The character of the focusing surface $s=0$ in colliding plane gravitational waves coupled to a supercritical stiff ($\epsilon=p$) fluid

When two plane gravitational waves collide in general relativity, the interaction region terminates at a focusing surface $s=1-u^2-v^2=0$, where $u$ and $v$ are null coordinates. For most known solutions the focus is a spacelike curvature singularity; exceptional solutions instead develop a Killing-Cauchy horizon there. Chandrasekhar and Xanthopoulos (1987) studied how matter sources modify this outcome and found, for a stiff perfect fluid (energy density $\epsilon$ equal to pressure $p$, equivalent to a massless scalar field), that the fate depends on the fluid amplitude $C$: below a critical amplitude $C_0$ the curvature scalars diverge on the whole of $s=0$; between $C_0$ and a second critical amplitude $C_1$ they diverge on a subarc; and for $C>C_1$ all curvature scalars vanish and the spacetime tends to flatness as $s\to0$, while contours of constant energy density become asymptotically null. The authors nevertheless conjectured that $s=0$ may still be a 'singularity' not unlike the one occurring along the null boundaries of the interaction region, and explicitly left the matter open. The problem is to determine the true geometric and causal character of $s=0$ in this supercritical regime - a regular boundary admitting an extension, a null or quasiregular singularity, or a surface carrying distributional curvature - and to establish the criteria that distinguish these alternatives. The same paper's companion conjecture, the 'transmutation' of null dust into stiff fluid, has since been refuted in the literature.

Background

Colliding plane waves (CPW) are exact solutions of Einstein's equations describing the head-on collision of two plane-fronted gravitational waves. The spacetime is divided by the null hypersurfaces $u=0$ and $v=0$, with $u$ and $v$ retarded and advanced null coordinates, into four regions: a flat region I before the waves meet, single-wave regions II and III, and an interaction region IV in which the waves gravitationally focus each other. In the Szekeres class of solutions (Szekeres, 1972) - the general framework containing the first exact colliding-wave solution of Khan and Penrose (1971) for impulsive waves (idealized wave fronts with a delta-function profile) - the interaction region is covered by coordinates in which the metric is regular only for $s=1-u^2-v^2>0$; the surface $s=0$ is the focusing surface (or focus) at which the two wave fronts are mutually focused, and it is a boundary of the coordinate patch. Generically, curvature scalars diverge at $s=0$ and the spacetime is inextendible there: the focus is a spacelike curvature singularity.

Exceptional solutions exist in which $s=0$ is instead a Killing-Cauchy horizon - a null boundary at which one of the spacetime's Killing vectors (here generating a translational plane symmetry) becomes null and across which the metric can be analytically extended, in close analogy with the inner horizon of a rotating or charged black hole. Chandrasekhar and Xanthopoulos (1986) found such a vacuum solution, locally isometric to the interior of the Kerr black hole (the rotating black hole exact solution), in which the horizon is followed, after extension, by timelike curvature singularities along hyperbolic arcs; the authors called this "a new type of singularity". Their 1987 paper on the effect of sources then asked how robust this structure is against physically motivated matter sources: it first constructs a one-parameter Einstein-Maxwell family from any vacuum solution $E$ of the Ernst equation - the standard complex-potential reformulation of the vacuum Einstein equations for spacetimes with two commuting Killing vectors, on which the generation of colliding-wave solutions is based - starting from the simple seed $E=p\eta-\mathrm{i}q\mu$ with $p^2+q^2=1$ and $\eta$ and $\mu$ hyperbolic coordinates of the interaction region: the resulting spacetime is locally isometric to the interior of the Kerr-Newman black hole (the charged, rotating black-hole solution) and, for non-parallel wave polarizations $q\neq0$, again develops a horizon on $s=0$ followed by timelike singularities, and then couples the underlying vacuum colliding-wave metric to a stiff perfect fluid and to null dust (pressureless matter moving at the speed of light, such as a beam of massless particles).

