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 (which we take to represent a preferred stationary background family of observers), the red-shift factor of observer relative to signals sent by observer is given by . That is, if observer sends a signal of frequency , observer will see it as having frequency .
We can also consider the infinitesimal version of the frequency shift. Taking the derivative of (calculate it when the observers are infinitesimally close to each other) we see that .
Now suppose we are at a Killing horizon. That is, suppose we have a null hypersurface and a Killing vector field such that (a) is time-like on one side of the hypersurface (b) degenerates to a null vector field along and is in fact tangent to its null generator. Therefore is parallel to along the horizon. Write . Using that is Killing, we have that along the horizon. Comparing this with the paragraph above (and using Taylor’s theorem for smooth functions), we see now the connection between , the surface gravity of a stationary black-hole, and the red-shift effect for observers extremely close to the event horizon: if , 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 is negative, and so we expect a red-shift.