Extensions of spacetimes with Killing horizons

Abstract

The authors consider spacetimes possessing a one-parameter group of isometries with a Killing horizon, N, i.e. an isometry-invariant null hypersurface to which the Killing field is normal. They assume further that the Killing orbits on N are diffeomorphic to R, and that N admits a smooth cross section Sigma , such that each orbit intersects Sigma precisely once. If the surface gravity, kappa , on a generator gamma of N is non-vanishing, then gamma will be null geodesically incomplete. It is proved that any such incomplete generator gamma must terminate in a parallelly propagated curvature singularity whenever the surface gravity has a non-vanishing gradient on gamma . If, however, kappa is constant throughout the horizon, the authors prove that one can extend a neighbourhood of N so that N is a proper subset of a regular bifurcate Killing horizon in the extended spacetime. Since constancy of kappa on N is implied by Einstein's equations and the dominant energy condition, these results indicate that the only physically relevant Killing horizons are bifurcate Killing horizons and horizons with kappa =0. They also prove that for a static or stationary axisymmetric spacetime with a bifurcate Killing horizon, the natural static or stationary axisymmetric hypersurfaces smoothly intersect the bifurcation surface.