened, then can you measure the change?

No – the ruler is length-contracted as well

Similarly, clocks slow down, so changes in time can’t be measured

But doesn’t relativity define c as a maximum absolute speed?

Not quite – this is a maximum relative speed, as light has the same speed (c) relative to any frame

The Lorentz Transformation

We can now derive a relativistic transformation

x

y

z

S S’

x’

y’

z’

v ms-1

this length is contracted in the frame S

The y- and z-coordinates will be the same in both frames, as before

From the Galilean transformation, x = x’+ vt

But in frame S, x’is length-contracted to -1x’

'xvtx vtxx '

The Lorentz Transformation

To get the time transformation is a little trickier

Notice that the transformations are linear

'xvtx

xvtx ''

vtxx '

xvtvtx ')(

2'

c

vxttLorentz time transformation

The Lorentz Transformation

We can now state the full Lorentz Transformation:

2'

'

'

'

c

vxtt

zz

yy

vtxx

tt

zz

yy

vtxx

'

'

'

'

This satisfies the conditions for the transformation

v << c

Can We Go Faster Than Light?

From Newton’s Second Law, F = ma (constant mass)

So if we provide a continuous force, we can achieve v > c

But momentum must be conserved

vp m vp 0mrelativistic momentum

m0 is the rest mass

The factor in effect increases the mass, as v increases

A greater force is needed to provide the same acceleration

To reach the speed of light, an infinite force would be required

Conclusions

We have looked at the theory of Special Relativity

This resolved the conflict between Newtonian mechanics, and Maxwell’s equations

It is a ‘special’ theory, as it doesn’t consider accelerations

This is dealt with in General Relativity:

Acceleration and gravity are equivalent

Geometrical interpretation of gravitation

Much more difficult theory!

Derivation left as an exercise to the student