Optical Metrics and Projective Equivalence

TL;DR

This paper studies the uniqueness of optical metrics derived from static spacetimes, revealing conditions under which different observers or parameters lead to projectively equivalent or distinct light ray geometries.

Contribution

It classifies Lorentzian metrics with multiple static observers and analyzes projective equivalence of optical metrics, especially in the context of $SL(2, R)$ symmetry and cosmological constants.

Findings

Optical metrics are projectively equivalent if generated by $SL(2, R)$ symmetries.

Different observers may perceive different light ray geometries in static spacetimes.

Optical metrics for varying cosmological constants are projectively equivalent.

Abstract

Trajectories of light rays in a static spacetime are described by unparametrised geodesics of the Riemannian optical metric associated with the Lorentzian spacetime metric. We investigate the uniqueness of this structure and demonstrate that two different observers, moving relative to one another, who both see the universe as static may determine the geometry of the light rays differently. More specifically, we classify Lorentzian metrics admitting more than one hyper--surface orthogonal time--like Killing vector and analyze the projective equivalence of the resulting optical metrics. These metrics are shown to be projectively equivalent up to diffeomorphism if the static Killing vectors generate a group $SL(2, \R)$, but not projectively equivalent in general. We also consider the cosmological $C$--metrics in Einstein--Maxwell theory and demonstrate that optical metrics corresponding to different values of the cosmological constant are projectively equivalent.