Global Hyperbolicity and Completeness

TL;DR

This paper establishes conditions under which spacetimes are globally hyperbolic and geodesically complete, focusing on regularity conditions of the metric and integrability of geometric quantities.

Contribution

It proves global hyperbolicity under generic regularity and shows geodesic completeness when certain geometric derivatives are integrable, with specific conditions for expanding universes.

Findings

Spacetimes are globally hyperbolic under generic regularity conditions.

Geodesic completeness is achieved if the gradient of the lapse and extrinsic curvature are integrable.

Specific conditions apply for expanding universes regarding the tracefree part of K.

Abstract

We prove global hyperbolicity of spacetimes under generic regularity conditions on the metric. We then show that these spacetimes are timelike and null geodesically complete if the gradient of the lapse and the extrinsic curvature $K$ are integrable. This last condition is required only for the tracefree part of $K$ if the universe is expanding.