Soliton propagation on a gravitational plane-wave collision spacetime

TL;DR

This paper introduces new exact solutions to Einstein's equations modeling soliton propagation in a plane-wave collision spacetime, revealing conditions for horizon formation and singularity development.

Contribution

It applies the Belinskii and Zakharov Inverse Scattering Method with a novel renormalization to generate and analyze solitonic solutions in a gravitational collision context.

Findings

Solutions contain two additional parameters compared to standard ISM.

A specific parameter relation yields a regular horizon instead of a singularity.

Curvature singularities appear in the extended spacetime due to complex poles.

Abstract

We present a new family of exact solutions of the Einstein equations, constructed through the Khan-Penrose procedure, that may be interpreted as representing the propagation of a pair of solitons, in the background of a plane-wave collision spacetime. The metric in the interaction region is obtained as a diagonal solitonic perturbation of Rindler's spacetime, applying the Belinskii and Zakharov Inverse Scattering Method (ISM), with two real poles and a pair of complex conjugate poles. We use a non-standard renormalization procedure, obtaining solutions that contain two more parameters than in the standard ISM. We analyze the asymptotic behaviour of the solutions in the limit where the determinant of the Killing part of the metric vanishes, finding in this limit a curvature singularity, except when the free parameters contained in the solutions satisfy a particular relation. Assuming this condition is satisfied, we show that the metric is regular in the above mentioned limit, and that the spacetimes contain in this case a Killing Cauchy horizon instead of a curvature singularity. The solutions are analytically extended through the horizon, and we find a curvature singularity in this extension, related to the presence and propagation of the complex poles.