Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations

TL;DR

This paper introduces a binary Darboux transformation within bidifferential calculus to generate exact solutions for integrable PDEs, applying it to vacuum Einstein equations to produce known and new black hole solutions.

Contribution

It develops a general solution-generating method using binary Darboux transformations in bidifferential calculus and applies it to Einstein equations, providing an alternative to existing formalisms.

Findings

Generated a large class of exact solutions for Einstein equations.

Reproduced known black hole solutions like Kerr-NUT and Myers-Perry.

Provided new insights into integrable reductions of Einstein equations.

Abstract

We present a general solution-generating result within the bidifferential calculus approach to integrable partial differential and difference equations, based on a binary Darboux-type transformation. This is then applied to the non-autonomous chiral model, a certain reduction of which is known to appear in the case of the D-dimensional vacuum Einstein equations with D-2 commuting Killing vector fields. A large class of exact solutions is obtained, and the aforementioned reduction is implemented. This results in an alternative to the well-known Belinski-Zakharov formalism. We recover relevant examples of space-times in dimensions four (Kerr-NUT, Tomimatsu-Sato) and five (single and double Myers-Perry black holes, black saturn, bicycling black rings).