Einstein--Maxwell equations: Solution-generating methods as ``coordinate'' transformations in the~solution spaces

G. A. Alekseev

Teoretičeskaâ i matematičeskaâ fizika, Tome 211 (2022) no. 3, pp. 502-534

Résumé

The solution-generating methods discovered for integrable reductions of the Einstein and Einstein–Maxwell field equations (soliton-generating techniques, Bäcklund transformations, HKX transformations, Hauser–Ernst homogeneous Hilbert problem, 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 “coordinates” for every local solution can be performed by monodromy data of fundamental solutions of the corresponding spectral problems. However, for large classes of fields, these can be the values of Ernst potentials on the boundaries that consist of degenerate orbits of the space–time isometry group such that space–time geometry and the electromagnetic fields behave regularly near these boundaries. In this paper, transformations of such “coordinates” corresponding to different known solution-generating procedures are described by relatively simple algebraic expressions that do not require 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 that arise in different solution-generating procedures and to determine some physical and geometrical properties of each generating solution even before a detail calculations of all its components.

Détail

Keywords: gravitational and electromagnetic fields, Einstein–Maxwell equations, integrability, solution-generating methods.