Structure of the singularities produced by colliding plane waves

Abstract

When gravitational plane waves propagating and colliding in an otherwise flat background interact, they produce spacetime singularities. If the colliding waves have parallel (linear) polarizations, the mathematical analysis of the field equations in the interaction region is especially simple. Using the formulation of these field equations previously given by Szekeres, we analyze the asymptotic structure of a general colliding parallel-polarized plane-wave solution near the singularity. We show that the metric is asymptotic to an inhomogeneous Kasner solution as the singularity is approached. We give explicit expressions which relate the asymptotic Kasner exponents along the singularity to the initial data posed along the wave fronts of the incoming, colliding plane waves. It becomes clear from these expressions that for specific choices of initial data the curvature singularity created by the colliding waves degenerates to a coordinate singularity, and that a nonsingular Killing-Cauchy horizon is thereby obtained. Our equations prove that these horizons are unstable in the full nonlinear theory against small but generic perturbations of the initial data, and that in a very precise sense, "generic" initial data always produce all-embracing, spacelike curvature singularities without Killing-Cauchy horizons. We give several examples of exact solutions which illustrate some of the asymptotic singularity structures that are discussed in the paper. In particular, we construct a new family of exact colliding parallel-polarized plane-wave solutions, which create Killing-Cauchy horizons instead of a spacelike curvature singularity. The maximal analytic extension of one of these solutions across its Killing-Cauchy horizon results in a colliding plane-wave spacetime, in which a Schwarzschild black hole is created out of the collision between two planesymmetric sandwich waves propagating in a cylindrical universe.

Spin-2 particles in gravitational fields

“…The metric (V.1) has a singularity at x 2 = x 3 = 0. More complete treatments of this problem show that this is a curvature singularity [17,18,19,20].…”

Type-IIB colliding plane waves

“…Finally, a case of particular interest is when the two colliding waves have identical impulsive profile: this is the higher dimensional analog of the Khan-Penrose solution. The solution may be obtained to order 4 by setting 24) in the formulae in appendix A.…”

“…A solution could be obtained using the Green function technique developed in [18,24], however for n > 1 the resulting integrals appear to be too difficult.…”

“…It is often useful then to change coordinates from (u, v) to (f, g). Solving for the other functions V, W, H i is in general a complicated non-linear problem, except for purely gravitational collinear waves (W = H i = 0), where V satisfies a linear Euler-Darboux equation, and can be determined by Green's function techniques [18,24]. When either W or H i are non-zero, the problem is more difficult, although integrability provides solution-generating techniques which in some cases allow to obtain exact solutions.…”

The characteristic initial value problem for colliding plane waves: the linear case

“…It is known [2]- [3] that, in the vacuum (linear) case in which the approaching waves have constant and aligned polarization, the characteristic initial value problem may be solved theoretically using Riemann's method. Unfortunately, this method involves integrals which cannot be evaluated explicitly for appropriate initial data.…”