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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