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The Penrose singularity theorem, MOTS stability, and horizon topology in weighted spacetimes

Eric Ling, Argam Ohanyan, Eric Woolgar

TL;DR

The paper extends Penrose's singularity theorem and Hawking horizon topology results to weighted spacetimes using $m$-Bakry--Émery curvature and a weighted null energy condition, defining $f$-trapped surfaces in arbitrary dimensions and synthetic settings. It develops the weighted null mean curvature $\theta_f=\theta-\nabla_\ell f$, derives a Bakry--Émery Raychaudhuri equation, and proves the ellipticity of the corresponding operator to enable PDE techniques. It proves a Bakry--Émery null splitting theorem and weighted Penrose theorems for various $m$, including a conformally invariant view at $m=2-n$, and analyzes marginally $f$-trapped surfaces via a weighted stability/second-variation theory yielding positive weighted scalar curvature under mild energy conditions. The work further interlinks warped-product geometry with these weighted notions, offering a DEC interpretation and topological implications for horizon cross-sections, thereby connecting curvature, energy conditions, and topology in weighted spacetime settings.

Abstract

We consider versions of the Penrose singularity theorem and the Hawking horizon topology theorem in weighted spacetimes that contain weighted versions of trapped surfaces, for arbitrary spacetime dimension and synthetic dimension. We find that suitable generalizations of the unweighted theorems hold under a weighted null energy condition. Our results also provide further evidence in favour of a weighted scalar curvature that differs from the trace of the weighted Ricci curvature. When the synthetic dimension is a positive integer, these weighted curvatures have a natural interpretation in terms of warped product metrics.

The Penrose singularity theorem, MOTS stability, and horizon topology in weighted spacetimes

TL;DR

The paper extends Penrose's singularity theorem and Hawking horizon topology results to weighted spacetimes using -Bakry--Émery curvature and a weighted null energy condition, defining -trapped surfaces in arbitrary dimensions and synthetic settings. It develops the weighted null mean curvature , derives a Bakry--Émery Raychaudhuri equation, and proves the ellipticity of the corresponding operator to enable PDE techniques. It proves a Bakry--Émery null splitting theorem and weighted Penrose theorems for various , including a conformally invariant view at , and analyzes marginally -trapped surfaces via a weighted stability/second-variation theory yielding positive weighted scalar curvature under mild energy conditions. The work further interlinks warped-product geometry with these weighted notions, offering a DEC interpretation and topological implications for horizon cross-sections, thereby connecting curvature, energy conditions, and topology in weighted spacetime settings.

Abstract

We consider versions of the Penrose singularity theorem and the Hawking horizon topology theorem in weighted spacetimes that contain weighted versions of trapped surfaces, for arbitrary spacetime dimension and synthetic dimension. We find that suitable generalizations of the unweighted theorems hold under a weighted null energy condition. Our results also provide further evidence in favour of a weighted scalar curvature that differs from the trace of the weighted Ricci curvature. When the synthetic dimension is a positive integer, these weighted curvatures have a natural interpretation in terms of warped product metrics.

Paper Structure

This paper contains 17 sections, 12 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Weighted null mean curvature
  3. Definition of weighted null mean curvature
  4. The Raychaudhuri equation
  5. Ellipticity of the weighted null mean curvature
  6. Penrose singularity theorems and null rigidity in weighted spacetimes
  7. The null splitting theorem
  8. Penrose singularity theorems
  9. Marginally outer trapped surfaces
  10. Stability formula
  11. A special case
  12. Warped products
  13. The proof of Theorem \ref{['theorem1.9']}
  14. Interpretation of the dominant energy condition and trapping
  15. Flowing the Bakry--Émery scalar curvature
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $(M,g)$ be a globally hyperbolic spacetime of dimension $\dim M = n \geq 3$ with noncompact Cauchy surfaces, $f \in C^2(M)$ and $m \in (-\infty, 2 - n] \cup (0,\infty]$. Suppose that $(M,g)$ satisfies the Bakry--Émery null energy condition $\mathop{\mathrm{Ric}}\nolimits^m_f(\ell,\ell) \geq 0$ f

Theorems & Definitions (38)

  • Theorem 1.1: Bakry--Émery Penrose singularity theorem
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Conjecture 1.7: J Rosenberg Rosenberg
  • Conjecture 1.8: see Chodosh Conjecture 7.28
  • Theorem 1.9
  • Remark 1.10
  • ...and 28 more