Cyclic spacetimes through singularity scattering maps. Plane-symmetric gravitational collisions

We study the plane-symmetric collision of two gravitational waves and describe the global spacetime geometry generated by this collision. We formulate the characteristic initial value problem for the Einstein equations, when Goursat data describing the incoming waves are prescribed on two null hypersurfaces. We construct a global solution representing a cyclic spacetime based on junction conditions associated with a prescribed singularity scattering map, as we call it. This amounts to a detailed analysis of the Goursat and Fuchsian initial value problems associated with singular hyperbolic equations, when junction conditions at interfaces are prescribed. We introduce a partition into monotonicity diamonds (that is, spacetime domains) and we construct the solution by concatenating domains across interfaces of timelike, null, or spacelike type.