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




This article is cited in 2 scientific papers (total in 2 papers)
Integrable and nonintegrable 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 twodimensional 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 socalled nondynamical 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 wellknown integrable structures. We find modifications of the known reduced forms of Einstein–Maxwell equations, namely, the Ernst equations and the selfdual Kinnersley equations, in which the presence of nondynamical degrees of freedom is taken into account, and consider the following subclasses of fields with different nondynamical 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 twodimensional 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 
145000005 
This work is supported by the Russian Science Foundation under grant 145000005. 
DOI:
https://doi.org/10.1134/S0371968516040014
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English version:
Proceedings of the Steklov Institute of Mathematics, 2016, 295, 1–26
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Article
UDC:
517.958 Received: September 28, 2016
Citation:
G. A. Alekseev, “Integrable and nonintegrable structures in Einstein–Maxwell equations with Abelian isometry group $\mathcal G_2$”, Modern problems of mechanics, Collected papers, Tr. 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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\serial Tr. Mat. Inst. Steklova
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\vol 295
\pages 733
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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