Скрытые симметрии уравнений Эйнштейна и конформные преобразования пространства потенциалов

Abstract

Hidden symmetries are global symmetries that arise in dimensional reduction of Einstein's equations (and their generalizations) that inherited from the group of diffeomorphisms of the original theory. In three-dimensional reduction all non-gravitational field are effectively reduced to scalar, with the broad classes of theories describe three-dimensional gravity, which has as a source a system of scalar fields, forming a nonlinear sigma-model. Hidden symmetries are used for obtaining new solutions of Einstein's equations depending on three variables (almost all nowadays analytically known solutions can be reduced to such a class). The "potential" space (target space) of the three-dimensional sigma-models obtained by reduction theory allowing solutions with a flat three-dimensional space (in models of supergravity this solution, saturating Bogomol'nyi-Prasad-Sommerfield bound (BPS)) has a metric of Lorentz signature, and contain isotropic geodesic. In particular, they correspond to extreme black holes with degenerate event horizon, and form an important sub-class of exact solutions. Since isotropic curves remain so under conformal transformations of the metric, there the question of usefulness of possible conformal symmetries of space the potential for the generation of BPS solutions. This work is attempt such a study. It is shown that some well-known sigma-models arising from the dimensional reduction, have conformally flat target space.