Areas and volumes for null cones

TL;DR

This paper establishes new monotonicity and comparison theorems for the area and volume of null cone slices in Lorentzian manifolds, with potential applications to controlling geometric properties in Ricci-flat spacetimes.

Contribution

It introduces novel area and volume comparison results for null cones, extending classical geometric theorems to Lorentzian geometry.

Findings

Proved monotonicity properties for null cone slice areas.

Established volume comparison results analogous to Bishop-Gromov and Guenther theorems.

Discussed applications to controlling null second fundamental form in Ricci-flat manifolds.

Abstract

Motivated by recent work of Choquet-Bruhat, Chrusciel, and Martin-Garcia, we prove monotonicity properties and comparison results for the area of slices of the null cone of a point in a Lorentzian manifold. We also prove volume comparison results for subsets of the null cone analogous to the Bishop-Gromov relative volume monotonicity theorem and Guenther's volume comparison theorem. We briefly discuss how these estimates may be used to control the null second fundamental form of slices of the null cone in Ricci-flat Lorentzian four-manifolds with null curvature bounded above.