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

Abstract

In this paper, we show that any globally hyperbolic space-time admits at least one globally defined locally semiconcave 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 and has some weak KAM properties, as the one in the Riemannian case. On the other hand, if the time orientation of a viscosity solution is non-consistent, it will exhibit some peculiar properties which makes this kind of viscosity solutions totally different from the ones in the Riemannian case.