General Relativity: An Introduction for Physicists · Problem 1.10

The spacetime of special relativity — Hobson, Efstathiou & Lasenby

Physics · transverse-doppler-effect

Problem

Two observers are moving along trajectories parallel to the

y

-axis in some inertial frame. Observer

A

emits a photon with frequency

\nu_A

that travels in the positive

x

-direction and is received by observer

B

with frequency

\nu_B

. Show that the Doppler shift

\nu_B/\nu_A

in the photon frequency is the same whether

A

and

B

travel in the same direction or opposite directions.