Special Relativity

I wonder, what would happen if I was travelling at the speed of light and looked in a mirror?

Gallileo

• laws of physics should be the same in all inertial frames of reference. ( moving at constant velocity )

• Galilean Ivariance If you were on a bus moving at

constant speed you would experience the laws in the same way as if you were at rest

Newton

• Introduced the idea of universal time and space.

• He believed that it was the same time at all points in the universe as it was on Earth, clocks tick at the same rate regardless of their movement. Clocks tick at the same rate relative to observers who may have a different motion to the clock.

Absolute Space

A static background against which all movement can be referenced

A train moving out of a station : which moves ? The station or the train ?

Newton said that the train moved BUT is the station stationary ?

Galilean Transformations

Train moves at 50 ms-1 to the left

I walk at 5 m s-1 to the front / back of the train . What is my velocity relative to a stationary observer ?

Galilean Transformations 2

• What about my velocity relative to someone stationary on the train ?

Galilean Transformations 3

Runner A , 5 ms-1, is moving left towards runner B, 10 ms-1 to the right.

What is the velocity of runner A relative to runner B ?

Some language

• The train ( or lab or …. ) is called a ‘frame of reference’

• The frame of reference is called an ‘inertial frame of reference’ when the movement is constant velocity

James Clerk Maxwell

Light is an electromagnetic wave Predicted the speed of light to be

3.0x108ms-1

If light is a wave ,then is it passing through something ? Something must be vibrating. This was called the ether.

Michelson and Morley set out to detect the motion of the earth through the ether !

No difference in speed of light as earth moves in different directions.

EinsteinSpeed of light is constant for all observers

The laws of Physics are the same for all observers in all parts of the universeLeads to a new theory of motion :

A spaceship travelling at 2.9 x 106 ms-1 approaches a stationary planet, it sends out a light signal to the planet. What speed does the observer on the planet measure the light to be travelling at ?

If speed = distance / time and speed = frequency x wavelength then if the speed is to remain constant ‘ something’ must happen to the distance and time

Time DilationConsider a person on a platform who shines a laser pulse upwards, reflecting the light off a mirror. The time interval for the pulse to travel up and down is t

Travellers in this different frame of reference observe the ‘event’ (eg out of the window of the train), which takes place in the platform frame of reference and measure a time t’.

platform

mirror

hPerson in same frame as ‘event’ measures a time t.Total distance travelled by pulse 2h = ct.

h

v

Person in same frame as event measures a time , t, Total distance travelled by pulse

2h = ct

Travellers in this different frame of reference observe the ‘event’ (eg out of the window of the train), which takes place in the platform frame of reference and measure a time t’.

Time Dilation 2Both observers measure the same speed for the speed of light

Distance = 2h = ct

Train frame of reference

Horizontal distance travelled by train = vt’

Total distance travelled by pulse = ct’

Platform frame of reference

Time dilation 3For the Train frame of reference: Draw a right angled triangle where the vertical side is height of pulse, ( 0.5ct), the horizontal side is half the distance , d , travelled by the train, 0.5 vt’ and the hypoteneuse is half he distance gone by the pulse as seen by travellers on the train( 0.5ct’ ).

0.5ct’0.5ct

0.5 vt’

Time Dilation 4

2'2

2

2

22 t

c

v

c

ct

2

2'222

c

tvct

2'2

22 1 t

c

vt

Apply Pythagorus

(½ct)2 + (½vt’)2 = (½ct’)2 (ct)2 + (vt’)2 = (ct’)2

(ct)2 = (ct’)2 –(vt’)2 c2t2 = (t’)2(c2 – v2)

2

2

22'

1cv

tt

2

2

'

1cv

tt

Time Dilation 5

2

2

'

1cv

tt

v = speed of train

c = speed of light

t = time for event as viewed by observer on platform ( stationary )

t’ = time for event as viewed by observer on train ( moving observer)

Assumptions

1)Two frames are moving relative to each other along the x axis

2)Two observers on train as start / finish points are different.

Time Dilation 6

2

2

'

1cv

tt

Example

Dr Who is moving at a constant 0.92c. , relative to earth. He measures the time between ticks,∆t, of his

clock to be 1.0 s. What is ∆t measured as at on earth by Sarah Jane ?

t = 1.0 s

V = 0.92c.

t’ = ?

2

2

'

92.01

1

cc

t

= 2.6 sFrom the point of view of the earth based observer ( Sarah Jane ) ,Dr Who’s clock runs slow.

Time Dilation 7The term is called the gamma factor. At

everyday speeds it is almost unity ( 1 ). For time dilation to be noticeable then ‘relative speed’ must be c.a. 10 % of c.

2

2

1c

v

Length Contraction

Consider a rod placed on the platform discussed previously. To measure the length :Fix a mirror to one end and fire a beam of light from the other, the time for the beam to reflect back is t.

Therefore 2l = ct

Beam of light

mirror

lt = 2l/c

Length Contraction 2

Observer on moving train with velocity , v :

The time for the light to travel from the source to the mirror will be t1

Distance from source to mirror = ct1 = l’ + vt1

Start point

mirror

l’ + vt1

l’

ct1 -vt1 =l’

t1(c-v) =l’

t1 = l’ / ( c-v)

Length Dilation 3

Consider now the beam of light after reflection :

The measured time is t2

Distance back from the mirror is ct2 = l’ –vt2

l’ - vt1

l’

ct2+vt2 = l’

t2(c+v) =l’

t2 = l’/(c+v)

Length Contraction 4

For the moving observer the total time to measure the length is t1 + t2 =t’

t’= l’ / ( c-v) + l’/(c+v)

22

''

22

'''''

'''

2

vc

clt

vc

vlclvlclt

vcvc

vclvclt

A bit of heavy algebra

Length Contraction 5The Algebra gets seriously scary here :

We end up with

2

2' 1

c

vll

l = length measured by stationary observer

l’ = length measured by observer moving at constant velocity ,v ,relative to stationary observer

Length Contraction 6

2

2' 1

c

vll

Example

Luke Skywalker, a stationary observer, measures a stationary racing pod to be 4m long. What length will ET travelling at a constant 0.5c, relative to Luke, measure the pod to be ?

l = 4m

v = 0.5c

mc

cl 5.325.014

5.014

2

2'

Equally Luke would observe ET to be shorter than Et measures himself to be .

Experimental VerificationCosmic rays collide with atoms in the earth’s upper atmosphere. This produces particles called MUONS, in the lab they have a mean lifetime of 2.2 μ s. The muons travel at 99.9%c.They would travel 650m.

BUT substantial numbers are detected at sea level . In our reference frame the lifetime has been extended to 9 μs AND we measure the distance they travel as 60 km but in the reference frame of the muons the distance travelled is considerably smaller .

When an atomic clock is flown around the world very fast, there is a measurable time dilation.

E = mc2

Newton considered mass to be conserved BUT……

If a Luke Sywalker crashes the Millenium Falcon into the stationary Death Star, the damage caused will depend on the momentum of the Millenium Falcon. The change in momentum measured by Luke will be the same as that measured by Darth Vader on the Death Star. BUT they will both measure different velocities for the Millenium Falcon, the change in momentum is the same therefore Luke and Darth Vader must consider the mass of the Millenium Falcon to be different………………

E = mc2

I showed that there is a mass /energy equivalence

As the inertial mass increases with speed it requires more and more energy to increase the speed. No mass can travel at the speed of light as it would require an infinite amount of energy .