Solution generating methods as "coordinate" transformations in the solution spaces

TL;DR

This paper describes how solution generating methods for Einstein and Einstein-Maxwell equations can be understood as coordinate transformations in the solution space, using algebraic expressions related to monodromy data and boundary values.

Contribution

It provides explicit algebraic forms of these transformations, enabling analysis of solution parameters and properties without detailed calculations.

Findings

Transformations expressed as simple algebraic formulas

Interrelations between different solution parameters established

Physical and geometrical properties inferred prior to detailed solutions

Abstract

The solution generating methods discovered earlier for integrable reductions of Einstein's and Einstein - Maxwell field equations (such as soliton generating techniques, B$\ddot{a}$cklund or symmetry transformations and other group-theoretical methods) can be described explicitly as transformations of especially defined "coordinates" in the infinite-dimensional solution spaces of these equations. In general, the role of such "coordi\-nates", which characterize every local solution, can be performed by the monodromy data of the fundamental solutions of the corresponding spectral problems. However for large subclasses of fields, these can be the values of the Ernst potentials on the boundaries which consist of such degenerate orbits of the space-time isometry group, in which neighbourhood the space-time geometry and electromagnetic fields possess a regular behaviour. In this paper, transformations of such "coordinates", corresponding to different known solution generating procedures are described by simple enough algebraic expressions which do not need any particular choice of the initial (background) solution. Explicit forms of these transformations allow us to find the interrelations between the sets of free parameters, which arise in different solution generating procedures, as well as to determine some physical and geometrical properties of each generating solution even before a detail calculations of all its components.