EXTENSIONS OF SPACETIMES WITH KILLING HORIZONS

We 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. We assume further that the Killing orbits on Ar 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, we 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. We 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.