Einstein vacuum field equations with a single non-null Killing vector

Details Einstein vacuum field equations with a single non-null Killing vector Einstein vacuum field equations with a single non-null Killing vector

Aug 1991

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1991gregr..23..861f&link_type=abstract

General Relativity and Gravitation, Volume 23, Issue 8, pp.861-876

Physics

3

Scientific paper

It is shown that, in the case where there is a single non-null Killing vector, the vacuum Einstein field equations imply that there is a Ricci collineation in the quotient 3-space. Using coordinates adapted to the collineation vector, we derive a fourth order partial differential equation involving the metric of the quotient 3-space 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 3-space 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 non-Painlevé type in two independent variables and obtain Petrov type III solution of Robinson-Trautman type.

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