General Relativity: An Introduction for Physicists · Problem 7.7

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

Physics · satellite-clock-gravitational-time-dilation

Problem

A satellite is in circular polar orbit of radius

r

around the Earth (radius

R

, mass

M

). A standard clock

C

on the satellite is compared with an identical clock

C_0

at the south pole on Earth. Show that the ratio of the rate of the orbiting clock to that of the clock on Earth is approximately

1 + \frac{GM}{Rc^2} - \frac{3GM}{2rc^2}.

Note that the orbiting clock is faster only if

r > \tfrac{3}{2}R

, i.e. if

r - R > 3184\,\text{km}