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 σ(p,p ′) to denote a suitable generalization of Synge's world function, conjugate points with multiplicity 1 convert Green function singularities involving δ(σ) into singularities involving ±1/πσ (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. Separately, we provide an extensive review of plane wave geometry that may be of independent interest. Explicit forms for bitensors such as Synge's function, the van Vleck determinant, and the parallel and Jacobi propagators are derived almost everywhere for all nonsingular four-dimensional plane waves. The asymptotic behaviors of various objects near caustics are also discussed.
|Original language||English (US)|
|Journal||Physical Review D - Particles, Fields, Gravitation and Cosmology|
|State||Published - Jun 19 2012|
All Science Journal Classification (ASJC) codes
- Nuclear and High Energy Physics
- Physics and Astronomy (miscellaneous)