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Local existence of dynamical and trapping horizons

Lars Andersson, Marc Mars, Walter Simon

TL;DR

Given a spacelike foliation of a spacetime and a marginally outer trapped surface S on some initial leaf, it is proved that under a suitable stability condition S is contained in "horizon" i.e., a smooth 3-surface foliated by marginally outer trapping slices which lie in the leaves of the given foliation.

Abstract

Given a spacelike foliation of a spacetime and a marginally outer trapped surface S on some initial leaf, we prove that under a suitable stability condition S is contained in a ``horizon'', i.e. a smooth 3-surface foliated by marginally outer trapped slices which lie in the leaves of the given foliation. We also show that under rather weak energy conditions this horizon must be either achronal or spacelike everywhere. Furthermore, we discuss the relation between ``bounding'' and ``stability'' properties of marginally outer trapped surfaces.

Local existence of dynamical and trapping horizons

TL;DR

Given a spacelike foliation of a spacetime and a marginally outer trapped surface S on some initial leaf, it is proved that under a suitable stability condition S is contained in "horizon" i.e., a smooth 3-surface foliated by marginally outer trapping slices which lie in the leaves of the given foliation.

Abstract

Given a spacelike foliation of a spacetime and a marginally outer trapped surface S on some initial leaf, we prove that under a suitable stability condition S is contained in a ``horizon'', i.e. a smooth 3-surface foliated by marginally outer trapped slices which lie in the leaves of the given foliation. We also show that under rather weak energy conditions this horizon must be either achronal or spacelike everywhere. Furthermore, we discuss the relation between ``bounding'' and ``stability'' properties of marginally outer trapped surfaces.

Paper Structure

This paper contains 6 theorems, 5 equations, 1 figure.

Key Result

Theorem 1

Let $(M, g_{\alpha\beta})$ be a smooth spacetime foliated by smooth spacelike hypersurfaces $\Sigma_t$. Assume that some leaf $\Sigma = \Sigma_0$ contains a smooth marginally outer trapped surface $S$ which is strictly stably outermost. Then, $S$ is contained in a smooth horizon $H$ whose marginally

Figures (1)

  • Figure 1: A horizon

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • Lemma 3
  • ...and 1 more