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

作者：Dimakis Aristophanes^*; Kanning Nils; Mueller Hoissen Folkert

来源：Symmetry Integrability and Geometry-Methods and Applications, 2011, 7: 118.

DOI：10.3842/SIGMA.2011.118

• bidifferential calculus
• chiral model
• Ernst equation
• Sylvester equation

The non-autonomous chiral model equation for an m x 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-Kerr-NUT and multi-Demianski-Newman metrics.

• 出版日期2011