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VOL. 12 | 2011 Einstein Metrics with Two-Dimensional Killing Leaves and Their Applications in Physics
Gaetano Vilasi

Editor(s) Ivaïlo M. Mladenov, Gaetano Vilasi, Akira Yoshioka

Geom. Integrability & Quantization, 2011: 329-341 (2011) DOI: 10.7546/giq-12-2011-329-341


Solutions of vacuum Einstein’s field equations, for the class of pseudo-Riemannian four-metrics admitting a non Abelian two dimensional Lie algebra of Killing fields, are explicitly described. When the distribution orthogonal to the orbits is completely integrable and the metric is not degenerate along the orbits, these solutions are parameterized either by solutions of a transcendental equation (the tortoise equation), or by solutions of a linear second order differential equation in two independent variables. Metrics, corresponding to solutions of the tortoise equation, are characterized as those that admit a three dimensional Lie algebra of Killing fields with two dimensional leaves. Metrics, corresponding to the case in which the commutator of the two Killing fields is isotropic, represent nonlinear gravitational waves.


Published: 1 January 2011
First available in Project Euclid: 13 July 2015

zbMATH: 1382.53010
MathSciNet: MR3087988

Digital Object Identifier: 10.7546/giq-12-2011-329-341

Rights: Copyright © 2011 Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences


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