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18 April 2024 |
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Thirty years of studies of integrable reductions of Einstein's field equations | | G.A. Alekseev
; | | Date: |
16 Nov 2010 | | Abstract: | More than thirty years passed since the first discoveries of various aspects
of integrability of the symmetry reduced vacuum Einstein equations and
electrovacuum Einstein - Maxwell equations were made and gave rise to
constructions of powerful solution generating methods for these equations. In
the subsequent papers, the inverse scattering approach and soliton generating
techniques, B"acklund and symmetry transformations, formulations of auxiliary
Riemann-Hilbert or homogeneous Hilbert problems and various linear integral
equation methods have been developed in detail and found different interesting
applications. Recently many efforts of different authors were aimed at finding
of generalizations of these solution generating methods to various (symmetry
reduced) gravity, string gravity and supergravity models in four and higher
dimensions. However, in some cases it occurred that even after the
integrability of a system was evidenced, some difficulties arise which do not
allow the authors to develop some effective methods for constructing of
solutions. The present survey includes some remarks concerning the history of
discoveries of some of the well known solution generating methods for these
equations, brief descriptions of various approaches and their scopes as well as
some comments concerning the possible difficulties of generalizations of
various approaches to more complicate gravity models and possible ways for
avoiding these difficulties. | | Source: | arXiv, 1011.3846 | | Services: | Forum | Review | PDF | Favorites | |
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