Black holes and black regions, horizons and barriers in Lorentzian manifolds

TL;DR

This paper proves that null hypersurfaces in Lorentzian manifolds act as barriers, separating spacetime regions, and introduces concepts like black regions, with implications for understanding horizons and black holes.

Contribution

It establishes that event horizons are a consequence of null hypersurfaces separating spacetime, leading to the definitions of barriers and black regions.

Findings

Null hypersurfaces enforce a consistent crossing direction for causal world-lines.

Barriers are defined by local nullity and global separation properties.

The results may simplify numerical horizon detection in spacetime simulations.

Abstract

We prove that if S is a time-oriented null hypersurface of a Lorentzian n-manifold (M, g), the causal world-lines, which intersect transversally S and are time-oriented in a compatible way, cross the hypersurface all in the same direction, the other being forbidden. Even if it is known that a smooth event horizon (in the sense of Penrose, Hawking and Ellis) is a null hypersurface and has the above semi-permeability property, at the best of our knowledge, in the literature it was not stated so far that the latter is a mere consequence of the former. Our result leads to the concepts of barriers (= null hypersurfaces separating the space-time into disjoint regions) and black regions (= time-oriented regions bounded by barriers). These objects naturally include (smooth) event horizons and (smoothly bounded) black holes. Since barriers are defined by two simple properties -- the merely local property of "nullity" combined with the global property of "separating the space-time" -- we expect they may be used to simplify computations for locating static and/or dynamic horizons in numerical computations.