Integrable and non-integrable structures in Einstein--Maxwell equations with Abelian isometry group~$\mathcal G_2$

G. A. Alekseev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mechanics, Tome 295 (2016), pp. 7-33

Résumé

We consider the classes of electrovacuum Einstein–Maxwell fields (with a cosmological constant) for which the metrics admit an Abelian two-dimensional isometry group $\mathcal G_2$ 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 $\mathcal G_2$ 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 $\mathcal G_2$ are diagonal, all field equations can be reduced to one nonlinear equation for one real function $\alpha(x^1,x^2)$ that characterizes the area element on these orbits. Simple examples of solutions for each of these classes are presented.