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A synthetic Gannon-Lee incompleteness theorem

Mathias Braun, Carlo Rotolo

Abstract

We prove the Gannon-Lee incompleteness theorem for globally hyperbolic spacetimes. We assume the synthetic null energy condition of Ketterer and a trappedness condition we call "synthetically asymptotically regular". Our result generalizes this classical result to the weighted case. It also motivates and indicates extensions to low regularity, which are deferred to future work.

A synthetic Gannon-Lee incompleteness theorem

Abstract

We prove the Gannon-Lee incompleteness theorem for globally hyperbolic spacetimes. We assume the synthetic null energy condition of Ketterer and a trappedness condition we call "synthetically asymptotically regular". Our result generalizes this classical result to the weighted case. It also motivates and indicates extensions to low regularity, which are deferred to future work.
Paper Structure (14 sections, 20 theorems, 53 equations)

This paper contains 14 sections, 20 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. The incompleteness theorems
  3. Synthetic curvature bounds
  4. Content of the work and main results
  5. Organization of the work
  6. Preliminary notions
  7. Lorentzian geometry
  8. Ketterer's synthetic NEC
  9. Statement of the synthetic Gannon--Lee theorem
  10. A local to global property for the synthetic NEC
  11. Proof of the theorem
  12. Focusing result
  13. Causality step
  14. Topological aspects

Key Result

Theorem 1.1

Let $(M,g, e^{-V}\mathrm{Vol}_M)$ be a past reflecting, null geodesically complete weighted spacetime of dimension $n+1$ which satisfies the synthetic $N$-null energy condition for $N\ge n-1$. Let $\Sigma$ be a synthetically asymptotically regular hypersurface admitting a piercing, with enclosing su Then the map $i_\#:\pi_1(S)\rightarrow \pi_1(\Sigma)$, induced by the inclusion $i:S\rightarrow \Si

Theorems & Definitions (39)

  • Theorem 1.1: Synthetic Gannon--Lee theorem
  • Theorem 1.2: Globally hyperbolic Gannon--Lee theorem
  • Proposition 2.1: Properties of $\ll$ and $\le$
  • Proposition 2.2: Push-up Lemma
  • Proposition 2.3: Achronal set with no edge is a topological hypersurface
  • Definition 2.4: Null coupling
  • Definition 2.5: Null connected probability measures
  • Definition 2.6: Dynamical null coupling
  • Lemma 2.7: Existence of a transport map, ket, Lemma 3.2
  • Remark 2.8
  • ...and 29 more