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Gravitational solitons and monodromy transform approach to solution of integrable reductions of Einstein equations. (English) Zbl 0982.83016

Summary: In this paper the well known Belinskij and Zakharov soliton generating transformations of the solution space of vacuum Einstein equations with two-dimensional Abelian groups of isometries are considered in the context of the so-called “monodromy transform approach”, which provides some general base for the study of various integrable space-time symmetry reductions of Einstein equations. Similarly to the scattering data used in the known spectral transform, in this approach the monodromy data for the solution of the associated linear system characterize completely any solution of the reduced Einstein equations, and many physical and geometrical properties of the solutions can be expressed directly in terms of the analytical structure on the spectral plane of the corresponding monodromy data functions. The Belinskij and Zakharov vacuum soliton generating transformations can be expressed in explicit form (without specification of the background solution) as simple (linear-fractional) transformations of the corresponding monodromy data functions with coefficients, polynomial in spectral parameter. This allows to determine many physical parameters of the generating soliton solutions without (or before) calculation of all components of the solutions. A similar characterization for electrovacuum soliton generating transformations is also presented.

MSC:

83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory
83C80 Analogues of general relativity in lower dimensions

References:

[1] Belinskii, V. A.; Zakharov, V. E., Sov. Phys. JETP, 48, 985 (1978)
[2] Belinskii, V. A.; Zakharov, V. E., Sov. Phys. JETP, 50, 1 (1979)
[3] Alekseev, G. A., Sov. Phys. Dokl. USA, 26, 158 (1981)
[4] Alekseev, G. A., Sov. Phys. Dokl. USA, 30, 565 (1985)
[5] G.A. Alekseev, Proc. Steklov Inst. Math. 3 (1988) 215.; G.A. Alekseev, Proc. Steklov Inst. Math. 3 (1988) 215.
[6] G.A. Aleksejev, in: Proceedings of the Eighth International Workshop on NEEDS’92, World Scientific, Singapore, 1992, p. 5.; G.A. Aleksejev, in: Proceedings of the Eighth International Workshop on NEEDS’92, World Scientific, Singapore, 1992, p. 5.
[7] G.A. Alekseev, in: Proceedings of the Workshop on Nonlinearity, Integrability and All That: Twenty Years after NEEDS’79, World Scientific, Singapore, pp. 12-18.; G.A. Alekseev, in: Proceedings of the Workshop on Nonlinearity, Integrability and All That: Twenty Years after NEEDS’79, World Scientific, Singapore, pp. 12-18. · Zbl 0965.35175
[8] Kinnersley, W.; Chitre, D. M., J. Math. Phys., 19, 1927 (1978)
[9] Alekseev, G. A., Pis’ma JETP, 32, 301 (1980)
[10] S. Miccichè, J.B. Griffiths, Class. Quantum Grav. 17 (1) (2000). gr-qc/990974.; S. Miccichè, J.B. Griffiths, Class. Quantum Grav. 17 (1) (2000). gr-qc/990974. · Zbl 0937.83005
[11] G.A. Alekseev, Abstracts of contributed papers, in: Proceedings of the 13th International Conference on General Relativity and Gravitation, Cordoba, Argentina, 1992 (1993), p. 3.; G.A. Alekseev, Abstracts of contributed papers, in: Proceedings of the 13th International Conference on General Relativity and Gravitation, Cordoba, Argentina, 1992 (1993), p. 3.
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