Hedgehog ansatz and its generalization for self-gravitating Skyrmions

TL;DR

This paper revisits and generalizes the hedgehog ansatz for self-gravitating Skyrmions, enabling the study of non-spherically symmetric spacetimes and leading to new exact solutions, including a black hole with a global monopole.

Contribution

It introduces a generalized hedgehog ansatz that accommodates non-spherical symmetries in self-gravitating Skyrme models, expanding the scope of analytical solutions.

Findings

Derived coupled differential equations for scalar fields.

Presented specific field configurations in stationary and axisymmetric spacetimes.

Obtained new exact solutions, including a black hole with a global monopole.

Abstract

The hedgehog ansatz for spherically symmetric spacetimes in self-gravitating nonlinear sigma models and Skyrme models is revisited and its generalization for non-spherically symmetric spacetimes is proposed. The key idea behind our construction is that, even if the matter fields depend on the Killing coordinates in a nontrivial way, the corresponding energy-momentum tensor can still be compatible with spacetime symmetries. Our generalized hedgehog ansatz reduces the Skyrme equations to coupled differential equations for two scalar fields together with several constraint equations between them. Some particular field configurations satisfying those constraints are presented in several physically important spacetimes, including stationary and axisymmetric spacetimes. Incidentally, several new exact solutions are obtained under the standard hedgehog ansatz, one of which represents a global monopole inside a black hole with the Skyrme effect.