Some criteria for Wind Riemannian completeness and existence of Cauchy hypersurfaces

TL;DR

This paper explores the properties of wind Riemannian structures, establishing criteria for their completeness and the existence of Cauchy hypersurfaces in certain spacetimes, linking Finslerian and Lorentzian geometries.

Contribution

It introduces new characterizations and criteria for the completeness of wind Riemannian structures and applies these to identify Cauchy hypersurfaces in specific stationary spacetimes.

Findings

Criteria for wind Riemannian completeness

Conditions for slices to be Cauchy hypersurfaces

Examples illustrating the Lorentzian-Finslerian link

Abstract

Recently, a link between Lorentzian and Finslerian Geometries has been carried out, leading to the notion of wind Riemannian structure (WRS), a generalization of Finslerian Randers metrics. Here, we further develop this notion and its applications to spacetimes, by introducing some characterizations and criteria for the completeness of WRS's. As an application, we consider a general class of spacetimes admitting a time function $t$ generated by the flow of a complete Killing vector field (generalized standard stationary spacetimes or, more precisely, SSTK ones) and derive simple criteria ensuring that its slices $t=$ constant are Cauchy. Moreover, a brief summary on the Finsler/Lorentz link for readers with some acquaintance in Lorentzian Geometry, plus some simple examples in Mathematical Relativity, are provided.