We construct solutions of higher-dimensional Einstein gravity coupled to nonlinear $$\sigma $$ σ -model with cosmological constant. The $$\sigma $$ σ -model can be perceived as exterior configuration of a spontaneously-broken $$SO(D-1)$$ S O ( D - 1 ) global higher-codimensional “monopole”. Here we allow the kinetic term of the $$\sigma $$ σ -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 $$\Lambda >0$$ Λ > 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 $$\Lambda <0$$ Λ < 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 ( $$M_2$$ M 2 , $$dS_2$$ d S 2 , or $$AdS_2$$ A d S 2 ) with (D-2)-sphere. We study all possible factorized channels.
General Relativity and Gravitation – Springer Journals
Published: Aug 9, 2017
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