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The area of horizons and the trapped region

Lars Andersson, Jan Metzger

TL;DR

The paper investigates marginally trapped surfaces (MOTS) and the boundary of the trapped region in Einstein initial data sets. It develops an area bound for outermost MOTS by combining curvature estimates for stable MOTS with a novel surgery construction that produces outer MOTS outside any given large-area MOTS, thereby enforcing a compactness-like control. Existence of MOTS is established via a Jang equation approach, using a regularized problem to obtain blow-up limits that project to MOTS on the physical slice, while weak barrier techniques extend the theory to less restrictive inner boundaries. A detailed analysis of the trapped region shows its outer boundary is a unique, smooth, outermost MOTS, with the interior boundary characterized by stability and regularity results, giving a complete quasi-local description of horizons in these initial data sets.

Abstract

This paper considers some fundamental questions concerning marginally trapped surfaces, or apparent horizons, in Cauchy data sets for the Einstein equation. An area estimate for outermost marginally trapped surfaces is proved. The proof makes use of an existence result for marginal surfaces, in the presence of barriers, curvature estimates, together with a novel surgery construction for marginal surfaces. These results are applied to characterize the boundary of the trapped region.

The area of horizons and the trapped region

TL;DR

The paper investigates marginally trapped surfaces (MOTS) and the boundary of the trapped region in Einstein initial data sets. It develops an area bound for outermost MOTS by combining curvature estimates for stable MOTS with a novel surgery construction that produces outer MOTS outside any given large-area MOTS, thereby enforcing a compactness-like control. Existence of MOTS is established via a Jang equation approach, using a regularized problem to obtain blow-up limits that project to MOTS on the physical slice, while weak barrier techniques extend the theory to less restrictive inner boundaries. A detailed analysis of the trapped region shows its outer boundary is a unique, smooth, outermost MOTS, with the interior boundary characterized by stability and regularity results, giving a complete quasi-local description of horizons in these initial data sets.

Abstract

This paper considers some fundamental questions concerning marginally trapped surfaces, or apparent horizons, in Cauchy data sets for the Einstein equation. An area estimate for outermost marginally trapped surfaces is proved. The proof makes use of an existence result for marginal surfaces, in the presence of barriers, curvature estimates, together with a novel surgery construction for marginal surfaces. These results are applied to characterize the boundary of the trapped region.

Paper Structure

This paper contains 14 sections, 31 theorems, 112 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Existence of MOTS
  4. Setup and Outline
  5. Preparing the Data
  6. Existence Proof
  7. Blowup surfaces are stable
  8. Weak barriers
  9. Surgery
  10. Neck geometry
  11. Point selection
  12. Surgery
  13. Results
  14. The trapped region

Key Result

Theorem 1.1

Let $(M,g,K)$ be a Cauchy data set. Assume that $M$ is compact with two boundary components, an inner and an outer boundary and assume that the inner boundary is outer trapped and the outer boundary is outer untrapped. Then $M$ contains a stable MOTS.

Figures (4)

  • Figure 1: A surface that nearly meets itself.
  • Figure 2: The $\delta$-standard neck.
  • Figure 3: Selecting the points $p$ and $q$ where a ball $B_\delta(O)$ touches $\Sigma$.
  • Figure 4: The surgery in geodesic normal coordinates.

Theorems & Definitions (61)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • ...and 51 more