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We revisit the physical effects of discrete $mathbb{Z}_p$ gauge charge on black hole thermodynamics, building on the seminal work of Coleman, Preskill, and Wilczek. Realising the discrete theory from the spontaneous breaking of an Abelian gauge theory, we consider the two limiting cases of interest, depending on whether the Compton wavelength of the massive vector is much smaller or much larger than the size of the black hole -- the so-called thin- and thick-string limits respectively. We find that the qualitative effect of discrete hair on the mass-temperature relationship is the same in both regimes, and similar to that of unbroken $U(1)$ charge: namely, a black hole carrying discrete gauge charge is always colder than its uncharged counterpart. In the thick-string limit, our conclusions bring into question some of the results of Coleman et al., as we discuss. Further, by considering the system to be enclosed within a finite cavity, we argue how the unbroken limit may be smoothly defined, and the unscreened electric field of the standard Reissner-Nordstrom solution recovered.
We show that a black hole surrounded by scalar dark matter develops scalar hair. This is the generalization of a phenomenon pointed out by Jacobson, that a minimally coupled scalar with a non-trivial time dependence far away from the black hole would
The specific nonlinear vector $sigma$-model coupled to Einstein gravity is investigated. The model arises in the studies of the gravitating matter in non-commutative geometry. The static spherically symmetric spacetimes are identified by direct solvi
We show that the nonlinear $sigma-$model in an asymptotically $AdS_3$ space-time admits a novel local symmetry. The field action is assumed to be quartic in the nonlinear $sigma-$model fields and minimally coupled to gravity. The local symmetry trans
We study a hairy black hole solution in the dilatonic Einstein-Gauss-Bonnet theory of gravitation, in which the Gauss-Bonnet term is non-minimally coupled to the dilaton field. Hairy black holes with spherical symmetry seem to be easily constructed w
Motivated by black hole solutions with matter fields outside their horizon, we study the effect of these matter fields in the motion of massless and massive particles. We consider as background a four-dimensional asymptotically AdS black hole with sc