Singularities and horizons in the collisions of gravitational waves

It is well known that when gravitational plane waves propagating and colliding in an otherwise flat background interact they produce singularities. In this paper we explore the structure of the singularities produced in the collisions of arbitrarily polarized gravitational plane waves and we consider the problem of whether (or under what conditions) singularities can be produced in the collisions of almost-plane gravitational waves with finite but very large transverse sizes. First we analyze the asymptotic structure of a general arbitrarily polarized colliding plane-wave spacetime near its singularity. We show that the metric is asymptotic to a generalized inhomogeneous-Kasner solution as the singularity is approached. In general, the asymptotic Kasner axes as well as the asymptotic Kasner exponents along the singularity are functions of the spatial coordinate that runs tangentially to the singularity in the non-plane-symmetric direction.

It becomes clear that for specific values of these asymptotic Kasner exponents and axes the curvature singularity created by the colliding waves degenerates to a coordinate singularity, and that a nonsingular Killing-Cauchy horizon is thereby obtained. Our analysis proves that these horizons are unstable in the full nonlinear theory against small but generic plane-symmetric perturbations of the initial data, and that in a very precise and rigorous sense, ‘‘generic’’ initial data for colliding arbitrarily polarized plane waves always produce all-embracing, spacelike curvature singularities without Killing-Cauchy horizons. Next we turn to the problem of colliding almost-plane gravitational waves, and by combining the results that we obtain in this paper and in other previous papers with the Hawking-Penrose singularity theorem and the Cauchy stability theorem, we prove that if the initial data for two colliding almost-plane waves are sufficiently close to being exactly plane symmetric across a sufficiently large but bounded region of the initial surface, then their collision must produce spacetime singularities.

Although our analysis proves the existence of these singularities rigorously, it does not give any information about either their global structure (e.g., whether they are hidden behind an event horizon) or their local asymptotic behavior (e.g., whether they are of Belinsky-Khalatnikov-Lifshitz generic-mixmaster type).