Properties of Discrete Black Hole Hair

TL;DR

This paper investigates how discrete gauge charges affect black hole thermodynamics, showing that black holes with discrete hair are colder, and explores the limits where the gauge field's Compton wavelength compares to the black hole size.

Contribution

It analyzes the impact of discrete $ ext{Z}_p$ gauge charge on black hole thermodynamics in different regimes, clarifying the effects on temperature and field screening.

Findings

Black holes with discrete charge are always colder than uncharged ones.

In the thick-string limit, some previous results are challenged.

The unbroken gauge limit can be smoothly approached within a finite cavity.

Abstract

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.