This thesis presents a study of the dynamical, nonlinear interaction of colliding gravitationalwaves, as described by classical general relativity. It is focused mainly on two fundamentalquestions: First, what is the general structure of the singularities and Killing-Cauchyhorizons produced in the collisions of exactly plane-symmetric gravitational waves? Second,under what conditions will the collisions of almost-plane gravitational waves (waves withlarge but finite transverse sizes) produce singularities?
In the work on the collisions of exactly-plane waves, it is shown that Killing horizons inany plane-symmetric spacetime are unstable against small plane-symmetric perturbations. Itis thus concluded that the Killing-Cauchy horizons produced by the collisions of some exactlyplane gravitational waves are nongeneric, and that generic initial data for the colliding planewaves always produce "pure" spacetime singularities without such horizons. This conclusionis later proved rigorously (using the full nonlinear theory rather than perturbation theory), inconnection with an analysis of the asymptotic singularity structure of a general collidingplane-wave spacetime. This analysis also proves that asymptotically the singularities createdby colliding plane waves are of inhomogeneous-Kasner type; the asymptotic Kasner axes andexponents of these singularities in general depend on the spatial coordinate that runs tangentiallyto the singularity in the non-plane-symmetric direction.
In the work on collisions of almost-plane gravitational waves, first some general propertiesof single almost-plane gravitational-wave spacetimes are explored. It is shown that, bycontrast with an exact plane wave, an almost-plane gravitational wave cannot have a propagationdirection that is Killing; i.e., it must diffract and disperse as it propagates. It is alsoshown that an almost-plane wave cannot be precisely sandwiched between two null wavefronts;i.e., it must leave behind tails in the spacetime region through which it passes. Next,the occurrence of spacetime singularities in the collisions of almost-plane waves is investigated. It is proved that if two colliding, almost-plane gravitational waves are initiallyexactly plane-symmetric across a central region of sufficiently large but finite transversedimensions, then their collision produces a spacetime singularity with the same local structureas in the exact-plane-wave collision. Finally, it is shown that a singularity still forms when thecentral regions are only approximately plane-symmetric initially. Stated more precisely, it isproved that if the colliding almost-plane waves are initially sufficiently close to being exactlyplane-symmetric across a bounded central region of sufficiently large transverse dimensions,then their collision necessarily produces spacetime singularities. In this case, nothing is nowknown about the local and global structures of the singularities.
