Accelerating Kerr solution in SU(1,1)/U(1)-sigma model

TL;DR

Problem: obtain accelerating Kerr solutions within the $SU(1,1)/U(1)$ nonlinear sigma-model formulation of vacuum Einstein gravity. Approach: apply the Belinski–Zakharov inverse scattering method in the coset setting, dress a pure Rindler seed by a pair of complex solitons to produce a 3-soliton solution, and impose reduction conditions to stay on the coset. Key result: after suitable parameter choices and a coordinate transformation, the 3-soliton solution is equivalent to the accelerating Kerr metric with parameters $m$, $a$, and $\alpha$. Significance: validates a coset-based reduction framework for generating accelerating spacetimes and lays groundwork for extending to Einstein–Maxwell and Einstein–dilaton–axion theories via further reductions and transformations.

Abstract

Under the coset formulation of pure Einstein spacetime, we solve the reduction problem of the nonlinear \(σ\)-model for this spacetime and present the process of the inverse scattering technique after simplification. Through the simplified inverse scattering process, taking the pure Rindler metric as the background, we introduce a pair of solitons on the imaginary axis and obtain an accelerating solution in the form of three solitons in this model. By means of appropriate parameter selection and coordinate transformation, we prove that this accelerating solution in the three-soliton formulation is equivalent to the accelerating Kerr solution.