Gravitational redshift

Remarks

The vector field $\xi = \partial_t$ is a Killing vector field, so $g(k, \xi)$ is constant along the null geodesic with tangent $k$ followed by the light. A static observer has four-velocity $u = \xi / \sqrt{-g(\xi, \xi)}$, and the frequency it measures is proportional to $-g(k, u)$. Since $-g(\xi, \xi) = c^2 (1 - r_s/r)$, $\nu_o/\nu_e$ is the value of $\sqrt{-g(\xi,\xi)}$ at the emitter divided by its value at the observer, which is the formula above. The same argument works in any static spacetime, with $\sqrt{-g(\xi, \xi)}$ in place of $c\sqrt{1 - r_s/r}$. The Pound–Rebka experiment, published in 1960, measured the shift of gamma rays over a height of about $22.5$ metres at the surface of the Earth. The Global Positioning System must correct its clocks for this effect.

Going deeper

For a student

The redshift follows already from the equivalence principle: in an accelerated laboratory, light sent from floor to ceiling is Doppler shifted by the motion of the receiver. So it tests the equivalence principle more than the full field equations of general relativity. Near a black hole the redshift becomes unbounded as the emitter approaches the horizon $r = r_s$.