Abstract
We consider a class of ansätze for the construction of exact solutions of the Einstein-nonlinear σ-model system with an arbitrary cosmological constant in (3+1) dimensions. Exploiting a geometric interplay between the SU(2) field and Killing vectors of the spacetime reduces the matter field equations to a single scalar equation (identically satisfied in some cases) and simultaneously simplifies Einstein’s equations. This is then exemplified over various classes of spacetimes, which allows us to construct stationary black holes with a NUT parameter and uniform black strings, as well as time-dependent solutions such as Robinson-Trautman and Kundt spacetimes, Vaidya-type radiating black holes and certain Bianchi IX cosmologies. In addition to new solutions, some previously known ones are rederived in a more systematic way.
Highlights
Nonlinear σ-models have many important applications, e.g., in quantum field theory and statistical mechanics, cf., e.g., [1, 2] and references therein
We have explored the generalized hedgehog ansatz and some extensions thereof in order to find exact solutions of the Einstein-nonlinear SU(2) σ-model
The toroidal and spherical ansatze contains static black holes along with their extremal limits. Extensions of these ansatze lead to new nutty generalizations and to larger classes of Robinson-Trautman and Kundt metrics, which describe time-dependent solutions
Summary
Nonlinear σ-models have many important applications, e.g., in quantum field theory and statistical mechanics, cf., e.g., [1, 2] and references therein. We will demonstrate that the same method can be used to construct some new solutions, and further allows one to drop the requirement of spacetime symmetries assumed in previous works (at least under certain circumstances, as we shall explain in the following). This enables one, in particular, to construct time-dependent Robinson-Trautman or Kundt spacetimes [23] sourced by the SU(2) field. An appendix contains a few remarks about test fields in the Kerr spacetime
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