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 Trudy Mat. Inst. Steklova, 2016, Volume 295, Pages 7–33 (Mi tm3789)

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

G. A. Alekseev

Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

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 $\mathcal G_2$ 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.

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 This work is supported by the Russian Science Foundation under grant 14-50-00005.

DOI: https://doi.org/10.1134/S0371968516040014

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English version:
Proceedings of the Steklov Institute of Mathematics, 2016, 295, 1–26

Bibliographic databases:

UDC: 517.958

Citation: G. A. Alekseev, “Integrable and non-integrable structures in Einstein–Maxwell equations with Abelian isometry group $\mathcal G_2$”, Modern problems of mechanics, Collected papers, Trudy Mat. Inst. Steklova, 295, MAIK Nauka/Interperiodica, Moscow, 2016, 7–33; Proc. Steklov Inst. Math., 295 (2016), 1–26

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3789
• https://doi.org/10.1134/S0371968516040014
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