When two exact plane gravitational waves collide in general relativity, their mutual focusing generically terminates the interaction region in an all-embracing space-like curvature singularity. Chandrasekhar and Xanthopoulos (1987) constructed, in the Einstein-Maxwell theory (gravity coupled to an electromagnetic field), a new class of colliding-wave solutions in which the collision instead produces a regular Killing-Cauchy horizon — a null surface generated by a Killing vector flow, beyond which the spacetime extends non-uniquely — followed by time-like singularities, leaving open what property of the initial data selects this alternative fate. The parallel-polarized vacuum case was largely settled by Yurtsever (1988): the asymptotic Kasner exponents at the terminal surface are explicit functionals of the initial data, degenerate exponent values characterize horizons, generic data produce space-like singularities, and the horizons are unstable against plane-symmetric perturbations. For the electrovac setting of the source paper — and a fortiori for non-parallel polarizations, for which no general solution representation exists — no classification criterion is known, and no rigorous instability result exists beyond the two-Killing, plane-symmetric reduction. The surviving open core is: (i) a necessary-and-sufficient criterion, in terms of the characteristic initial data (wave profiles, relative polarization, matter content), deciding between an all-embracing space-like singularity and a Killing-Cauchy horizon with unrestricted post-horizon content; and (ii) the stability of the horizon branch under generic perturbations.

Background

An exact plane gravitational wave in general relativity is a gravitational field with planar wavefronts propagating at the speed of light along a null direction. When two such waves traveling in opposite directions collide, each focuses the other (it contracts the transverse directions of the other wave), and the interaction region that develops between the two null wavefronts admits two commuting spacelike Killing vectors — generator fields of translational symmetries along the wavefront planes — so the local problem reduces to functions of two variables. Mathematically this is a characteristic or Goursat initial-value problem: an initial-value problem for hyperbolic partial differential equations with data prescribed on two intersecting null (characteristic) surfaces, namely the two wavefronts.

The prototypical result is due to Khan and Penrose (1971), who solved the collision of two impulsive gravitational waves (waves with a delta-function profile on a null hypersurface) in vacuum: the interaction region terminates at a scalar-polynomial curvature singularity — a surface on which curvature invariants such as the Kretschmann scalar RabcdRabcdR_{abcd}R^{abcd} diverge — of space-like character, i.e., a terminal caustic lying to the future of every observer in the region. Szekeres (1972) embedded this in a wider family, and Bell and Szekeres (1974) solved the corresponding Einstein-Maxwell problem (gravity coupled to an electromagnetic field) for colliding electromagnetic shock waves, finding an interaction region that ends not in a curvature singularity but in a horizon.

These two possibilities define the dichotomy at the center of the problem. A space-like singularity is unavoidable and terminal. The alternative is a Killing-Cauchy horizon: a null surface, swept out by the orbits of a Killing vector, across which the metric is regular but the future evolution from the initial data is not unique; it is the colliding-wave analogue of the inner (Cauchy) horizon of a charged or rotating black hole. Beyond such a horizon the extended spacetime may contain time-like singularities — singular boundaries reachable along non-spacelike curves and hence visible from a distance — and the extension itself is generally non-unique.

Chandrasekhar and Xanthopoulos discovered that the second fate occurs in explicit families. Their vacuum solution (1986), whose interaction region is locally isometric to part of the interior of the Kerr black hole (as reviewed by Bicák 2000), and its Einstein-Maxwell generalization (1987), the source paper, produce a regular Killing-Cauchy horizon followed by two- and three-dimensional time-like singularities. Technically, the 1987 paper works in the Z=1Z=1 sector of the Einstein-Maxwell equations with two commuting spacelike Killing vectors, a hypersurface-orthogonal (diagonal-metric) reduction in which each Killing vector is orthogonal to a family of two-surfaces; the field equations are treated through the Ernst equation, a standard complex-potential first-order reformulation of the two-Killing Einstein and Einstein-Maxwell equations, and an Ehlers transformation — a one-parameter transformation mapping any solution of the Ernst equation to a one-parameter family of new solutions — applied to the Bell-Szekeres solution as seed. The resulting colliding-wave metrics are of Petrov type D (the algebraically special class in which the Weyl curvature tensor admits two double principal null directions, as do the Kerr and Schwarzschild fields). Generically the interaction region approaches, at its terminal surface, an inhomogeneous Kasner form — a vacuum cosmological metric whose expansion exponents vary with position — and Yurtsever (1988) showed that in the parallel-polarized vacuum case the values of these exponents encode which fate occurs.

The broader context sharpens the stakes. By the local-isometry results of Ferrari and Ibáñez (1987) and Yurtsever (1988), colliding-wave interaction regions are locally isometric to interiors of static or stationary axisymmetric black-hole solutions, so the fate of the Killing-Cauchy horizons is the plane-wave-collision counterpart of the celebrated instability problem of black-hole inner horizons. Quantum-field-theoretic work (Yurtsever 1989; Dorca and Verdaguer 1993, 1994, 1997; Feinstein and Pérez Sebastián 1995) established that these horizons radiate thermally, with temperature inversely proportional to the focusing time, and that the renormalized stress-energy tensor diverges on them, suggesting quantum effects destroy the horizon. What is still missing is the classical classification criterion for the source paper's own electrovac setting and a rigorous statement of the horizons' stability.

Problem Statement

The problem raised by the horizon-forming solutions of Chandrasekhar and Xanthopoulos (1987) — under what conditions the collision of waves produces a space-time curvature singularity or a Killing-Cauchy horizon, a question explicitly described as still open in the later literature (Gurtug and Halilsoy 2009) — is to classify the possible fates of the colliding-plane-wave interaction region and to decide, from the initial data, which fate occurs.

The inputs are of two kinds: the characteristic (Goursat) data on the two intersecting null wavefronts — the profiles of the two waves (impulsive or shock, purely gravitational or coupled to an electromagnetic field) and their relative polarization — and the matter model of the interaction region (vacuum; electrovac, i.e. the Einstein-Maxwell field; stiff fluid, with pressure equal to energy density; or null dust, i.e. pressureless incoherent radiation). The two possible fates are: (a) an all-embracing space-like curvature singularity, across which no extension of the spacetime is possible; or (b) a regular Killing-Cauchy horizon through which the spacetime extends — non-uniquely — with unrestricted post-horizon content.

The core objective has two linked parts.

(i) Criterion. State and prove a necessary-and-sufficient condition on the initial data that decides between outcomes (a) and (b), for the Einstein-Maxwell colliding-wave problem as posed in the source paper (collinearly polarized waves in its hypersurface-orthogonal diagonal-metric reduction with two commuting spacelike Killing vectors, the so-called Z=1Z=1 sector), and for non-parallel polarizations and other matter models. The criterion must be consistent with all known examples: Khan-Penrose, the Szekeres family, Bell-Szekeres, the source paper's family, and the outcome branches of the Papacostas-Xanthopoulos family, in which the extension beyond the Cauchy horizon is singularity-free, or contains a two-dimensional space-like, or a two-dimensional time-like, curvature singularity.

(ii) Stability. Determine whether the horizon-forming branch identified in (i) is stable: does the Killing-Cauchy horizon persist under small generic perturbations of the initial data — including perturbations that break plane symmetry or collinear polarization — or is it nonlinearly unstable, so that outcome (b) is non-generic in the full (not two-Killing-reduced) Einstein-Maxwell theory?

The auxiliary questions raised in the same source (the explicit integration of its Section-5 family, quantum radiation from and backreaction on these horizons, and the stationary-axisymmetric analogue) lie outside the scope of this problem.

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