Microscopic entropy of higher-dimensional nonminimally dressed Lifshitz black holes

Abstract

In arbitrary dimension, we consider a theory described by the most general quadratic curvature corrections of Einstein gravity together with a self-interacting nonminimally coupled scalar field. This theory is shown to admit five different families of Lifshitz black holes dressed with a nontrivial scalar field. The entropy of these configurations is microscopically computed by means of a higherdimensional anisotropic Cardy-like formula where the role of the ground state is played by the soliton obtained through a double analytic continuation. This involves calculating the correct expressions for the masses of the higher-dimensional Lifshitz black hole as well as their corresponding soliton. The robustness of this Cardy-like formula is checked by showing that the microscopic entropy is in perfect agreement with the gravitational Wald entropy. Consequently, the calculated global charges are compatible with the first law of thermodynamics as well as an anisotropic higher-dimensional version of the Smarr formula. Some of these configurations exist on Lifshitz critical points of the theory where all their extensive thermodynamic quantities vanish.

Cardy-like formula for the Schwarzschild black hole entropy

“…As already mentioned in the Introduction, generalizations of the Cardy formula where the ground state is identified with a gravitational soliton have been given and explicitly tested in the current literature [21][22][23][24][25][26][27][28][29][30][31] For these previously mentioned cases, a precise formula for the density of states at fixed energy M was obtained under the assumption that the bulk gravitational soliton (identified with the ground state of the putative field theory) has a negative mass denoted by M soliton < 0, ensuring in turn the existence of a gap in the spectrum. We have noticed that the different Cardy-like formulas reported in Refs.…”

“…More recently, great progress in the microscopic derivation of black hole entropy has been made possible thanks to the study of three-dimensional black holes by highlighting the prominent role played by the gravitational soliton in the generalizations of the Cardy formula [21][22][23][24][25][26][27][28][29][30][31]. This point of view is not in contradiction with the standard Cardy formula where the ground state is implicitly assumed to be the three-dimensional AdS spacetime.…”

“…In this context, a Lifshitz version of the Cardy formula was proposed in [21] where the S-modular invariance or equivalently the low-high temperature duality arises as a direct consequence of the isomorphism between the two-dimensional Lifshitz algebras with dynamical exponents z and z −1 . Using this duality, a precise formula for the asymptotic growth of the number of states was obtained in [21], and the robustness of this formula was tested in a wide variety of Lifshitz black holes [21][22][23].…”

“…This puzzle can be circumvented by first rewriting the Cardy formula (2) in terms of the vacuum energy rather than the central charge, and by identifying the vacuum energy with the mass of a gravitational soliton obtained from the black hole by means of a double Wick rotation as it occurs for the AdS soliton [32]. The importance played by the soliton to correctly reproduce the semiclassical entropy has also been put in light in the case of anisotropic black brane solutions [21][22][23]27,30], and for AdS black branes in arbitrary dimension [24][25][26]28,29,31]. This is precisely the road that we propose to explore in this paper by presenting a Cardy-like formula that perfectly reproduces the Bekenstein-Hawking formula (1) in the case of the Schwarzschild black hole solution.…”

Mass and angular momentum of charged rotating Gödel black holes in five-dimensional minimal supergravity

“…Consequently, in some sense, the offshell generalized ADT formalism could also be viewed as the cousin of both the BBC and phase space approaches. Till now, there have existed a series of developments and applications of the off-shell ADT formulation in a range of gravity theories regardless of the asymptotic behavior of the spacetime metrics [32,33,34,35,36,37,38,39,40,41,42,43,44,45,46].…”