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Metric Convergence of Sequences of Static Spacetimes with the Null Distance

Brian Allen

TL;DR

The paper advances the program of defining metric convergence for sequences of spacetimes by leveraging the null distance $\hat{d}_{t,g}$ on static spacetimes. It establishes a Hölder-type control that yields uniform and Gromov-Hausdorff convergence under suitable $L^p$ bounds, and develops intrinsic-flat convergence results for spacetime sequences with the null distance, under volume, area, and lower-bound hypotheses. A key contribution is an analog of the Volume Above Distance Below (VADB) theorem tailored to static spacetimes and a framework to estimate SWIF distances between spacetimes by embedding them into a joining space $Z$. The work clarifies when GH versus SWIF limits occur, supported by illustrative examples, and proposes a conjecture extending these ideas to globally hyperbolic spacetimes with the null distance, highlighting the broader impact for Lorentzian geometry and spacetime convergence studies.

Abstract

How should one define metric space notions of convergence for sequences of spacetimes? Since a Lorentzian manifold does not define a metric space directly, the uniform convergence, Gromov-Hausdorff (GH) convergence, and Sormani-Wenger Intrinsic Flat (SWIF) convergence does not extend automatically. One approach is to define a metric space structure, which is compatible with the Lorentzian structure, so that the usual notions of convergence apply. This approach was taken by C. Sormani and C. Vega when defining the null distance. In this paper, we study sequences of static spacetimes equipped with the null distance under uniform, GH, and SWIF convergence, as well as Hölder bounds. We use the results of the Volume Above Distance Below (VADB) theorem of the author, R. Perales, and C. Sormani to prove an analog of the VADB theorem for sequences of static spacetimes with the null distance. We also give a conjecture of what the VADB theorem should be in the case of sequences of globally hyperbolic spacetimes with the null distance.

Metric Convergence of Sequences of Static Spacetimes with the Null Distance

TL;DR

The paper advances the program of defining metric convergence for sequences of spacetimes by leveraging the null distance on static spacetimes. It establishes a Hölder-type control that yields uniform and Gromov-Hausdorff convergence under suitable bounds, and develops intrinsic-flat convergence results for spacetime sequences with the null distance, under volume, area, and lower-bound hypotheses. A key contribution is an analog of the Volume Above Distance Below (VADB) theorem tailored to static spacetimes and a framework to estimate SWIF distances between spacetimes by embedding them into a joining space . The work clarifies when GH versus SWIF limits occur, supported by illustrative examples, and proposes a conjecture extending these ideas to globally hyperbolic spacetimes with the null distance, highlighting the broader impact for Lorentzian geometry and spacetime convergence studies.

Abstract

How should one define metric space notions of convergence for sequences of spacetimes? Since a Lorentzian manifold does not define a metric space directly, the uniform convergence, Gromov-Hausdorff (GH) convergence, and Sormani-Wenger Intrinsic Flat (SWIF) convergence does not extend automatically. One approach is to define a metric space structure, which is compatible with the Lorentzian structure, so that the usual notions of convergence apply. This approach was taken by C. Sormani and C. Vega when defining the null distance. In this paper, we study sequences of static spacetimes equipped with the null distance under uniform, GH, and SWIF convergence, as well as Hölder bounds. We use the results of the Volume Above Distance Below (VADB) theorem of the author, R. Perales, and C. Sormani to prove an analog of the VADB theorem for sequences of static spacetimes with the null distance. We also give a conjecture of what the VADB theorem should be in the case of sequences of globally hyperbolic spacetimes with the null distance.

Paper Structure

This paper contains 14 sections, 12 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Pointwise Convergence on Manifolds with Boundary
  4. $L^p$ bounds on Riemannian Manifolds and Hölder Distance Bounds
  5. Uniform Distance Bounds on a Good Set
  6. Uniform Convergence of Metric Spaces
  7. Sormani-Wenger Intrinsic Flat Convergence
  8. Examples
  9. No uniform control from below in the space direction
  10. No uniform control from above in the time direction
  11. Volume bound without volume convergence produces bubbles
  12. Volume convergence without $L^{\frac{p}{2}}$, $p >n$ bound separates GH from SWIF
  13. Estimating the SWIF Distance Between Spacetimes
  14. Proof of Main Theorems

Key Result

Theorem 1.1

Let $0<C, C'<\infty$, $M$ a compact, connected, and oriented manifold, $(M,\sigma_1)$ a continuous Riemannian manifold, $(M,\sigma_0)$ a smooth Riemannian manifold, $L=[t_0,t_1]\times M$, $h_i:M \rightarrow (0,\infty)$, $g_i=-h_i^2dt^2+\sigma_j$, $i \in \{0,1\}$. If $p > n$, then then for all $x,y \in L$ we find

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Conjecture 1.8
  • Theorem 2.1: Theorem 4.5 of Allen-Bryden
  • Theorem 2.2: Theorem 1.1 of Allen-Holder
  • ...and 25 more