Geometrical optics for scalar, electromagnetic and gravitational waves in curved spacetime

TL;DR

This paper develops a unified geometrical optics framework for scalar, electromagnetic, and gravitational waves in curved spacetime, revealing common propagation properties and providing new methods to analyze wave behavior beyond conjugate points.

Contribution

It introduces a second-order transport equation approach to determine optical and Newman-Penrose scalars, improving analysis of wave propagation in curved spacetime.

Findings

All wave types propagate at light speed along null rays.

Wave amplitude varies inversely with beam cross section.

The polarization is parallel transported along rays.

Abstract

The geometrical-optics expansion reduces the problem of solving wave equations to one of solving transport equations along rays. Here we consider scalar, electromagnetic and gravitational waves propagating on a curved spacetime in general relativity. We show that each is governed by a wave equation with the same principal part. It follows that: each wave propagates at the speed of light along rays (null generators of hypersurfaces of constant phase); the square of the wave amplitude varies in inverse proportion to the cross section of the beam; and the polarization is parallel-propagated along the ray (the Skrotskii/Rytov effect). We show that the optical scalars for a beam, and various Newman-Penrose scalars describing a parallel-propagated null tetrad, can be found by solving transport equations in a second-order formulation. Unlike the Sachs equations, this formulation makes it straightforward to find such scalars beyond the first conjugate point of a congruence, where neighbouring rays cross, and the scalars diverge. We discuss differential precession across the beam which leads to a modified phase in the geometrical-optics expansion.