Global Extensions of Spacetimes Describing Asymptotic Final States of Black Holes

TL;DR

This paper proves conditions under which spacetimes with black holes can be extended to include bifurcate horizons, showing that static or symmetric black holes naturally have constant surface gravity, supporting their role as final states of gravitational collapse.

Contribution

It establishes necessary and sufficient conditions for extending black hole spacetimes to include bifurcate horizons and characterizes when surface gravity is constant without using Einstein's equations.

Findings

Spacetimes with non-zero constant surface gravity can be globally extended to bifurcate horizons.

Conditions for matter field extendibility are identified and are automatically satisfied in symmetric cases.

Constancy of surface gravity is linked to the vanishing of the twist of the horizon Killing field.

Abstract

We consider a globally hyperbolic, stationary spacetime containing a black hole but no white hole. We assume, further, that the event horizon, $\tn$, of the black hole is a Killing horizon with compact cross-sections. We prove that if surface gravity is non-zero constant throughout the horizon one can {\it globally} extend such a spacetime so that the image of $\cal N$ is a proper subset of a regular bifurcate Killing horizon in the enlarged spacetime. The necessary and sufficient conditions are given for the extendibility of matter fields to the enlarged spacetime. These conditions are automatically satisfied if the spacetime is static (and, hence ``$t$"-reflection symmetric) or stationary-axisymmetric with ``$t-\phi$" reflection isometry and the matter fields respect the reflection isometry. In addition, we prove that a necessary and sufficient condition for the constancy of the surface gravity on a Killing horizon is that the exterior derivative of the twist of the horizon Killing field vanish on the horizon. As a corollary of this, we recover a result of Carter that constancy of surface gravity holds for any black hole which is static or stationary- axisymmetric with the ``$t-\phi$" reflection isometry. No use of Einstein's equation is made in obtaining any of the above results. Taken together, these results support the view that any spacetime representing the asymptotic final state of a black hole formed by gravitational collapse may be assumed to possess a bifurcate Killing horizon or a Killing horizon with vanishing surface gravity.