Lightlike hypersurfaces and time-minimizing geodesics in cone structures

TL;DR

This paper extends Lorentzian geometry concepts to cone structures, defining lightlike hypersurfaces and demonstrating their role in understanding time-minimizing geodesics without relying on a metric.

Contribution

It introduces a framework for cone structures that generalizes Lorentzian causality and geodesics, including the novel definition of lightlike hypersurfaces and their foliation by cone geodesics.

Findings

Lightlike hypersurfaces admit a unique foliation by cone geodesics.

In globally hyperbolic spacetimes, achronal boundaries are lightlike hypersurfaces under certain conditions.

Time-minimization properties of cone geodesics are established among causal curves from hypersurfaces.

Abstract

Some well-known Lorentzian concepts are transferred into the more general setting of cone structures, which provide both the causality of the spacetime and the notion of cone geodesics without making use of any metric. Lightlike hypersurfaces are defined within this framework, showing that they admit a unique folitation by cone geodesics. This property becomes crucial after proving that, in globally hyperbolic spacetimes, achronal boundaries are lightlike hypersurfaces under some restrictions, allowing one to easily obtain some time-minimization properties of cone geodesics among causal curves departing from a hypersurface of the spacetime.