On General Solutions of Einstein Equations

Abstract

We show how the Einstein equations with cosmological constant (and/or various types of matter field sources) can be integrated in a very general form following the anholonomic deformation method for constructing exact solutions in four and five dimensional gravity (S. Vacaru, IJGMMP 4 (2007) 1285). In this letter, we prove that such a geometric method can be used for constructing general non-Killing solutions. The key idea is to introduce an auxiliary linear connection which is also metric compatible and completely defined by the metric structure but contains some torsion terms induced nonholonomically by generic off-diagonal coefficients of metric. There are some classes of nonholonomic frames with respect to which the Einstein equations (for such an auxiliary connection) split into an integrable system of partial differential equations. We have to impose additional constraints on generating and integration functions in order to transform the auxiliary connection into the Levi-Civita one. This way, we extract general exact solutions (parametrized by generic off-diagonal metrics and depending on all coordinates) in Einstein gravity and five dimensional extensions.