An algebraically special subclass of vacuum metrics admitting a Killing motion

Abstract

The subclass of vacuum metrics with a Killing vector field (which may be either timelike or spacelike), having two of the nonzero eigenvalues of the Ricci subtensor equal, is investigated. Invariant methods are used. A special triad is chosen for the associated space V3 which gives rise to a special tetrad for the space–time V4, and this choice simplifies the expressions for the Weyl tensor and Newman–Penrose coefficients for V4. This subclass of vacuum metric reduces to two cases depending on whether or not the complex dilatation vanishes. In the first case the metric reduces to an example of plane-fronted waves. In the second case the problem reduces to a difficult pair of partial differential equations which has not been solved in the fullest generality. However, it has been shown that this case includes Robinson–Trautman metrics, Held–Robinson metrics, and some additional new Petrov type-III metrics with twisting rays.