Generalizing Optical Geometry

TL;DR

This paper extends the concept of optical geometry to a broader class of spacetimes using generalized measures of spatial curvature, enabling simplified descriptions of photon paths and inertial forces in dynamic settings.

Contribution

It introduces a new generalization of optical geometry applicable to more spacetimes, with simplified photon and gyroscope behavior, and develops an inertial force formalism for this framework.

Findings

Photon paths are spatial geodesics with unit speed in the generalized geometry.

Gyroscopes do not precess when moving along spatial geodesics.

The sideways force on particles is velocity-independent in the generalized framework.

Abstract

We show that by employing the standard projected curvature as a measure of spatial curvature, we can make a certain generalization of optical geometry (Abramowicz and Lasota 1997, Class. Quantum Grav. 14 (1997) A23). This generalization applies to any spacetime that admits a hypersurface orthogonal shearfree congruence of worldlines. This is a somewhat larger class of spacetimes than the conformally static spacetimes assumed in standard optical geometry. In the generalized optical geometry, which in the generic case is time dependent, photons move with unit speed along spatial geodesics and the sideways force experienced by a particle following a spatially straight line is independent of the velocity. Also gyroscopes moving along spatial geodesics do not precess (relative to the forward direction). Gyroscopes that follow a curved spatial trajectory precess according to a very simple law of three-rotation. We also present an inertial force formalism in coordinate representation for this generalization. Furthermore, we show that by employing a new sense of spatial curvature (Jonsson, Class. Quantum Grav. 23 (2006) 1) closely connected to Fermat's principle, we can make a more extensive generalization of optical geometry that applies to arbitrary spacetimes. In general this optical geometry will be time dependent, but still geodesic photons move with unit speed and follow lines that are spatially straight in the new sense. Also, the sideways experienced (comoving) force on a test particle following a line that is straight in the new sense will be independent of the velocity.