General Relativity: An Introduction for Physicists · Problem 5.17

Special relativity revisited — Hobson, Efstathiou & Lasenby

Physics · doppler-shift-covariant

Problem

In Minkowski spacetime, consider an emitter

\mathcal{E}

moving at speed

v

along the positive

x^1

-axis of the frame

S

in which a receiver

\mathcal{R}

is at rest. Prove the Doppler shift formula

\frac{\lambda_{\mathcal{R}}}{\lambda_{\mathcal{E}}} = \gamma_v\left(1 - \frac{v}{c}\cos\theta\right),

where

\theta

is the angle made by the photon trajectory with the

x^1

-axis of

S

. Show that this expression can be written in the manifestly covariant way

\frac{\lambda_{\mathcal{R}}}{\lambda_{\mathcal{E}}} = \frac{u^\mu_\mathcal{E} k_\mu}{u^\nu_{\mathcal{R}} k_\nu},

where

\boldsymbol{k}

is the photon 4-wavevector and

\boldsymbol{u}_\mathcal{E}

and

\boldsymbol{u}_{\mathcal{R}}

are the 4-velocities of

\mathcal{E}

and

\mathcal{R}

respectively.