Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 118, 27 pages      arXiv:1106.4122      http://dx.doi.org/10.3842/SIGMA.2011.118

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

Aristophanes Dimakis a, Nils Kanning b and Folkert Müller-Hoissen c
a) Department of Financial and Management Engineering, University of the Aegean, 41, Kountourioti Str., 82100 Chios, Greece
b) Institute for Mathematics and Institute for Physics, Humboldt University, Rudower Chaussee 25, 12489 Berlin, Germany
c) Max-Planck-Institute for Dynamics and Self-Organization, Bunsenstrasse 10, 37073 Göttingen, Germany

Received August 31, 2011, in final form December 16, 2011; Published online December 23, 2011

Abstract
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.

Key words: bidifferential calculus; chiral model; Ernst equation; Sylvester equation.

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