Special Relativity Revision

First Year CP1

Trinity Term

Dr. Robert A. Taylor

What you need to KNOW

The special theory of relativity restricted throughout to

problems in one space dimension. The constancy of the

speed of light and simultaneity. The Lorentz transformation

(derivation not required). Time dilation and length

contraction. The addition of velocities. Invariance of the

space-time interval. Energy, momentum rest may and their

relationship for a single particle. Conservation of energy

and momentum. Elementary kinematics of the scattering

and decay of sub-atomic particles, including the photon.

Relativistic Doppler effect (longitudinal only).

Specimen Section A Questions

Specimen Section B Questions

Lorentz Transformations

( )x x v ty yz z

t t v xc

'''

'

= −

==

= −⎡⎣⎢

⎤⎦⎥

γ

γ 2

( )x x v ty yz z

t t v xc

= +

==

= +⎡⎣⎢

⎤⎦⎥

γ

γ

' ''

'

' '2

12 2

21 vcγ

−⎡ ⎤= −⎢ ⎥⎣ ⎦where

and v is the velocity of S´ as measured in S

Inverse:

TIME DILATIONIn the Lorentz Transformations the interval between 2 events is INVARIANT.

Δ Δ Δ Δ Δ Δ Δ Δx y z c t x y z c t2 2 2 2 2 2 2 2 2 2+ + − = + + −' ' ' '

Now for 2 events in S’ at the same place (e.g.clock ticks), we have Δx’ = 0. This is the clock rest frame and the time interval between 2 such events is the proper time denoted by Δτ. So:

c c t x y z2 2 2 2 2 2 2Δτ Δ Δ Δ Δ= − − −

Divide by Δt2 then:

ΔτΔt

v c⎛⎝⎜

⎞⎠⎟ = −

22 21 /

ddt S

x v'=⎛

⎝⎜⎞⎠⎟

∴ = −∴ <Δτ ΔΔτ Δ

t v ct

1 2 2/

Therefore the time interval is longer than that in the rest frame - TIME DILATION

LORENTZ CONTRACTIONShould be called Lorentz-Fitzgerald contraction

Consider a rigid rod of length l0 at rest in frame S’, moving with velocity v w.r.t. to S.

x1 x2

S

O x

x’1 x’

2

S’

O’ x’

l0

v

In S , use light signals as shown to measure x1 and x2 at the SAME TIME

In S’ use a metre rule to measure l0.

' '2 1 0'x x x lΔ = − =

( ) ( )' '1 1 2 2

0

' /

x x vt x x vt

x x x l

γ γ

γ γ

= − = −

Δ = Δ Δ =From the Lorentz transformations:

Relativistic Doppler Shift

(Observer moving away from source) (Observer moving towards source)

ν ν ββ

' = +−

⎡⎣⎢

⎤⎦⎥0

121

1ν ν β

β' = −

+⎡⎣⎢

⎤⎦⎥0

121

1

Useful way to remember formula: ( )ν ν γ β' = 0 1∓

- sign: moving APART

+ sign: moving TOGETHER

Relativistic Energy

( )( )2

2

0

2 2 212

d d

d d 1 1 1

d d1

t

vc

W F x Fv t

vFv mv m v W mc mc

t tγ

γ γ

= =

⎡ ⎤⎢ ⎥⎢ ⎥= + = − = −⎢ ⎥⎢ ⎥−⎣ ⎦

∫ ∫Then:

This is the Relativistic Kinetic EnergyRelativistic Kinetic Energy.

E W mc mc

E p c m c

= + =

− =

2 2

2 2 2 2 4

γThen Thus E2 - p2c2 is an INVARIANTINVARIANT

Transformation of E and p

( )

( )

'

'

'

'

x x

y y

z z

x

p p E c

p p

p p

E c E c p

γ β

γ β

= −

=

=

= −

Therefore once more we may define a 4-vector such that:

X L Xμ μν ν' =

where X is a 4-vector & Lμν is the Lorentz Transformation matrix.

( )

( )( )

, , ,

, , ,

, , ,

x y z

x y z

x x y z ict

p p p p iE c

k k k k i c

μ

μ

μ ω

=

=

=

and

0 0

0 1 0 0

0 0 1 0

0 0

i

L

i

γ βγ

βγ γ

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜= ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟− ⎟⎜⎜⎝ ⎠⎟⎜ ⎟

Relativistic KinematicsParticle physics units:

m = MeV/c2 p = MeV/c E = MeV 1 eV = 1.6 x 10-19 J

Then 2 2 2E p m= +

In particle physics β ª 1 therefore

2

2

&

& E

E mc mc

E mc c

γ γ β

γ β

= =

= =

p

pthus

Therefore in particle physics units:

& E m Eγ β= = p partpart part

Et

m

τγτ= =and

Centre of Mass or Centre of Momentum

S E cii

ii

= ⎡

⎣⎢⎤

⎦⎥− ⎡

⎣⎢⎤

⎦⎥∑ ∑

2 2

p

Is INVARIANTINVARIANT for a group of particles.

In C of M frame: S E Eii

= ⎡

⎣⎢⎤

⎦⎥=∑ *

22cm

Where is the energy of the ith particle in that frame.Ei*

cm cmcm

& i i

i i

ii

E c

E Eγ β= =

∑ ∑∑

pThen:

Compton Scattering

( ) ( )222 2

, ' ',

'e

E h E h

E p c mc

ν ν= =

= +

Using conservation of energy and momentum we get:

( )

( )

( )

2

2

2

' 1 cos

'1 cos

cot 1 tan2

hmc

EmcE

E mc

Emc

λ λ θ

θ

θϕ

− = −

=− +

= +

Oxϕ

θν, E

q

q’ν’, E’

p’ Ee