Type D gravitational fields

Abstract

The Newman-Penrose tetrad equations are set up for the principal tetrad of a Type D gravitational field in vacuum. With no further assumptions, the equations are integrated, yielding an exhaustive list of Type D vacuum metrics. The solutions all possess two commuting Killing vectors and depend on from one to four arbitrary constants. The Type D fields with expanding rays are six closely related versions of Kerr-NUT space, the EhlersKundt C metric, and a new generalization of the C metric possessing rotation. For zero expansion we find the three EhlersKundt B metrics, plus rotating generalizations of each. The six Kerr-NUT metrics are interpreted as spinning particles with timelike, lightlike, or spacelike momentum and angular momentum vectors occurring in all possible combinations. The C metric is tentatively identified as a gravitational analog of the runaway solutions encountered in electrodynamics, i. e. , a point mass executing hyperbolic motion. Next we consider Type D fields with electromagnetism present. We find that all of the above vacuum metrics can be readily charged by adding a non-null electromagnetic field whose principal null vectors coincide with the gravitational ones. We also discuss some interesting generalizations of the Schwarzschild and C metrics containing the geometrical optics limit of a null electromagnetic field which propagates along one principal null congruence. In the Schwarzschild case they generalize Vaidya's shining star metric, to include the field of a particle traveling along an arbitrarily accelerated world-line.