A perturbation analysis of the Bell-Szekeres space-time

Abstract

A perturbation analysis of the Bell-Szekeres space-time is provided both in the region where the scattering of the colliding plane-fronted waves occurs (Region I) and in the regions prior to the instant of collision (Regions II and III). A complete set of normal modes, expressed in terms of spin-weighted spherical harmonics and bounded in the entire Region I, is obtained. These modes exhibit a behaviour of ever-increasing frequency as the horizon (that develops as a result of the collision) is approached. Region II, bounded by the null lines, u = 0(0 ≼ v ≼ 1) and v = 1( u ≼ 0) (where u and v define the directions of propagation of the colliding waves) does not allow u -independent perturbations to all orders; while all the u -dependent perturbations exhibit strong diver­gences along the entire line, v = 1, u < 0. This same description applies to Region III with the roles of u and v interchanged. These behaviours of the perturbations, in the regions of the space-time prior to the instant of collision, raise some serious conceptual questions.