Cauchy Horizon Endpoints and Differentiability

TL;DR

This paper investigates the differentiability properties of Cauchy horizons, establishing conditions under which endpoints and differentiability are related, and showing that under certain curvature conditions, horizons are smooth and endpoint-free.

Contribution

It provides new theoretical results linking the differentiability of Cauchy horizons to their null generator endpoints and curvature conditions, advancing understanding of horizon regularity.

Findings

Endpoints occur only where a single null generator leaves the horizon.

A horizon is differentiable on an open subset iff it has no endpoints there.

A C^2 almost everywhere horizon with null convergence condition has no endpoints.

Abstract

Cauchy horizons are shown to be differentiable at endpoints where only a single null generator leaves the horizon. A Cauchy horizon fails to have any null generator endpoints on a given open subset iff it is differentiable on the open subset and also iff the horizon is (at least) of class C^1 on the open subset. Given the null convergence condition, a compact horizon which is of class C^2 almost everywhere has no endpoints and is (at least) of class C^1 at all points.