Gravitational waves in general relativity XVI. Standing waves

Abstract

The superposition of gravitational waves travelling in different directions can readily be treated in the linear approximation, which, however, is not very revealing. Attempting a fuller analysis of the apparently simple case of plane waves inevitably leads to major singularities. It is therefore interesting to study a mathematically tractable system in which superposition occurs without generating space singularities. Here I study the gravitational waves associated with an unsymmetric body rotating about the z –axis, all features being independent of z . Diverging and converging waves combine to make the field, viewed from the co–rotating frame, time independent. Only the wave zone is studied. Equations defining solutions in terms of a single function are found and a simple approximate solution is derived. There is no singularity in the space containing these standing waves, which are ascribed to a superposition of inward and outward travelling waves. These standing waves are therefore taken to have physical significance and to resolve a difficulty in the theory.