Rigid Singularity Theorem in Globally Hyperbolic Spacetimes

TL;DR

This paper proves a rigidity result for globally hyperbolic spacetimes with trapped sets, showing they are either incomplete or split as a product, contributing to the understanding of spacetime singularities and Lorentzian geometry.

Contribution

It establishes a new rigidity theorem linking trapped sets and spacetime splitting under energy conditions, advancing Lorentzian geometry and singularity theory.

Findings

Spacetimes with trapped sets are either incomplete or split as space × time.

The result relates to Yau's Lorentzian splitting conjecture.

Provides conditions under which spacetime splitting occurs.

Abstract

We show the rigid singularity theorem, that is, a globally hyperbolic spacetime satisfying the strong energy condition and containing past trapped sets, either is timelike geodesically incomplete or splits isometrically as space $\times$ time. This result is related to Yau's Lorentzian splitting conjecture.