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Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems

Stephanie B. Alexander, Melanie Graf, Michael Kunzinger, Clemens Sämann

Abstract

We study generalizations of Lorentzian warped products with one-dimensional base of the form $I\times_f X$, where $I$ is an interval, $X$ is a length space and $f$ is a positive continuous function. These generalized cones furnish an important class of Lorentzian length spaces in the sense of [Kunzinger, Sämann; Ann. Glob. Anal. Geom. 54(3):399--447, 2018], displaying optimal causality properties that allow for explicit descriptions of all underlying notions. In addition, synthetic sectional curvature bounds of generalized cones are directly related to metric curvature bounds of the fiber $X$. The interest in such spaces comes both from metric geometry and from General Relativity, where warped products underlie important cosmological models (FLRW spacetimes). Moreover, we prove singularity theorems for these spaces, showing that non-positive lower timelike curvature bounds imply the existence of incomplete timelike geodesics.

Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems

Abstract

We study generalizations of Lorentzian warped products with one-dimensional base of the form , where is an interval, is a length space and is a positive continuous function. These generalized cones furnish an important class of Lorentzian length spaces in the sense of [Kunzinger, Sämann; Ann. Glob. Anal. Geom. 54(3):399--447, 2018], displaying optimal causality properties that allow for explicit descriptions of all underlying notions. In addition, synthetic sectional curvature bounds of generalized cones are directly related to metric curvature bounds of the fiber . The interest in such spaces comes both from metric geometry and from General Relativity, where warped products underlie important cosmological models (FLRW spacetimes). Moreover, we prove singularity theorems for these spaces, showing that non-positive lower timelike curvature bounds imply the existence of incomplete timelike geodesics.

Paper Structure

This paper contains 11 sections, 52 theorems, 74 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main results and outline of the paper
  3. Alexandrov spaces
  4. Lorentzian length space s
  5. Minkowski cones over metric spaces
  6. Generalized cones
  7. Generalized cones as Lorentzian length space s
  8. Curvature bounds
  9. Synthetic singularity theorems for generalized cones
  10. Conclusion and outlook
  11. Appendix: Lorentzian length structures

Key Result

Theorem 2.5

Let $Y=\mathrm{Cone}(X)$ be the Minkowski cone over a geodesic length space $X$. Then $Y$ has timelike curvature bounded below (above) by $0$ if and only if $X$ is an Alexandrov space of curvature bounded below (above) by $-1$.

Theorems & Definitions (118)

  • Theorem 2.5
  • Proposition 3.22
  • Theorem 5.3
  • Theorem 5.7
  • Corollary 6.4
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 108 more