Gravitational Solitons and Monodromy Transform Approach to Solution of Integrable Reductions of Einstein Equations

TL;DR

This paper explores the monodromy transform approach to Einstein equations, providing explicit forms for soliton generating transformations and enabling the determination of physical parameters directly from spectral data.

Contribution

It introduces a unified framework for soliton transformations in Einstein equations using monodromy data, simplifying the analysis of solutions without explicit metric calculations.

Findings

Explicit linear-fractional form of soliton transformations

Direct determination of physical parameters from spectral data

Unified approach for vacuum and electrovacuum solutions

Abstract

In this paper the well known Belinskii 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 solution of 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 Belinskii 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. The similar characterization for electrovacuum soliton generating transformations is also presented.