Complete integrability of higher-dimensional Einstein equations with additional symmetry, and rotating black holes

TL;DR

The paper addresses the challenge of finding exact black-hole solutions in higher-dimensional general relativity by extending the Belinski-Zakharov inverse scattering method to D-dimensional spacetimes with D-2 commuting Killing vectors. It shows that the Einstein equations reduce to a two-dimensional integrable system for the metric block and a conformal factor, enabling the construction of N-soliton solutions via a dressing procedure. As a concrete result, it derives the five-dimensional Myers-Perry black hole with two independent angular momenta as a two-soliton solution on a static background, clarifying the rod-structure interpretation and the role of soliton positions. The work suggests that this integrable framework can be used to explore black-hole uniqueness in higher dimensions and to pursue regular black ring solutions as future directions.

Abstract

A new derivation of the five-dimensional Myers-Perry black-hole metric as a 2-soliton solution on a non-flat background is presented. It is intended to be an illustration of how the well-known Belinski-Zakharov method can be applied to find solutions of the Einstein equations in D-dimensional space-time with D-2 commuting Killing vectors using the complete integrability of this system. The method appears also to be promising for the analysis of the uniqueness questions for higher-dimensional black holes.