Spacetime and Geometry · Problem 6.4

Perturbation Theory and Gravitational Radiation — Sean Carroll

Physics — General Relativity · de-sitter-space-killing-horizon

Problem

Consider de Sitter space in static coordinates:

ds^2 = -\left(1 - \frac{\Lambda}{3}r^2\right)\mathrm{d}t^2 + \frac{dr^2}{1 - \frac{\Lambda}{3}r^2} + r^2\mathrm{d}\Omega^2.

This space has a Killing vector

\partial_t

that is timelike near

r=0

and null on a Killing horizon. Locate the radial position of the Killing horizon,

r_\mathrm{K}

. What is the surface gravity,

\kappa

, of the horizon? Consider the Euclidean signature version of de Sitter space obtained by making the replacement

t \to i\tau

. Show that a coordinate transformation can be made to make the Euclidean metric regular at the horizon, so long as

\tau

is made periodic.