Thirty years of studies of integrable reductions of Einstein's field equations - General Relativity and Quantum CosmologyReport as inadecuate




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Abstract: More than thirty years passed since the first discoveries of various aspectsof integrability of the symmetry reduced vacuum Einstein equations andelectrovacuum Einstein - Maxwell equations were made and gave rise toconstructions of powerful solution generating methods for these equations. Inthe subsequent papers, the inverse scattering approach and soliton generatingtechniques, B\-acklund and symmetry transformations, formulations of auxiliaryRiemann-Hilbert or homogeneous Hilbert problems and various linear integralequation methods have been developed in detail and found different interestingapplications. Recently many efforts of different authors were aimed at findingof generalizations of these solution generating methods to various symmetryreduced gravity, string gravity and supergravity models in four and higherdimensions. However, in some cases it occurred that even after theintegrability of a system was evidenced, some difficulties arise which do notallow the authors to develop some effective methods for constructing ofsolutions. The present survey includes some remarks concerning the history ofdiscoveries of some of the well known solution generating methods for theseequations, brief descriptions of various approaches and their scopes as well assome comments concerning the possible difficulties of generalizations ofvarious approaches to more complicate gravity models and possible ways foravoiding these difficulties.



Author: G.A. Alekseev

Source: https://arxiv.org/







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