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Comparison theory for Lipschitz spacetimes

Mathias Braun, Marta Sálamo Candal

Abstract

We prove a globally hyperbolic spacetime with locally Lipschitz continuous metric and timelike distributional Ricci curvature bounded from below obeys the timelike measure contraction property. The remarkable class of examples of spacetimes that are covered by this result includes impulsive gravity waves, thin shells, and matched spacetimes. As applications, we get new comparison theorems for Lipschitz spacetimes in sharp form: d'Alembert, timelike Brunn-Minkowski, and timelike Bishop-Gromov. Under appropriate nonbranching assumptions (conjectured to hold in even lower regularity), our results also yield the timelike curvature-dimension condition, a volume incompleteness theorem, as well as exact representation formulas and sharp comparison estimates for d'Alembertians of Lorentz distance functions from general spacelike submanifolds. Moreover, we establish the sharp timelike Bonnet--Myers inequality ad hoc using the localization technique from convex geometry. Alongside, we prove a timelike diameter estimate for spacetimes whose timelike Ricci curvature is positive up to a "small" deviation (in an $L^p$-sense). This adapts prior theorems for Riemannian manifolds by Petersen-Sprouse and Aubry to Lorentzian geometry, a transition the former two anticipated almost 30 years ago.

Comparison theory for Lipschitz spacetimes

Abstract

We prove a globally hyperbolic spacetime with locally Lipschitz continuous metric and timelike distributional Ricci curvature bounded from below obeys the timelike measure contraction property. The remarkable class of examples of spacetimes that are covered by this result includes impulsive gravity waves, thin shells, and matched spacetimes. As applications, we get new comparison theorems for Lipschitz spacetimes in sharp form: d'Alembert, timelike Brunn-Minkowski, and timelike Bishop-Gromov. Under appropriate nonbranching assumptions (conjectured to hold in even lower regularity), our results also yield the timelike curvature-dimension condition, a volume incompleteness theorem, as well as exact representation formulas and sharp comparison estimates for d'Alembertians of Lorentz distance functions from general spacelike submanifolds. Moreover, we establish the sharp timelike Bonnet--Myers inequality ad hoc using the localization technique from convex geometry. Alongside, we prove a timelike diameter estimate for spacetimes whose timelike Ricci curvature is positive up to a "small" deviation (in an -sense). This adapts prior theorems for Riemannian manifolds by Petersen-Sprouse and Aubry to Lorentzian geometry, a transition the former two anticipated almost 30 years ago.
Paper Structure (40 sections, 54 theorems, 215 equations)

This paper contains 40 sections, 54 theorems, 215 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Sharp timelike Bonnet--Myers inequality
  4. Analytic and synthetic spacetime geometry
  5. Applications to comparison theory
  6. Organization
  7. Analytic timelike Ricci curvature lower bounds
  8. Lipschitz spacetime geometry
  9. Regularization
  10. Tensor distributions
  11. Good approximation
  12. Properties of good approximation
  13. Construction of compact and uniformly timelike sets
  14. Regularity of time separation function
  15. Synthetic timelike Ricci curvature lower bounds
  16. ...and 25 more sections

Key Result

Theorem 1.1

We fix $K> 0$. In addition, assume $(M,g)$ is a globally hyperbolic Lipschitz spacetime such that ${\mathrm{Ric}_{g}\geq K}$ timelike distributionally, cf. Def:Bounds. Then

Theorems & Definitions (96)

  • Theorem 1.1: Timelike Bonnet--Myers diameter estimate, \ref{['Th:TimelikeBonnetMyers']}
  • Theorem 1.2: Timelike Petersen--Sprouse diameter estimate, \ref{['Cor:PetersenSprouse']}
  • Definition 1.3: Timelike measure contraction property, \ref{['Def:TMCP']}
  • Theorem 1.4: Timelike measure contraction property, \ref{['Th:ToTMCP']}
  • Remark 1.5: Improvements of \ref{['Th:TLTMCPintro']}
  • Theorem 1.6: Timelike Brunn--Minkowski inequality, \ref{['Th:TimelikeBrunnMinkowski']}
  • Theorem 1.7: Timelike Bishop--Gromov inequality, \ref{['Th:TimelikeBishopGromov']}
  • Theorem 1.8: D'Alembert comparison theorem for powers of Lorentz distance functions, \ref{['Th:Dalembert powers']}
  • Theorem 1.9: D'Alembert comparison theorem for Lorentz distance functions, \ref{['Th:DAlembert']}
  • Definition 2.2: Global hyperbolicity
  • ...and 86 more