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A note on the Lorentzian splitting theorem

Gregory J. Galloway

TL;DR

The paper addresses the Lorentzian splitting problem under a weakened Ricci curvature condition, requiring $\liminf_{t \to \infty} \int_0^{t} \mathrm{Ric}(\alpha'(\hat{r}),\alpha'(\hat{r}))\, d\hat{r} \ge 0$ along timelike rays and the existence of a timelike line. It adopts a causal-geometric approach using achronal limits and ray horospheres, alongside mean-curvature in the support sense and the geometric maximum principle, to derive a local product near a timelike line and then a global splitting. It yields a global isometry $(M,g) \cong (\mathbb{R} \times S, -dt^2 + h)$ with $S$ a complete spacelike Cauchy hypersurface, determining structure via foliations by totally geodesic hypersurfaces. By removing the bounded Ricci condition required in prior work, the result broadens Lorentzian splitting to globally hyperbolic spacetimes under weaker energy conditions and suggests extensions to non-globally hyperbolic or low-regularity settings.

Abstract

We present a version of the Lorentzian splitting theorem under a weakened Ricci curvature condition. The proof makes use of basic properties of achronal limits [19], [20], together with the geometric maximum principle for $C^0$ spacelike hypersurfaces in [1]. Our version strengthens a related result in [29] in the globally hyperbolic setting by removing a certain boundedness condition on the Ricci curvature.

A note on the Lorentzian splitting theorem

TL;DR

The paper addresses the Lorentzian splitting problem under a weakened Ricci curvature condition, requiring along timelike rays and the existence of a timelike line. It adopts a causal-geometric approach using achronal limits and ray horospheres, alongside mean-curvature in the support sense and the geometric maximum principle, to derive a local product near a timelike line and then a global splitting. It yields a global isometry with a complete spacelike Cauchy hypersurface, determining structure via foliations by totally geodesic hypersurfaces. By removing the bounded Ricci condition required in prior work, the result broadens Lorentzian splitting to globally hyperbolic spacetimes under weaker energy conditions and suggests extensions to non-globally hyperbolic or low-regularity settings.

Abstract

We present a version of the Lorentzian splitting theorem under a weakened Ricci curvature condition. The proof makes use of basic properties of achronal limits [19], [20], together with the geometric maximum principle for spacelike hypersurfaces in [1]. Our version strengthens a related result in [29] in the globally hyperbolic setting by removing a certain boundedness condition on the Ricci curvature.

Paper Structure

This paper contains 2 sections, 6 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. A Lorentzian splitting theorem

Key Result

Theorem 1

Let $M$ be a globally hyperbolic, timelike geodesically complete spacetime, satisfying the strong energy condition, $\mathrm{Ric}(X,X) \ge 0$, for all timelike $X$. If $M$ admits a timelike line, then $M$ splits as a metric product, i.e. $(M, g)$ is isometric to $({\mathbb R} \times S, -dt^2 + h)$ w

Theorems & Definitions (7)

  • Theorem 1: Lorentzian Splitting Theorem
  • Conjecture 2: Bartnik Splitting Conjecture
  • Theorem 3
  • Proposition 4: horo1
  • Proposition 5
  • Theorem 6
  • Theorem 7