Past exam of the symmetry and relativity course of the University of Oxford / 2017 / 3 / a / Solution

Three relevant observable effects are the relativistic Doppler effect in spectral-line shifts, including the transverse time-dilation contribution; relativistic aberration, observable as a change of apparent light direction with observer motion; and relativistic beaming of moving emitters, associated with strongly asymmetric jet brightness. Apparent superluminal motion supplies a related light-travel-time test:

v_{app} / c = β sin ϑ / (1 - β cos ϑ)

can exceed one without a material speed exceeding

c

. These interpretations require an emission and source-motion model; a cosmological or gravitational redshift by itself is not a direct test of special-relativistic kinematics.

For Compton scattering from an initially stationary electron, let incoming and outgoing photon energies be

ε

and

ε^{'}

and their mutual angle be

ψ

. Four-momentum conservation gives

P + Q = P^{'} + Q^{'}

. Squaring

P^{'} = P + Q - Q^{'}

and using both electron mass shells and both photon null norms gives

2 P \cdot (Q - Q^{'}) = 2 Q \cdot Q^{'}, m_{e} (ε - ε^{'}) = \frac{ε ε ^{'}}{c ^{2}} (1 - cos ψ) .

Consequently the outgoing angular frequency and shift are

ω^{'} = \frac{ω}{1 + \frac{ℏ ω}{m _{e} c ^{2}} ( 1 - cos ψ )}, Δ ω = - ω \frac{\frac{ℏ ω}{m _{e} c ^{2}} ( 1 - cos ψ )}{1 + \frac{ℏ ω}{m _{e} c ^{2}} ( 1 - cos ψ )} .

Equivalently, the Compton wavelength shift is

λ^{'} - λ = h (1 - cos ψ) / (m_{e} c)

. The electron recoil is essential; treating its energy as unchanged would suppress the effect.

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