We describe a generating algorithm to construct exact solutions of the vacuum Einstein field equations which can be interpreted as the interaction regions produced in the collision between two gravitational waves with varible polarisation and different wavefronts.
We present a large family of exact vacuum solutions of Einstein's equations which describe gravitational waves with a distinct spherical wavefront and which propagate into a Minkowski background. The gravitational waves may have either shock wavefronts or fronts on which the metric and the Weyl tensor components possess n continuous derivatives for any n >= 0. The wave region behind the spherical wavefront contains a line singularity.
Three families of exact vacuum solutions of Einstein's equations are presented which, when considered locally or in some finite spacetime region, describe gravitational waves with distinct nonplane wavefronts propagating into a Minkowski background. The wavefronts may be cylindrical, spherical or toroidal and may include impulsive or shock components or be characterized by arbitrarily weak discontinuities. Considered globally, without matching to any material sources but extending maximally as vacuum solutions, particular curvature singularities arise in the outer regions as the sources of these waves. Specifically, we find waves with a half-cylindrical wavefront driven by two singular parallel half-planes moving apart at the speed of light, waves with a complete exact spherical wavefront created by an expanding line singularity joining opposite poles of the wavefront, and waves with toroidal fronts produced by two parallell coaxial disc-like singularities and an additional line singularity joining their centres. In each case, the character of the gravitational wave on the wavefront is analysed and the collision and subsequent interaction of these waves is considered.
Exact solutions of Einstein's vacuum equations are considered which describe gravitational waves with distinct wavefronts. A family of such solutions presented recently in which the wavefronts have various geometries and which propagate into a number of physically significant backgrounds is here related to an integral representation which is a generalisation of the Rosen pulse solution for cylindrical waves. A nondiagonal solution is also constructed which is a generalisation of the Rosen pulse, being a cylindrical pulse wave with two states of polarization propagating into a Minkowski background. The solution is given in a complete and explicit form. A further generalisation to include electromagnetic waves with a distinct wavefront of the same type is also discussed.
The full metric corresponding to the colliding wave solutions with variable polarization of the type considerd by Bretón et al. [Class. Quant. Grav. 9, 2437 (1992)] is presented.
For space-times with two spacelike isometries, we present infinite hierarchies of exact solutions of the Einstein equations for vacuum and the Einstein-Maxwell equations for electrovacuum fields as represented by their Ernst potentials. These solutions are parametrized by three arbitrary rational functions of an auxiliary complex parameter and, therefore, depend on an arbitrarily large number of free parameters. They are constructed using a linear singular integral equation. This arises in the context of the so called "monodromy transform" approach, and is equivalent to the usual partial derivative form of space-time symmetry reduced Einstein equations. The solutions presented can be used to describe various inhomogeneous cosmological models, or gravitational and electromagnetic waves and their interactions. They include a number of important known solutions as particular cases.
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