Classical defects in higher-dimensional Einstein gravity coupled to nonlinear $\sigma$-models

TL;DR

This paper constructs and analyzes higher-dimensional black hole solutions coupled with nonlinear sigma-model defects, revealing unique horizon structures, asymptotic behaviors, and conditions for extremal solutions in various cosmological constant scenarios.

Contribution

It introduces noncanonical kinetic terms in higher-dimensional sigma-model coupled gravity, generalizing known monopole solutions and exploring their black hole and factorized spacetime configurations.

Findings

Existence of black hole solutions with deficit angles and multiple horizons.

Classification of extremal black holes in different dimensions and cosmological constants.

Identification of factorized solutions with product topologies.

Abstract

We construct solutions of higher-dimensional Einstein gravity coupled to nonlinear $\sigma$-model with cosmological constant. The $\sigma$-model can be perceived as exterior configuration of a spontaneously-broken $SO(D-1)$ global higher-codimensional "monopole". Here we allow the kinetic term of the $\sigma$-model to be noncanonical; in particular we specifically study a quadratic-power-law type. This is some possible higher-dimensional generalization of the Bariola-Vilenkin (BV) solutions with $k$-global monopole studied recently. The solutions can be perceived as the exterior solution of a black hole swallowing up noncanonical global defects. Even in the absence of comological constant its surrounding spacetime is asymptotically non-flat; it suffers from deficit solid angle. We discuss the corresponding horizons. For $\Lambda>0$ in $4d$ there can exist three extremal conditions (the cold, ultracold, and Nariai black holes), while in higher-than-four dimensions the extremal black hole is only Nariai. For $\Lambda<0$ we only have black hole solutions with one horizon, save for the $4d$ case where there can exist two horizons. We give constraints on the mass and the symmetry-breaking scale for the existence of all the extremal cases. In addition, we also obtain factorized solutions, whose topology is the direct product of two-dimensional spaces of constant curvature ($M_2$, $dS_2$, or $AdS_2$) with (D-2)-sphere. We study all possible factorized channels.