Exact solutions for gravitational waves with cylindrical, spherical and toroidal wavefronts

Abstract

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 parallel 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.