Caustics and wave propagation in curved spacetimes

TL;DR

This paper studies how light cone caustics affect scalar wave propagation in curved spacetimes, revealing how the singular structure of Green functions changes at conjugate points, with implications for understanding wave behavior near caustics.

Contribution

It introduces a method to analyze Green function singularities near caustics using Penrose limits and conjugate point multiplicities, providing a simple rule for their transformation.

Findings

Green functions are globally singular along null geodesics.

Singularity types change at conjugate points depending on their multiplicity.

A rule relates conjugate point multiplicity to singularity transformation.

Abstract

We investigate the effects of light cone caustics on the propagation of linear scalar fields in generic four-dimensional spacetimes. In particular, we analyze the singular structure of relevant Green functions. As expected from general theorems, Green functions associated with wave equations are globally singular along a large class of null geodesics. Despite this, the "nature" of the singularity on a given geodesic does not necessarily remain fixed. It can change character on encountering caustics of the light cone. These changes are studied by first deriving global Green functions for scalar fields propagating on smooth plane wave spacetimes. We then use Penrose limits to argue that there is a sense in which the "leading order singular behavior" of a (typically unknown) Green function associated with a generic spacetime can always be understood using a (known) Green function associated with an appropriate plane wave spacetime. This correspondence is used to derive a simple rule describing how Green functions change their singular structure near some reference null geodesic. Such changes depend only on the multiplicities of the conjugate points encountered along the reference geodesic. Using sigma(p,p') to denote a suitable generalization of Synge's world function, conjugate points with multiplicity 1 convert Green function singularities involving delta(sigma) into singularities involving 1/pi sigma (and vice-versa). Conjugate points with multiplicity 2 may be viewed as having the effect of two successive passes through conjugate points with multiplicity 1.