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

Oscar Meneses Rojas

TL;DR

This work reframes the problem of black hole horizon births by casting the crease set as part of a BigFront in a Lorentzian contact-geometry setting, where horizon formation is studied through Legendrian projections and their wavefronts. Using Arnol'd singularity theory, it identifies the local stable structures of crease sets for spherical horizons, and refines the stability analysis of the horizon-birth component that is not connected to null infinity, while acknowledging additional crease-set components related to null infinity. The analysis highlights a unique stable cusp-caustic pattern, notably a cuspidal set of type $A_3$, and provides a geometric framework connecting Penrose's classical picture to modern contact-geometric methods. This approach offers precise local models for horizon births and implications for the causal structure and diffraction phenomena at caustics.

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.

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

TL;DR

This work reframes the problem of black hole horizon births by casting the crease set as part of a BigFront in a Lorentzian contact-geometry setting, where horizon formation is studied through Legendrian projections and their wavefronts. Using Arnol'd singularity theory, it identifies the local stable structures of crease sets for spherical horizons, and refines the stability analysis of the horizon-birth component that is not connected to null infinity, while acknowledging additional crease-set components related to null infinity. The analysis highlights a unique stable cusp-caustic pattern, notably a cuspidal set of type , and provides a geometric framework connecting Penrose's classical picture to modern contact-geometric methods. This approach offers precise local models for horizon births and implications for the causal structure and diffraction phenomena at caustics.

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.
Paper Structure (4 sections, 7 theorems, 1 equation, 1 figure)

This paper contains 4 sections, 7 theorems, 1 equation, 1 figure.

Table of Contents

  1. Introduction
  2. Contact Geometry in Lorentzian Geometry
  3. Black Hole event horizons as BigFronts
  4. Acknowledgments

Key Result

Lemma 2.2

Let $q\in \mathscr{M}$ and consider a null vector $l\in T_q\mathscr{M}$. The kernel of the 1-form $\alpha_{q,l}:=g_q(l,\cdot)$ is a tangent hyperplane to the null cone in $T_q\mathscr{M}$.

Figures (1)

  • Figure 1: Classification of singularities and self-intersection sets in wavefronts.

Theorems & Definitions (9)

  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Theorem 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 3.1
  • Theorem 3.2