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Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes

Lars Andersson, Marc Mars, Walter Simon

TL;DR

This work develops a rigorous framework for quasi-local black-hole horizons by analyzing marginally outer trapped surfaces (MOTS) and their propagation into marginally outer trapped tubes (MOTT). Central to the approach is the stability operator $L_v$ acting on normal deformations of a MOTS, with its principal eigenvalue $ ext{λ}$ governing stability and the ability to extend MOTS into a MOTT. The authors establish a graph-based, elliptic-perturbation method to prove local existence of open MOTTs around strictly stable MOTS and derive barrier and symmetry properties that constrain the evolution. Under standard energy conditions, the resulting MOTT is either spacelike or null near the MOTS, providing a robust, quasi-local description of black-hole boundaries. The results extend previous short-communication findings and offer a (4D) global perspective on the causal structure and topology of MOTTs in general spacetimes.

Abstract

The present work extends our short communication Phys. Rev. Lett. 95, 111102 (2005). For smooth marginally outer trapped surfaces (MOTS) in a smooth spacetime we define stability with respect to variations along arbitrary vectors v normal to the MOTS. After giving some introductory material about linear non self-adjoint elliptic operators, we introduce the stability operator L_v and we characterize stable MOTS in terms of sign conditions on the principal eigenvalue of L_v. The main result shows that given a strictly stable MOTS S contained in one leaf of a given reference foliation in a spacetime, there is an open marginally outer trapped tube (MOTT), adapted to the reference foliation, which contains S. We give conditions under which the MOTT can be completed. Finally, we show that under standard energy conditions on the spacetime, the MOTT must be either locally achronal, spacelike or null.

Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes

TL;DR

This work develops a rigorous framework for quasi-local black-hole horizons by analyzing marginally outer trapped surfaces (MOTS) and their propagation into marginally outer trapped tubes (MOTT). Central to the approach is the stability operator acting on normal deformations of a MOTS, with its principal eigenvalue governing stability and the ability to extend MOTS into a MOTT. The authors establish a graph-based, elliptic-perturbation method to prove local existence of open MOTTs around strictly stable MOTS and derive barrier and symmetry properties that constrain the evolution. Under standard energy conditions, the resulting MOTT is either spacelike or null near the MOTS, providing a robust, quasi-local description of black-hole boundaries. The results extend previous short-communication findings and offer a (4D) global perspective on the causal structure and topology of MOTTs in general spacetimes.

Abstract

The present work extends our short communication Phys. Rev. Lett. 95, 111102 (2005). For smooth marginally outer trapped surfaces (MOTS) in a smooth spacetime we define stability with respect to variations along arbitrary vectors v normal to the MOTS. After giving some introductory material about linear non self-adjoint elliptic operators, we introduce the stability operator L_v and we characterize stable MOTS in terms of sign conditions on the principal eigenvalue of L_v. The main result shows that given a strictly stable MOTS S contained in one leaf of a given reference foliation in a spacetime, there is an open marginally outer trapped tube (MOTT), adapted to the reference foliation, which contains S. We give conditions under which the MOTT can be completed. Finally, we show that under standard energy conditions on the spacetime, the MOTT must be either locally achronal, spacelike or null.

Paper Structure

This paper contains 11 sections, 21 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Some basic definitions
  3. Varying the expansion
  4. Properties of linear elliptic operators
  5. Stability of MOTS
  6. The graph representation of MOTS
  7. Barrier properties of stable MOTS
  8. Symmetries
  9. Existence and properties of MOTTs
  10. Proof of Lemma \ref{['gv']}
  11. Proof of Lemma 4.1 (sketch)

Key Result

Lemma 3.1

Let ${\cal S}$, $l^{\alpha}$, ${\theta}$ and $q^{\alpha}= {q}^{\, \| \alpha}+ b {l}^{\alpha} - \frac{u}{2} {k}^{\alpha}$ be as before. Then, the variation of ${\theta}$ along ${q}$ is where $\Delta_{{\cal S}} = D_A D^A$ is the Laplacian on $({\cal S},h)$ and $a = - \frac{1}{2} k_{\alpha} \partial_{\sigma} l^{\alpha}_{\sigma} |_{\sigma =0}$.

Theorems & Definitions (43)

  • Definition 2.1
  • Definition 2.2
  • Remark
  • Lemma 3.1
  • Definition 3.1
  • Lemma 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Remark
  • proof
  • ...and 33 more