General Relativity: An Introduction for Physicists · Problem 7.6

The equivalence principle and spacetime curvature — Hobson, Efstathiou & Lasenby

Physics · gravitational-redshift-clocks

Problem

An experimenter

A

drops a pebble of rest mass

m

in a uniform gravitational field

g

. At a distance

h

below

A

, experimenter

B

converts the pebble (with no energy loss) into a photon of frequency

\nu_B

. The photon passes by

A

, who observes it to have frequency

\nu_A

. Use simple physical arguments to show that to a first approximation

\frac{\nu_B}{\nu_A} = 1 + \frac{gh}{c^2}.

Use this result to argue that for two stationary observers

A

and

B

in a weak gravitational field with potential

\Phi

, the ratio of the rates at which their laboratory clocks run is

1 + \Delta\Phi/c^2

, where

\Delta\Phi

is the potential difference between

A

and

B