Classification and stability of black hole event horizon births: a contact geometry approach

TL;DR

This paper applies contact geometry and singularity theory to classify and analyze the stability of black hole event horizon formation, focusing on crease sets and caustics using Legendrian projections.

Contribution

It extends the classification of black hole crease sets by incorporating stability analysis within a contact geometry framework, refining prior approaches.

Findings

Classified crease sets and caustics using Legendrian projections.

Refined stability analysis of horizon birth components.

Identified additional crease set components connected to null infinity.

Abstract

A classical result by Penrose establishes that null geodesics generating a black hole event horizon can only intersect at their entrance to the horizon in ``crossover'' points. This points together with limit points of this set, namely caustics, form the so-called "crease set". Light rays enter into the horizon through the crease set, characterizing the latter as the birth of the horizon. A natural question in this context refers to the classification and stability of the structural possibilities of black hole crease sets. In this work we revisit the strategy adopted by Gadioux & Reall for such a classification in the setting of singularity theory in contact geometry. Specifically, in such contact geometry setting, the event horizon is identified as a component (not connected to null infinity) of a so-called ``BigFront''. The characterization of BigFronts as Legendrian projections of Legendrian submanifolds permits to classify the crease sets and ``cuspidal sets'' (or caustics in Penrose's terminology) by applying classical results established by V.I. Arnol'd. Here we refine the stability discussion presented by Gadioux & Reall of that connected component of the crease set that is not causally connected to null infinity and that constitutes the event horizon birth. In addition, we identify the existence of other components of the crease set that lie in the part of the BigFront that is causally connected to null infinity.