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Dynamics of Apparent Horizon and a Null Comparison Principle

Xinliang An, Taoran He

Abstract

This paper investigates the global dynamics of the apparent horizon. We present an approach to establish its existence and its long-term behaviors. Our apparent horizon is constructed by solving the marginally outer trapped surface (MOTS) along each incoming null hypersurface. Based on the nonlinear hyperbolic estimates established in [24] by Klainerman-Szeftel under polarized axial symmetry, we prove that the corresponding apparent horizon is smooth, asymptotically null and converging to the event horizon eventually. To further address the local achronality of the apparent horizon, a new concept, called the null comparison principle, is introduced in this paper. For three typical scenarios of gravitational collapse, our null comparison principle is tested and verified, which guarantees that the apparent horizon must be piecewise spacelike or piecewise null. In addition, we also validate and provide new proofs for several physical laws along the apparent horizon.

Dynamics of Apparent Horizon and a Null Comparison Principle

Abstract

This paper investigates the global dynamics of the apparent horizon. We present an approach to establish its existence and its long-term behaviors. Our apparent horizon is constructed by solving the marginally outer trapped surface (MOTS) along each incoming null hypersurface. Based on the nonlinear hyperbolic estimates established in [24] by Klainerman-Szeftel under polarized axial symmetry, we prove that the corresponding apparent horizon is smooth, asymptotically null and converging to the event horizon eventually. To further address the local achronality of the apparent horizon, a new concept, called the null comparison principle, is introduced in this paper. For three typical scenarios of gravitational collapse, our null comparison principle is tested and verified, which guarantees that the apparent horizon must be piecewise spacelike or piecewise null. In addition, we also validate and provide new proofs for several physical laws along the apparent horizon.
Paper Structure (32 sections, 32 theorems, 310 equations, 3 figures)

This paper contains 32 sections, 32 theorems, 310 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Key Points and New Ingredients of the Paper
  4. Choice of Coordinate System
  5. Existence and Asymptotics of Apparent Horizon
  6. Null Comparison Principle and its Applications
  7. Connections to Black Hole Thermodynamics
  8. Acknowledgements
  9. Setup
  10. Incoming Geodesic Foliation in Klainerman-Szeftel's Work
  11. Deformation Equation
  12. An Elliptic Operator Associated with the MOTS
  13. Existence and Asymptotics of the Apparent Horizon in the Setting of Klainerman-Szeftel
  14. Coordinates transformation
  15. A Priori Estimates
  16. ...and 17 more sections

Key Result

Theorem 1.1

With initial data prescribed in K-S, the solved Einstein vacuum spacetime contains an apparent horizon that is asymptotically null and approaches the timelike infinity. Specifically, along each incoming null hypersurface $\underline{H}_{\underline{u}}$, there exists a unique MOTS $M_{\underline{u}}$

Figures (3)

  • Figure 1: Null comparison principle for MOTS $M_{\underline{u}}$ along $\underline{H}_{\underline{u}}$.
  • Figure 2:
  • Figure 3: Relevant region for applying the null comparison principle.

Theorems & Definitions (66)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1
  • Theorem 1.4
  • Remark 2
  • Theorem 2.1: Klainerman-Szeftel K-S
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 56 more