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A Splitting Theorem for non-positively curved Lorentzian spaces

Joe Barton, Tobias Beran, Mauricio Che, Sebastian Gieger, Jona Röhrig, Felix Rott

TL;DR

The paper extends splitting theorems to non-smooth Lorentzian spaces with timelike curvature upper bounds. It develops a first variation formula for time separation, a rigidity theory for upper curvature bounds, and a theory of parallel lines and rays in Lorentzian pre-length spaces. The main result is a Lorentzian splitting theorem: under global timelike curvature bounded above by $0$, the space decomposes as a Lorentzian product $(-\mathbb{R})\times S$ when covered by synchronised-parallel timelike lines, with $S$ a CAT(0) space. This provides a robust structural decomposition applicable to Lorentzian geometry and general relativity in a non-smooth setting.

Abstract

We prove a splitting theorem for Lorentzian pre-length spaces with global non-positive timelike curvature. Additionally, we extend the first variation formula to spaces with any timelike curvature bound, either from above or below, and different from 0.

A Splitting Theorem for non-positively curved Lorentzian spaces

TL;DR

The paper extends splitting theorems to non-smooth Lorentzian spaces with timelike curvature upper bounds. It develops a first variation formula for time separation, a rigidity theory for upper curvature bounds, and a theory of parallel lines and rays in Lorentzian pre-length spaces. The main result is a Lorentzian splitting theorem: under global timelike curvature bounded above by , the space decomposes as a Lorentzian product when covered by synchronised-parallel timelike lines, with a CAT(0) space. This provides a robust structural decomposition applicable to Lorentzian geometry and general relativity in a non-smooth setting.

Abstract

We prove a splitting theorem for Lorentzian pre-length spaces with global non-positive timelike curvature. Additionally, we extend the first variation formula to spaces with any timelike curvature bound, either from above or below, and different from 0.
Paper Structure (6 sections, 23 theorems, 135 equations, 11 figures)

This paper contains 6 sections, 23 theorems, 135 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. First variation formula
  4. Rigidity for upper curvature bounds
  5. Parallel lines and rays
  6. Splitting theorem

Key Result

Theorem 1.1

Let $X$ be a Lorentzian pre-length space with timelike curvature globally bounded above by $0$. Let $\gamma$ be a complete timelike line and define Then $S$ admits a metric that makes it into a $\mathrm{CAT}(0)$ space and the set $\bigcup_{\alpha\in S}\alpha(\mathbb R)$ is isometric to the Lorentzian product $\prescript{-}{}{\mathbb R}\times S$. In particular, if $X=\bigcup_{\alpha\in S}\alpha(\m

Figures (11)

  • Figure 1: Set up for the first variation formula for $\sigma=1$.
  • Figure 2: The triangle considered as a closed loop, and the outer angles.
  • Figure 3: Setup of the proof depending on $\sigma$.
  • Figure 4: Labelling of the comparison configuration.
  • Figure 5: Setup of the quadrangles, first case.
  • ...and 6 more figures

Theorems & Definitions (55)

  • Theorem 1.1
  • Definition 2.1: Timelike triangles
  • Definition 2.2: Timelike curvature bounds
  • Proposition 2.3: Triangle inequality for angles
  • Lemma 2.4: Angle sum for triangles in Minkowski space
  • proof
  • Definition 2.5: Structure preserving maps
  • Remark 2.6: Isometries, topology and causality
  • Definition 2.7: Lorentzian product
  • Proposition 2.8: Properties of Lorentzian products
  • ...and 45 more