Global viscosity solutions to Lorentzian eikonal equation on globally hyperbolic space-times

TL;DR

This paper proves the existence of globally defined viscosity solutions to the Lorentzian eikonal equation on globally hyperbolic space-times, classifies solutions based on time orientation, and explores their properties.

Contribution

It establishes the existence of distance-like viscosity solutions on globally hyperbolic space-times and analyzes their properties depending on time orientation.

Findings

Existence of globally defined viscosity solutions on globally hyperbolic space-times.

Local semiconcavity of solutions with consistent time orientation.

Distinct properties of solutions with non-consistent time orientation.

Abstract

In this paper, we show that any globally hyperbolic space-time admits at least one globally defined distance-like function, which is a viscosity solution to the Lorentzian eikonal equation. According to whether the time orientation is changed, we divide the set of viscosity solutions into some subclasses. We show if the time orientation is consistent, then a viscosity solution has a variational representation locally. As a result, such a viscosity solution is locally semiconcave, as the one in the Riemannian case. Also, if the time orientation of a viscosity solution is non-consistent, we analyse its peculiar properties which make this kind of viscosity solutions are totally different from the ones where the Hamiltonians are convex.