Globally Hyperbolic Spacetimes With Singularities

Details Globally Hyperbolic Spacetimes With Singularities Globally Hyperbolic Spacetimes With Singularities

2011-08-25

arxiv.org/abs/1108.5099v3

Mathematics

Differential Geometry

19 pages, 14 figures

Scientific paper

Black hole singularities are usually considered to break the time evolution, in (classical) General Relativity. It is shown here that, in fact, the standard black hole solutions are compatible with global hyperbolicity. For this, globally hyperbolic spacetimes containing singularities are constructed. The first step is to extend solutions of the Einstein equation to the singularities, in a way which avoids infinities. The stationary black hole solutions are then analytically extended beyond the singularities. Next, the topology of the spacetime at the singularity is repaired, by using the new extensions. Then, solutions are constructed, which are shown to be globally hyperbolic by foliating the spacetime with Cauchy hypersurfaces. The foliations are constructed by applying a Schwarz-Christoffel mapping to the Penrose-Carter representations of the black hole solutions. The results are applied to non-stationary black holes, including evaporating ones.

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