Limits on the Statistical Description of Charged de Sitter Black Holes

TL;DR

This paper addresses the thermodynamics of charged de Sitter black holes by adopting the observer-based Bousso-Hawking normalization, resolving ambiguities from dual horizons and lacking a global timelike Killing vector. It derives observer-normalized first laws and a redshifted mass $ ilde{M}$, then analyzes the black hole heat capacity across near-extremal limits, finding finite behavior in the near-Nariai regime away from ultracold points but vanishing in cold and ultracold limits. The work highlights that, under this normalization, log-$T$ corrections are not generically needed in the Nariai region, and it clarifies how Schwinger pair production and charge exchange affect thermodynamic stability. The results have implications for the microscopic/statistical interpretation of de Sitter black holes and motivate explicit one-loop calculations in the physically meaningful normalization. The study emphasizes the observer's role in de Sitter thermodynamics and provides a framework to assess quantum corrections in near-extremal regimes.

Abstract

The thermodynamics of de Sitter black holes is complicated by the presence of two horizons and the absence of a globally defined timelike Killing vector. The standard choice of the Gibbons-Hawking Killing vector is at odds with the interpretation of the surface gravity as an acceleration measured by a physical observer at rest. Focusing on four-dimensional Reissner-Nordström de Sitter black holes we show that this issue can be resolved by adopting a normalization originally proposed by Bousso and Hawking, which defines thermodynamic quantities relative to the unique freely-falling observer at a fixed radial coordinate. Within this framework, we derive new first laws for the black hole and cosmological horizon and re-examine the black hole's heat capacity. We find that the heat capacity remains finite in the near-extremal Nariai limit, thus averting a breakdown of the semi-classical thermodynamic description. However, the heat capacity does vanish in the cold limit, as expected, and for Nariai black holes in the ultracold limit, indicating that fundamental limitations on the statistical description persist in these regimes. We discuss the implications of our results for log-$T$ corrections to near-extremal de Sitter black holes.

Figures (5)

• Figure 1: Shark fin-shaped diagram for Reissner-Nordström-de Sitter black holes. The shaded region corresponds to black hole solutions, and the white region to solutions with naked singularities. The upper boundary of the shark fin corresponds to cold black holes $(r_a=r_b)$ and the lower curved boundary to Nariai black holes $(r_b=r_c)$. The two lines meet at the ultracold point $r_a=r_b=r_c$.
• Figure 2: There is a unique linear trajectory (purple, dashed) that remains within the sharkfin and ends at the ultracold point given by ${\cal Q} = \sqrt{2}{\cal M}-\frac{\ell_4}{6\sqrt{3}}$.
• Figure 3: Heat capacity of an uncharged black hole in de Sitter space for $\gamma=1$ normalization and Bousso-Hawking normalization as a function of radius. We used $\ell_4=G_4=1$.
• Figure 4: Heat capacity at fixed charge as a function of radius of charged Nariai black holes in de Sitter space. We used $\ell_4=G_4=1$.
• Figure 5: Heat capacity of lukewarm black holes in various ensembles specified by $\mathcal{Z}$. From top to bottom, the three curves correspond to $\mathcal{Z}=2.2,2,1.8$ respectively. When $\mathcal{Z}>2$, $\tilde{C}^z_b\rightarrow \infty$ at the lukewarm-Nariai intersection point where as for $\mathcal{Z}<2$, $\tilde{C}^z_b\rightarrow -\infty$. For the later case, $\tilde{C}^z_b$ has a zero at the point $1-\frac{\mathcal{Z} r_c}{\ell_4}=0$ which corresponds to $\delta S_b =0$.