The Non-Autonomous Chiral Model and the Ernst Equation of General Relativity in the Bidifferential Calculus Framework

Abstract

The non-autonomous chiral model equation for an $m\times m$ matrix function on a two-dimensional space appears in particular in general relativity, where for $m=2$ a certain reduction of it determines stationary, axially symmetric solutions of Einstein's vacuum equations, and for $m=3$ solutions of the Einstein-Maxwell equations. Using a very simple and general result of the bidifferential calculus approach to integrable partial differential and difference equations, we generate a large class of exact solutions of this chiral model. The solutions are parametrized by a set of matrices, the size of which can be arbitrarily large. The matrices are subject to a Sylvester equation that has to be solved and generically admits a unique solution. By imposing the aforementioned reductions on the matrix data, we recover the Ernst potentials of multi-Kerr-NUT and multi-Demianski-Newman metrics.

References in corpus (6)

• Complete integrability of higher-dimensional Einstein equations with additional symmetry, and rotating black holes
• Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions
• 5D Einstein-Maxwell solitons and concentric rotating dipole black rings
• Solitonic generation of vacuum solutions in five-dimensional General Relativity
• Thirty years of studies of integrable reductions of Einstein's field equations
• Soliton Nature of Equilibrium State of Two Charged Masses in General Relativity

Cited by in corpus (2)

• Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations
• Integrable discretization of the vector/matrix nonlinear Schrödinger equation and the associated Yang-Baxter map