The generic causal character of the singularity produced by colliding plane gravitational waves (Chandrasekhar–Xanthopoulos, 1986)

When two plane gravitational waves collide in general relativity, they focus each other so strongly that a spacetime singularity forms a finite 'time' after the collision. In every solution known before 1986 this singularity is spacelike and terminates the spacetime. Chandrasekhar and Xanthopoulos (1986) then found the first vacuum exception: colliding plane impulsive waves accompanied by shock waves whose focus is a regular null surface — an event (Killing-Cauchy) horizon — beyond which the analytic extension reveals timelike curvature singularities along two hyperbolic arcs, in an interaction region locally isometric to the Kerr interior. They asked which structure is generic: a spacelike singularity, or a horizon followed by timelike singularities. Later work settled the constant-aligned-polarization, plane-symmetric case — generic data produce all-embracing spacelike singularities and the horizons are unstable (Yurtsever 1987–1988; Griffiths 2005) — but the question remains open without those restrictions. For non-collinear polarizations and for non-plane-symmetric perturbations it is not known whether the generic outcome is a spacelike singularity, a weak null singularity, or a horizon of the Chandrasekhar–Xanthopoulos type, and the vacuum solutions admitting a regular null focusing surface are not classified.

Background

A plane gravitational wave is an exact solution of the vacuum Einstein equations whose wavefronts are flat planes, so that the spacetime admits two commuting spacelike Killing vectors (one for each translation along the wavefront). In 1970–1971 Szekeres, and Khan and Penrose, solved the idealized problem in which two such waves propagating in opposite directions collide: writing null coordinates $u$ and $v$ that vanish on the two wavefronts and meet at the collision event, the mutual focusing of the waves drives the metric to a 'focus' on the surface $u^2+v^2=1$. In all colliding-plane-wave solutions known before 1986 — beginning with the Khan–Penrose solution for two impulsive waves (an impulsive wave carries its curvature as a Dirac delta distribution on a single null hypersurface, while a shock wave carries a finite step discontinuity) and the Szekeres solutions for waves of finite duration — this surface is a spacelike curvature singularity that seals off the spacetime, in the way the big-crunch singularity does inside a Schwarzschild black hole. A singularity is called spacelike or timelike according to the causal character of the endpoints it imposes on incomplete geodesics: a spacelike singularity, like the Schwarzschild one, is reached by essentially all observers, whereas a timelike one, like the Kerr ring singularity, can be avoided and sits at a definite place in the extended spacetime.

The standard machinery for these spacetimes reduces the vacuum Einstein equations, for metrics with two commuting Killing vectors, to the Ernst equation — a nonlinear equation for a complex potential encoding the metric functions (the same reduction used for stationary axisymmetric black holes). Chandrasekhar and Xanthopoulos (1986) took a linear Ernst potential, $E=p\eta+iq\mu$ with $p^2+q^2=1$, where $\eta$ measures time from the collision and $\mu$ the distance across the wavefronts, and found that for $q\neq 0$ the Weyl curvature scalars (the complex components into which the Weyl tensor — the free, tidal part of the gravitational field not locally determined by matter — decomposes) remain finite on $u^2+v^2=1$. The focus is then not a curvature singularity but a regular null surface acting as an event horizon — more precisely a Killing-Cauchy horizon, a null surface across which the spacetime is analytically extendible but beyond which the extension is no longer determined by the data on the wavefronts. In coordinates suggested by Penrose, the extension reveals timelike curvature singularities along two hyperbolic arcs $\xi\zeta=-q^2/p^2$, reminiscent of the Kerr ring singularity; the interaction region is of Petrov type D (the algebraic class of the Weyl tensor shared by Schwarzschild and Kerr, in which only the 'Coulomb' component $\Psi_2$ survives in a suitable frame) and is locally isometric to the Kerr spacetime interior to the ergosphere. The extension across the wavefronts is only $C^0$, with Heaviside-step and delta-function discontinuities in the curvature corresponding to the impulsive and shock-wave content.

The paper thus confronted the field with a dichotomy: every previously known colliding-wave singularity was spacelike, yet here was a vacuum collision producing a horizon plus timelike singularities. A theorem of Tipler (1980) implies that singularities are unavoidable in generic colliding plane waves, but it does not determine the causal character of the singularity, and the question of which member of the dichotomy — spacelike singularity or horizon plus timelike singularity — is generic was posed explicitly by the source paper.

Problem Statement

Determine the generic structure that the collision of two plane gravitational waves produces at the focus, without restricting the initial waves to constant aligned polarizations or the allowed perturbations to plane symmetry — the question posed by Chandrasekhar and Xanthopoulos (1986) as the 'generic nature' of the singularity, 'whether it is space-like or time-like'. The single objective is to decide, in the full colliding-plane-wave problem rather than in a symmetric subclass, whether the horizon-plus-timelike-singularity structure of the Chandrasekhar–Xanthopoulos type is generic or non-generic, and to identify what the generic outcome of the focusing is: an all-embracing spacelike curvature singularity, a null weak singularity, or a Killing-Cauchy horizon followed by timelike singularities. The question concerns, in particular, (1) colliding plane waves with non-aligned (non-collinear) polarizations — the general case of the hyperbolic (wave-type) Ernst equation, posed as a Goursat problem, i.e., with initial data given on the two intersecting null wavefronts — for which it is open whether generic data develop an all-embracing spacelike curvature singularity (as proved only for parallel polarizations by Yurtsever), a Killing-Cauchy horizon followed by timelike singularities, or an intermediate null weak singularity; and (2) the stability of the Killing-Cauchy horizons of the known horizon-forming solutions (Chandrasekhar–Xanthopoulos; the Ferrari–Ibañez type-D family) against generic perturbations that are not plane-symmetric — every existing instability proof is restricted to plane-symmetric perturbations of constant-aligned-polarization waves — the stability of these horizons against general bounded perturbations being the means by which the genericity of the horizon-plus-timelike-singularity structure is decided. 'Generic' is used in the perturbative-stability sense of the settled aligned-polarization results (Yurtsever 1988; Griffiths 2005): a structure of the interaction region is generic if it is not destroyed by arbitrarily small bounded perturbations of the characteristic initial data on the two null wavefronts, applied independently on either characteristic, and a Killing-Cauchy horizon is non-generic (unstable) if arbitrarily small bounded perturbations of the incoming waves produce a curvature singularity in its place.

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