Gravitational waves in general relativity. XII. Correspondence between toroidal and cylindrical waves

Abstract

A device making use of the linearity of one of the field equations for Rosen’s cylindrically symmetric metric is employed to demonstrate a correspondence between some outgoing toroidal gravitational waves and imploding-exploding cylindrical waves. An exact form of axially symmetric metric for a smooth toroidal pulse in the region ｜ z ｜ < t (in coordinates which approximate to cylindrical polars when the field is weak) is constructed, and extensions to the region ｜ z ｜ < t are considered. The solution remains valid throughout the reflexion of the wave as it meets itself on the symmetry axis, and displays this process clearly. As a particular example, a solution of Weber & Wheeler for a non-singular cylindrical pulse is converted into one for a time-symmetric toroidal pulse, and the distant field at the moment of time-symmetry is briefly examined.