Convex regions of stationary spacetimes and Randers spaces. Applications to lensing and asymptotic flatness

TL;DR

This paper uses the Stationary-to-Randers correspondence to characterize convexity and causal properties of stationary spacetimes, with applications to lensing phenomena and asymptotic flatness in general relativity.

Contribution

It provides a new characterization of convexity and causal structure in stationary spacetimes via Randers spaces, linking geometric properties to physical phenomena.

Findings

Characterization of light and time-convexity in stationary spacetimes.

Conditions for causal simplicity and existence of geodesics.

Analysis of lensing effects and focalization in Randers manifolds.

Abstract

By using Stationary-to-Randers correspondence (SRC), a characterization of light and time-convexity of the boundary of a region of a standard stationary (n+1)-spacetime is obtained, in terms of the convexity of the boundary of a domain in a Finsler n or (n+1)-space of Randers type. The latter convexity is analyzed in depth and, as a consequence, the causal simplicity and the existence of causal geodesics confined in the region and connecting a point to a stationary line are characterized. Applications to asymptotically flat spacetimes include the light-convexity of stationary hypersurfaces which project in a spacelike section of an end onto a sphere of large radius, as well as the characterization of their time-convexity with natural physical interpretations. The lens effect of both light rays and freely falling massive particles with a finite lifetime, (i.e. the multiplicity of such connecting curves) is characterized in terms of the focalization of the geodesics in the underlying Randers manifolds.