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Null distance and convergence of Lorentzian length spaces

Michael Kunzinger, Roland Steinbauer

Abstract

The null distance of Sormani and Vega encodes the manifold topology as well as the causality structure of a (smooth) spacetime. We extend this concept to Lorentzian length spaces, the analog of (metric) length spaces, which generalize Lorentzian causality theory beyond the manifold level. We then study Gromov-Hausdorff convergence based on the null distance in warped product Lorentzian length spaces and prove first results on its compatibility with synthetic curvature bounds.

Null distance and convergence of Lorentzian length spaces

Abstract

The null distance of Sormani and Vega encodes the manifold topology as well as the causality structure of a (smooth) spacetime. We extend this concept to Lorentzian length spaces, the analog of (metric) length spaces, which generalize Lorentzian causality theory beyond the manifold level. We then study Gromov-Hausdorff convergence based on the null distance in warped product Lorentzian length spaces and prove first results on its compatibility with synthetic curvature bounds.

Paper Structure

This paper contains 6 sections, 29 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The null distance in Lorentzian length spaces
  4. Warped products
  5. Null distance on generalized cones
  6. Convergence of generalized cones and curvature bounds

Key Result

Lemma 3.5

Let $(X,d,\ll,\le,\rho)$ be an scc Lorentzian pre-length space. Then for any $p, q\in X$ there exists a piecewise causal curve from $p$ to $q$.

Theorems & Definitions (61)

  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Definition 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • proof
  • Lemma 3.7
  • proof
  • ...and 51 more