Two-soliton solutions of the Einstein-Maxwell equations

Abstract

The authors present a family of solutions of the Einstein-Maxwell equations obtained as a two-soliton transformation of a Minkowskian seed, using Alekseev's inverse scattering method (AISM). For general values of the arbitrary parameters that arise from the AISM, the metrics are of Petrov type I, and represent cylindrically symmetric perturbations of a conical spacetime ('thin cosmic string'), that preserve the asymptotic flatness of the background, up to an additional deficit angle. The metrics can be made regular on the symmetry axis by an adequate choice of parameters. In the limit in which the C-energy goes to infinity, the metric is singular, but can be 'renormalized', obtaining either (i) a family of metrics where the symmetry axis contains a curvature singularity, or (ii) a family of cylindrically symmetric metrics with a regular axis, that can be interpreted as simple solitonic perturbations of an unstable Melvin universe. These contain a subfamily of diagonal metrics. They include a comparison with other metrics, obtained as limiting cases and, in the case of vacuum solutions, they show, by an explicit calculation, that the results obtained are equivalent to the Belinski-Zakharov transformation for two pairs of complex-conjugate soliton poles for the same seed.