Einstein vacuum field equations with a single nonnull Killing vector
Abstract
It is shown that, in the case where there is a single nonnull Killing vector, the vacuum Einstein field equations imply that there is a Ricci collineation in the quotient 3space. Using coordinates adapted to the collineation vector, we derive a fourth order partial differential equation involving the metric of the quotient 3space and we show that if this equation is satisfied, the Ernst potential may be obtained by integrating a total Riccati equation and a straightforward set of total differential equations. We also show that if the collineation vector is null, the metric of the quotient 3space may be expressed in terms of two real Clebsch potentials. Finally in the special case where the collineation vector is the generator of a timelike homothetic motion we reduce the field equations to a single second order partial differential equation of nonPainlevé type in two independent variables and obtain Petrov type III solution of RobinsonTrautman type.
 Publication:

General Relativity and Gravitation
 Pub Date:
 August 1991
 DOI:
 10.1007/BF00756909
 Bibcode:
 1991GReGr..23..861F