A stiff perfect fluid is one whose energy density equals its pressure, $\epsilon=p$; by a classical equivalence it can be replaced in the field equations by a free massless scalar field, which is how most later work interprets it. Chandrasekhar and Xanthopoulos had introduced such fluids into colliding waves in 1985. Coupling the fluid multiplies the interaction-region metric by a conformal factor $\mathrm{e}^{f(u,v)}$ determined by a potential $\phi$ satisfying

$$\phi_{,uv}+\frac{v\,\phi_{,u}+u\,\phi_{,v}}{1-u^{2}-v^{2}}=0 ,$$

together with the first-order equations $f_{,u}=2(\phi_{,u})^{2}/[u(1-u^{2}-v^{2})]$ and $f_{,v}=2(\phi_{,v})^{2}/[v(1-u^{2}-v^{2})]$. For the vacuum seed that develops a horizon, the 1987 paper analyzes the separable solution $\phi=C\alpha sK_{1}(\alpha s)\sinh(\alpha r)$, where $r=u^{2}-v^{2}$, $K_{1}$ is a modified Bessel function of order 1, $C>0$ is the fluid amplitude, and $\alpha>0$ is a separation constant. As $s\to0$ one has $\mathrm{e}^{-f}\sim A(r)\,s^{k(r)}$ with $k(r)=4C^{2}\alpha^{2}\cosh^{2}(\alpha r)$, so that the energy density and the Weyl and Ricci curvature scalars behave as $s^{k-2}$ for non-parallel wave polarizations ($q\neq0$, where $q$ is the seed polarization parameter) and as $s^{k-6}$ for parallel polarizations ($q=0$). This yields two critical amplitudes,

$$C_{0}=\frac{\sqrt{k_{\mathrm{c}}}}{2\alpha\cosh\alpha},\qquad C_{1}=\frac{\sqrt{k_{\mathrm{c}}}}{2\alpha},$$

with $k_{\mathrm{c}}=2$ for $q\neq0$ and $k_{\mathrm{c}}=6$ for $q=0$: for $C<C_{0}$ the scalars diverge on the whole of $s=0$ (the horizon is destroyed); for $C_{0}<C<C_{1}$ they diverge only on a subarc; and for $C>C_{1}$ all curvature scalars vanish and the spacetime tends to flatness as $s\to0$, while contours of constant energy density approach null character. By contrast, any nonzero null-dust flux destroys the horizon in all cases and forces a spacelike singularity.

The supercritical regime $C>C_{1}$ is the focus of the present problem, because the vanishing of curvature scalars does not by itself settle the character of $s=0$. Classifying a surface on which all scalar curvature invariants stay finite requires finer notions: a quasiregular singularity is a boundary where curvature invariants remain bounded yet geodesics terminate or the metric admits no regular limit (so the spacetime is still incomplete); a null singularity is a singular boundary of null tangent structure on which only some tidal components diverge; and distributional curvature refers to curvature concentrated on a hypersurface as a distribution (a generalized function), as happens for impulsive gravitational waves. The null boundaries $u=0$ and $v=0$ of the interaction region themselves carry the classic fold singularities of impulsive-wave spacetimes - quasiregular, non-curvature singularities where the spacetime is nonextendible because of the fold structure of the wave fronts - and it is to these that the authors' conjecture refers. They conjectured (Chandrasekhar and Xanthopoulos, 1987) that "even though the space-time becomes flat in the limit, the surface $s=0$ may, in fact, be a 'singularity' not unlike the one that occurs along the null boundaries", adding that "the matter clearly requires further investigation". The broader classification question - under what conditions a colliding-wave spacetime develops a curvature singularity rather than a horizon - was analyzed in this context by Yurtsever (1989).

Problem Statement

Determine the geometric and causal character of the focusing surface $s=0$ in the colliding-plane-wave spacetimes of Chandrasekhar and Xanthopoulos (1987) coupled to a stiff ($\epsilon=p$) perfect fluid in the supercritical regime $C>C_{1}$, in which all curvature scalars vanish as $s\to0$, the spacetime tends to flatness, and contours of constant energy density approach null character. Concretely: (1) Is $s=0$ in this regime a regular boundary across which the spacetime admits an extension of stated differentiability class (as the vacuum solution does at its Killing-Cauchy horizon), or does it retain a nontrivial singular character - for example a null-type or quasiregular boundary singularity like the fold singularities along the null boundaries, or a concentration of curvature in the distributional sense - notwithstanding the vanishing of the scalar invariants? (2) Which precise geometric or causal criterion - the decisive one among geodesic completeness, extendibility of the metric in a stated regularity class, distributional curvature, and the limiting behavior of the energy density and tidal components - decides among these alternatives, and what does it decide for the explicit supercritical solutions built on the potential $\phi=C\alpha sK_{1}(\alpha s)\sinh(\alpha r)$ with $r=u^{2}-v^{2}$? The outcome bears on the authors' conjecture that $s=0$ may be a 'singularity' not unlike the one occurring along the null boundaries even though the spacetime becomes flat in the limit.

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