Caustics and wave propagation in curved spacetimes

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 $\sigma (p,p^{\prime })$ 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 $\pm 1/\pi \sigma$ (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. 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.

Figure 1

Fixing any $u^{\prime }\in \mathbb{R}$, a plane wave spacetime naturally divides into a set of 3-surfaces $S_{{\tau }_n(u^{\prime })}$ and open 4-volumes ${\mathcal{N}}_n(u^{\prime })$ in between them. Every point in $S_{u^{\prime }}$ is connected to every point in ${\mathcal{N}}_n(u^{\prime })$ by exactly one geodesic. Such points are never conjugate. Points in $S_{u^{\prime }}$ can be connected to points in $S_{{\tau }_n(u^{\prime })}$ by either an infinite number of geodesics or by none. In the former case, both points are conjugate along every connecting geodesic. This justifies the description of $(S_{u^{\prime }},S_{{\tau }_n(u^{\prime })})$ as a pair of conjugate hyperplanes.

Figure 2

Schematic illustration of geodesic focusing in a plane wave spacetime. Three $u$ coordinates conjugate to $u^{\prime }$ are indicated together with the projection of three geodesics onto the $x^1\mathrm{\text{−}}u$ coordinate plane. The focusing here is assumed to act (at least) in the $x^1$ direction. The regions ${\mathcal{N}}_0(u^{\prime })$ and ${\mathcal{N}}_1(u^{\prime })$ are also illustrated. See Figs. 3, 4 for a different projection.

Figure 3

A collection of null geodesics emanating from a point on $S_{u^{\prime }}$ (the lower plane) and focusing back to a single point on a conjugate hyperplane $S_{{\tau }_n(u^{\prime })}$ with multiplicity 2 (the upper plane). One spatial coordinate has been suppressed.

Figure 4

A collection of null geodesics emanating from a point on $S_{u^{\prime }}$ (the lower plane) and focusing to a parabola on a conjugate hyperplane $S_{{\tau }_n(u^{\prime })}$ with multiplicity 1 (the upper plane). One spatial coordinate aligned with ${\widehat{\mathbf{p}}}_n(u^{\prime })$ is displayed. The other [aligned with ${\widehat{\mathbf{q}}}_n(u^{\prime })$] is suppressed.

Figure 5

$\det \mathbf{B}(u,0)$ for a plane wave spacetime with $\mathbf{H}(u)$ given by (3.95) and $h=2/3$. The zeros correspond to locations of conjugate hyperplanes. They occur in closely spaced pairs separated by approximately $3.8\pi$. Note that $\mathbf{H}(u)$ has the shorter period $2\pi$.

Figure 6

Schematic illustrating that Penrose limits map conjugate points to conjugate points. The inset shows the reference null geodesic in the spacetime $(\check{M},{\check{g}}_{ab})$ along with a number of nearby geodesics. The points $\check{z}(u^{\prime })$ and $\check{z}(u^{\prime \prime })$ are conjugate on the reference geodesic. They are mapped to the conjugate hyperplanes $S_{u^{\prime }}$ and $S_{u^{\prime \prime }}$ in the associated plane wave spacetime.

Figure 7

The effect of a Penrose limit on a null curve which intersects the reference geodesic at the points $\check{z}(u^{\prime })$ and $\check{z}(u^{\prime \prime })$. Such a curve is mapped to null geodesics on the hyperplanes $S_{u^{\prime }}$ and $S_{u^{\prime \prime }}$. Note that one continuous curve in the original spacetime $(\check{M},{\check{g}}_{ab})$ is mapped into two disconnected curves in the associated plane wave spacetime $(M,g_{ab})$.