Horizon classification via Riemannian flows

TL;DR

This paper explores the geometry of null hypersurfaces in Lorentzian manifolds, connecting it to Riemannian flows, and derives new theorems about the structure of compact horizons without relying on degeneracy assumptions.

Contribution

It introduces a novel approach by applying Riemannian flow theory to null hypersurfaces, providing new insights into horizon geometry and dynamics.

Findings

Results on the dynamical structure of compact horizons.

Clarification of the relation between isometric flows and non-degeneracy.

Positive results on zero surface gravity fields in degenerate cases.

Abstract

We point out that the geometry of connected totally geodesic compact null hypersurfaces in Lorentzian manifolds is only slightly more specialized than that of Riemannian flows over compact manifolds, the latter mathematical theory having been much studied in the context of foliation theory since the work by Reinhart (Ann Math 69:119, 1959). We are then able to import results on Riemannian flows to the horizon case, so obtaining theorems on the dynamical structure of compact horizons that do not rely on (non-)degeneracy assumptions. Furthermore, we clarify the relation between isometric/geodesible Riemannian flows and non-degeneracy conditions. This work also contains some positive results on the possibility of finding, in the degenerate case, lightlike fields tangent to the horizon that have zero surface gravity.