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

Symmetry Integrability and Geometry-Methods and Applications

article

Aristophanes Dimakis¹  Nils Kanning²  Folkert Müller-Hoissen³ 

[1] Department of Financial and Management Engineering, University of the Aegean;Institute for Mathematics and Institute for Physics, Humboldt University;Max-Planck-Institute for Dynamics and Self-Organization

关键词: bidif ferential calculus;    chiral model;    Ernst equation;    Sylvester equation;   

DOI  :  10.3842/SIGMA.2011.118

来源: National Academy of Science of Ukraine

【 摘 要 】

The non-autonomous chiral model equation for an m × 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-Deminski-Newman metrics.

【 授权许可】

Unknown