Black Hole Scan

TL;DR

This paper studies a family of higher-curvature gravities in $d$ dimensions, described by the action $I_k$, that share a unique AdS vacuum with radius $l$ and cosmological constant $\Lambda=-\frac{(d-1)(d-2)}{2l^2}$. For each $k$, static, spherically symmetric black holes are constructed (including charged versions), revealing that odd $k$ yield physically acceptable black holes while even $k$ generically admit naked singularities; charged solutions exhibit a minimum source size $r_e$ except in General Relativity. The thermodynamics of these AdS black holes is developed, showing a well-defined canonical ensemble due to the AdS regulator, with a Schwarzschild-AdS–like phase structure for generic $k$ and a mass-gap/positive specific heat for CS theories. The results highlight CS theories as exceptional, with a mass gap and robust thermodynamic stability, suggesting a special role in quantum gravity and holography within this Lovelock-family framework.

Abstract

Gravitation theories selected by requiring that they have a unique anti-de Sitter vacuum with a fixed cosmological constant are studied. For a given dimension d, the Lagrangians under consideration are labeled by an integer k=1,2,...,[(d-1)/2]. Black holes for each d and k are found and are used to rank these theories. A minimum possible size for a localized electrically charged source is predicted in the whole set of theories, except General Relativity. It is found that the thermodynamic behavior falls into two classes: If d-2k=1, these solutions resemble the three dimensional black hole, otherwise, their behavior is similar to the Schwarzschild-AdS_4 geometry.

Figures (5)

• Figure 1: The black hole temperature is plotted as a function of the horizon radius $r_{+}$. For $d-2k\neq 1$ the temperature reaches an absolute minimum $T_{c}$ at $r_{+}=r_{c}$.
• Figure 2: The specific heat $C_{k}$ is plotted as a function of the horizon radius. For a generic theory, $d-2k\neq 1$, $C_{k}$ has a simple pole at $r_{+}=r_{c}$. For the exceptional case, $d=2k+1$ (CS), the specific heat is a continuous, monotonically increasing, positive function of $r_{+}$.
• Figure 3: In the generic case, $d-2k\neq 1$, the black hole can reach thermal equilibrium with a bath of temperature higher than $T_{c},$ provided the horizon radius satisfies $r_{+}>r_{u}$.
• Figure 4: In the generic case, $d-2k\neq 1,$ the black hole cannot reach thermal equilibrium with a bath of temperature lower than $T_{c}$.
• Figure 5: Black Hole Scan: Summary of all theories described by $I_{k}$ up to eleven dimensions. The integer $k=1,...,[\frac{d-1}{2}]$ represents the highest power of curvature in the action. The columns with odd $k$ are singled out by cosmic censorship. The supersymmetric extensions of EH and CS theories are known. The supergravities for the remaining $I_{k}$'s are unknown.