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Rigidity aspects of Penrose's singularity theorem

Gregory J. Galloway, Eric Ling

TL;DR

This work extends Penrose's singularity theorem by examining rigidity when the trapped surface is only weakly trapped and the spacetime is null geodesically complete. Utilizing the MOTS stability operator $L$, inverse function arguments, and null maximum principles, the authors establish a local foliation by MOTS near a weakly trapped $\Sigma$ and show that outward null generators generate totally geodesic null hypersurfaces. They develop global rigidity results for spacetimes with compact-with-boundary regions and for cosmological settings with compact Cauchy surfaces, as well as topological censorship rigidity results that constrain spacetime topology under completeness assumptions. The main dichotomy is that either the spacetime is null geodesically incomplete, or it exhibits a highly rigid geometric structure characterized by MOTS/MITS foliations and totally geodesic null hypersurfaces, with concrete implications in four dimensions and cosmological models. The methods blend geometric analysis of MOTS, covering space arguments, and continuity/maximum-principle techniques to derive broad rigidity conclusions with clear physical interpretations.

Abstract

In this paper, we study rigidity aspects of Penrose's singularity theorem. Specifically, we aim to answer the following question: if a spacetime satisfies the hypotheses of Penrose's singularity theorem except with weakly trapped surfaces instead of trapped surfaces, then what can be said about the global spacetime structure if the spacetime is null geodesically complete? In this setting, we show that we obtain a foliation of MOTS which generate totally geodesic null hypersurfaces. Depending on our starting assumptions, we obtain either local or global rigidity results. We apply our arguments to cosmological spacetimes (i.e., spacetimes with compact Cauchy surfaces) and scenarios involving topological censorship.

Rigidity aspects of Penrose's singularity theorem

TL;DR

This work extends Penrose's singularity theorem by examining rigidity when the trapped surface is only weakly trapped and the spacetime is null geodesically complete. Utilizing the MOTS stability operator , inverse function arguments, and null maximum principles, the authors establish a local foliation by MOTS near a weakly trapped and show that outward null generators generate totally geodesic null hypersurfaces. They develop global rigidity results for spacetimes with compact-with-boundary regions and for cosmological settings with compact Cauchy surfaces, as well as topological censorship rigidity results that constrain spacetime topology under completeness assumptions. The main dichotomy is that either the spacetime is null geodesically incomplete, or it exhibits a highly rigid geometric structure characterized by MOTS/MITS foliations and totally geodesic null hypersurfaces, with concrete implications in four dimensions and cosmological models. The methods blend geometric analysis of MOTS, covering space arguments, and continuity/maximum-principle techniques to derive broad rigidity conclusions with clear physical interpretations.

Abstract

In this paper, we study rigidity aspects of Penrose's singularity theorem. Specifically, we aim to answer the following question: if a spacetime satisfies the hypotheses of Penrose's singularity theorem except with weakly trapped surfaces instead of trapped surfaces, then what can be said about the global spacetime structure if the spacetime is null geodesically complete? In this setting, we show that we obtain a foliation of MOTS which generate totally geodesic null hypersurfaces. Depending on our starting assumptions, we obtain either local or global rigidity results. We apply our arguments to cosmological spacetimes (i.e., spacetimes with compact Cauchy surfaces) and scenarios involving topological censorship.
Paper Structure (7 sections, 12 theorems, 11 equations)

This paper contains 7 sections, 12 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local rigidity
  4. Compact-with-boundary rigidity
  5. Cosmological rigidity
  6. Topological censorship rigidity
  7. Proof of Lemma \ref{['lem: IVT argument']}

Key Result

Lemma 1

Let $\Sigma$ be a stable MOTS in an initial data set $(V,h,K)$. If $\lambda_1 = 0$, then there is a neighborhood $U$ of $\Sigma$ in $V$ such that, up to isometry, where $\phi = \phi(t,x)$ is a smooth function on $U$ and $\gamma_t$ is the induced metric on $\Sigma_t = \{t\} \times \Sigma$. Moreover, the outward null expansion of each $\Sigma_t$ is constant, i.e., $\theta^+(t)$ is constant on $\Sig

Theorems & Definitions (12)

  • Lemma 1
  • Theorem 2
  • Proposition 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 2 more