No abstract

Using the Newman-Penrose formalism, the vacuum field equations are solved for Petrov type D. An exhaustive set of ten metrics is obtained, including among them a new rotating solution closely related to the Ehlers-Kundt ``C'' metric. They all possess at least two Killing vectors and depend only on a small number of arbitrary constants.

The Einstein equations for stationary axially symmetric gravitational fields are written in several extremely simple forms. Using a tensor generalization of the Ernst potential, we give forms that are manifestly covariant under (i) the external group G of coordinate transformations, (ii) the internal group H of Ehlers transformations and gage transformations, and (iii) the infinite parameter group K of Geroch which combines both. We then show how the same thing can be done to the Einstein–Maxwell equations. The enlarged internal group H′ now includes the Harrison transformations, and is isomorphic to SU(2,1). The enlarged group K′ contains even more parameters, and generates even more potentials and conservation laws.

A metric of the Kerr-Schild type is derived which contains four arbitrary functions of time. It is a generalization of Vaidya's shining-star metric, and permits arbitrary acceleration of the source.

We generalize Crapper's exact solution for capillary waves on fluid of infinite depth. We find two finite-depth solutions involving elliptic functions. We show they can also be interpreted as large amplitude symmetrical and antisymmetrical waves on a fluid sheet. Particularly interesting are the waves obtained from our solution in the limit when the fluid sheet is extremely thin.