Singularities in the collisions of almost-plane gravitational waves

It is well known that when gravitational plane waves propagating on an otherwise flat background collide, they produce spacetime singularities. In this paper we consider the problem of whether (or under what conditions) singularities can be produced by the collision of gravitational waves with finite but very large transverse sizes. On the basis of (nonrigorous) order-of-magnitude considerations, we discuss the outcome of the collision in two fundamentally different regimes for the parameters of the colliding waves; these parameters are the transverse sizes (${\mathit{L}}_{\mathit{T}}$ ${)}_{\mathit{i}}$, typical amplitudes ${\mathit{h}}_{\mathit{i}}$, typical reduced wavelengths λ${\mathrm{/}}_{\mathit{i}}$≡${\lambda }_{\mathit{i}}$/2π, thickneses ${\mathit{a}}_{\mathit{i}}$, and focal lengths ${\mathit{f}}_{\mathit{i}}$∼λ${\mathrm{/}}_{\mathit{i}}^2$/${\mathit{a}}_{\mathit{i}}$ ${\mathit{h}}_{\mathit{i}}^2$ (i=1,2) of the waves 1 and 2.