Gravitational redshift and Killing horizon

by Willie Wong

In a previous post I wrote about the phenomenon of gravitational red-shift in stationary space-times. Of particular importance is the following fact:

Given a time-like Killing vector field X (which we take to represent a preferred stationary background family of observers), the red-shift factor of observer A relative to signals sent by observer B is given by H = \sqrt{\frac{g(X,X)|_B}{g(X,X)|_A}}. That is, if observer B sends a signal of frequency f, observer A will see it as having frequency Hf.

We can also consider the infinitesimal version of the frequency shift. Taking the derivative of H (calculate it when the observers A,B are infinitesimally close to each other) we see that \nabla H = -\nabla \log (-g(X,X)).

Now suppose we are at a Killing horizon. That is, suppose we have a null hypersurface \mathfrak{h} and a Killing vector field X such that (a) X is time-like on one side of the hypersurface (b) X degenerates to a null vector field along \mathfrak{h} and is in fact tangent to its null generator. Therefore \nabla_XX is parallel to X along the horizon. Write \nabla_XX = \kappa X. Using that X is Killing, we have that \kappa X = \nabla( - g(X,X)) along the horizon. Comparing this with the paragraph above (and using Taylor’s theorem for smooth functions), we see now the connection between \kappa, the surface gravity of a stationary black-hole, and the red-shift effect for observers extremely close to the event horizon: if \kappa > 0, for observers very close to the event horizon, but sits to the past of the black hole (i.e. outside the black hole), this tells us that the out-ward gradient of H is negative, and so we expect a red-shift.