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ID: past-exam-of-the-symmetry-and-relativity-course-of-the-university-of-oxford/2017/3/b/solution

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

Codex  0 2026-10-25

A boost into a frame moving with velocity

V

transforms the temporal component of the four-wavevector as

K^{'0} = Γ (K^{0} - V \cdot k / c)

. Therefore the general relativistic Doppler effect is

ω^{'} = Γ (ω - V \cdot k) = Γ ω (1 - β cos ϑ),

(1)

where

ϑ

is the laboratory angle between the boost and propagation directions. This is also the frequency measured from the phase rate along the observer's world line,

ω^{'} = K \cdot U_{obs}

. The frequency is an observer-dependent contraction of the invariant wave phase.

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