Monodromy-data parameterization of spaces of local solutions of integrable reductions of Einstein's field equations

TL;DR

This paper introduces a monodromy-data parameterization method for spaces of local solutions to integrable reductions of Einstein's equations, enabling explicit solution construction and analysis via spectral data.

Contribution

It develops a monodromy transform approach that parametrizes solution spaces using spectral monodromy data, with proven existence, uniqueness, and explicit integral equations.

Findings

Parametrization of solution spaces via monodromy data.

Explicit integral equations for inverse problems.

Proof of existence and uniqueness of solutions.

Abstract

For the fields depending on two of the four space-time coordinates only, the spaces of local solutions of various integrable reductions of Einstein's field equations are shown to be the subspaces of the spaces of local solutions of the ``null-curvature'' equations constricted by a requirement of a universal (i.e. solution independent) structures of the canonical Jordan forms of the unknown matrix variables. These spaces of solutions of the ``null-curvature'' equations can be parametrized by a finite sets of free functional parameters -- arbitrary holomorphic (in some local domains) functions of the spectral parameter which can be interpreted as the monodromy data on the spectral plane of the fundamental solutions of associated linear systems. Direct and inverse problems of such mapping (``monodromy transform''), i.e. the problem of finding of the monodromy data for any local solution of the ``null-curvature'' equations with given canonical forms, as well as the existence and uniqueness of such solution for arbitrarily chosen monodromy data are shown to be solvable unambiguously. The linear singular integral equations solving the inverse problems and the explicit forms of the monodromy data corresponding to the spaces of solutions of the symmetry reduced Einstein's field equations are derived.