Caustics and Breakdown of Null Coordinates

Null coordinates are built by labeling generators and following them. They fail when those labels cease to define a one-to-one map, even if the metric and curvature remain completely regular. Focal points make the local Jacobian singular; crossovers make the generator map globally noninjective. Both are geometric limits of a characteristic chart rather than automatic spacetime singularities.

Required background. Timelike Congruences and Spatial Projection supplies connecting fields and the distinction between congruence breakdown and continuation of individual curves.

Helpful background. We use The Einstein Equation, Hypersurface Normals and Tangential Projectors, Formulations of General Relativity, Causality and Global Structure, and PDEs and Green Functions for Gravitation.

Generator map and coordinate Jacobian

Let $X^\mu(\lambda,y^A)$ follow a future-directed null generator $\ell=\partial_\lambda$ from an initial screen. The connecting vectors

$$E_A^\mu=\frac{\partial X^\mu}{\partial y^A}$$

span the screen while the labels are regular. Their Gram matrix is $q_{AB}=g(E_A,E_B)$. Relative to the initial screen, the transverse area Jacobian is

$$J(\lambda,y) =\frac{\sqrt{\det q_{AB}(\lambda,y)}} {\sqrt{\det q_{AB}(0,y)}}.$$

It obeys

$$\frac{d}{d\lambda}\ln J=\theta.$$

A focal point occurs when the connecting vectors lose rank after projection to the screen, equivalently modulo the generator direction, and $J\to0$. A crossover can occur when generators with distinct labels reach the same event even though the local Jacobian on each branch has not yet vanished. In either case $(\lambda,y^A)\mapsto X^\mu$ is not a valid single chart beyond that event.

The successive screen sections below isolate the local focal mechanism. Watch one singular value of the Jacobi map shrink to zero: the connecting vectors lose rank before any conclusion about curvature can be drawn.

The Jacobi map carries initial transverse labels to screen separations. Before the caustic its singular values are the beam ellipse semiaxes and $B=\dot D D^{-1}$; when one stretch vanishes, $D$ loses rank and the inverse-based optical tensor fails although the Jacobi equation may remain regular. Schematic and not to scale.

The worldtube–nullcone problem therefore requires a caustic-free exterior domain, not merely smooth initial data. This limitation and the contrast with spacelike evolution are emphasized in Winicour 2012, § 2.1.

Focusing from the optical equation

For hypersurface-orthogonal generators, twist vanishes. In an affine parameter, the four-dimensional Raychaudhuri equation is

$$\frac{d\theta}{d\lambda} =-\frac12\theta^2 -\sigma_{AB}\sigma^{AB} -8\pi G\,T_{\ell\ell}.$$

If the null energy condition holds, $T_{\ell\ell}\ge0$, then

$$\frac{d\theta}{d\lambda}\le-\frac12\theta^2.$$

For initial $\theta_0\lt 0$, divide by $\theta^2$ while the expansion remains finite:

$$\frac{d}{d\lambda}\left(\frac1\theta\right)\ge\frac12.$$

As long as the congruence is regular, $\theta$ stays negative and its reciprocal cannot remain negative beyond $\lambda=2/|\theta_0|$. If the geodesic and its Jacobi fields extend smoothly to that affine distance, a focal point occurs by then. If the spacetime or geodesic ends earlier, the inequality alone does not supply a focal point beyond the endpoint. Shear and positive $T_{\ell\ell}$ can only shorten the bound. This existence qualification is essential when focusing is used in singularity theorems.

For a nonaffine generator, a term $\kappa\theta$ appears. This does not move the geometric caustic. Under a constant rescaling $\ell'=a\ell$ and $\lambda'=\lambda/a$, one has $\theta'=a\theta$ and $\theta'\,d\lambda'=\theta\,d\lambda$, so $J$ and its zeros are unchanged.

Smooth spacetime, broken formulation

Flat spacetime already supplies a caustic. Future ingoing radial null rays from a sphere meet at its center. Curvature vanishes everywhere, but the angular labels all identify the same event, the screen area goes to zero, and an areal characteristic coordinate cannot continue through the center as one chart.

This distinction guides recovery strategies. One may stop the null domain before the first caustic and attach a Cauchy region across a timelike worldtube, cover the spacetime with several null patches, relaunch cones from a different worldline, or change to a formulation whose coordinates are not tied to that congruence. Continuing the same multivalued generator labels is not a valid remedy.

The converse caution also matters: finite curvature invariants do not guarantee a regular null chart, while a diverging optical scalar in a chosen congruence does not alone prove curvature blow-up. Diagnose the rank and injectivity of the coordinate map separately from curvature.

Exercises

In Minkowski spacetime, take future ingoing null generators with affine parameter $\lambda$ and radius $r(\lambda)=R-\lambda$, starting from a round sphere of radius $R$. Compute $J$, $\theta$, and the focusing time, and verify Raychaudhuri.

Solution

The screen metric is $q_{AB}=r^2\Omega_{AB}$, so

$$J=\frac{r^2}{R^2} =\left(1-\frac{\lambda}{R}\right)^2.$$

Therefore

$$\theta=\frac{d}{d\lambda}\ln J =-\frac{2}{R-\lambda} =-\frac2r.$$

It diverges and $J$ vanishes at $\lambda=R$. Since the shear, matter term, and curvature all vanish,

$$\frac{d\theta}{d\lambda} =-\frac{2}{r^2} =-\frac12\theta^2.$$

The general bound gives a focal point by $2/|\theta_0|=R$, and here it is saturated. The event $r=0$ is regular Minkowski space; only the spherical null labels collapse there.

References

• Winicour, Jeffrey. “Characteristic Evolution and Matching.” Living Reviews in Relativity 15 (2012): article 2. arXiv:0810.1903. doi:10.12942/lrr-2012-2.