Integrable and non-integrable structures in Einstein-Maxwell equations with Abelian isometry group G2

Abstract

We consider the classes of electrovacuum Einstein–Maxwell fields (with a cosmological constant) for which the metrics admit an Abelian two-dimensional isometry group G₂ with nonnull orbits and electromagnetic fields possess the same symmetry. For the fields with such symmetries, we describe the structures of the so-called non-dynamical degrees of freedom, whose presence, just as the presence of a cosmological constant, changes (in a strikingly similar way) the vacuum and electrovacuum dynamical equations and destroys their well-known integrable structures. We find modifications of the known reduced forms of Einstein–Maxwell equations, namely, the Ernst equations and the self-dual Kinnersley equations, in which the presence of non-dynamical degrees of freedom is taken into account, and consider the following subclasses of fields with different non-dynamical degrees of freedom: (i) vacuum metrics with cosmological constant; (ii) space–time geometries in vacuum with isometry groups G₂ that are not orthogonally transitive; and (iii) electrovacuum fields with more general structures of electromagnetic fields than in the known integrable cases. For each of these classes of fields, in the case when the two-dimensional metrics on the orbits of the isometry group G₂ are diagonal, all field equations can be reduced to one nonlinear equation for one real function α(x1, x2) that characterizes the area element on these orbits. Simple examples of solutions for each of these classes are presented.