Twistfree and Papapetrou vacuum spacetimes with a spacelike killing vector

Abstract

We show that, for the case of vacuum solutions of the Einstein equations with a spacelike hypersurface orthogonal Killing vector ∂/∂ϰ³ and associated metricds ² =e ^2U (dx ³)² +e ^−2U γ _ab dx ^a dx ^b whereU is not a constant, there exists at every point of the quotient 3-space a plane of vectorsK ^a such that £_KR_ab=0 andK ^a R_ab=0 whereR{inab} is the Ricci tensor formed fromγ _ab . Then in the case whereU{in,a} is a timelike or spacelike vector in the quotient 3-space, Petrov type I solutions of the vacuum field equations are obtained. In the simpler case whereU{in,a} is a null vector in the quotient 3-space, the complete solution of the vacuum field equations is obtained. It is shown that this solution is Petrov type III of Kundt's class. For the case of Papapetrou solutions where there is a twist potentialψ which is a function ofU, solutions corresponding to the twistfree solutions are given.