Convex Functions and Spacetime Geometry

TL;DR

This paper explores how the existence of convex functions in spacetime geometries imposes restrictions on their properties, potentially impacting the study of General Relativity.

Contribution

It introduces a framework for understanding the role of convex functions in spacetime geometry and their implications in General Relativity.

Findings

Convex functions restrict spacetime geometric properties

Existence of convex functions influences initial data sets

Provides new tools for analyzing spacetime structures

Abstract

Convexity and convex functions play an important role in theoretical physics. To initiate a study of the possible uses of convex functions in General Relativity, we discuss the consequences of a spacetime $(M,g_{\mu \nu})$ or an initial data set $(\Sigma, h_{ij}, K_{ij})$ admitting a suitably defined convex function. We show how the existence of a convex function on a spacetime places restrictions on the properties of the spacetime geometry.