Cardy formula for charged black holes with anisotropic scaling

Abstract

We first observe that for Lifshitz black holes whose only charge is the mass, the resulting Smarr relation is a direct consequence of the Lifshitz Cardy formula. From this observation, we propose to extend the Cardy formula to the case of electrically charged Lifshitz black holes satisfying as well a Smarr relation. The expression of our formula depends on the dynamical exponent, the energy and the charge of the ground state which is played by a magnetically charged soliton obtained through a double Wick rotation. The expression also involves a factor multiplying the chemical potentials which varies in function of the electromagnetic theory considered. This factor is precisely the one that appears in the Smarr formula for charged Lifshitz black holes. We test the validity of this Cardy formula in different situations where electrically Lifshitz charged black holes satisfying a Smarr relation are known. We then extend these results to electrically charged black holes with hyperscaling violation. Finally, an example in the charged AdS case is also provided.

Warped AdS3 black holes in higher derivative gravity theories

“…Remarkably, the horizon entropy of the black hole solutions belonging to the WAdS 3 phase space was seen to exactly match the field theory expression (1.4). Note that similar counting formulas and matchings have been observed in Lifshitz [60,61] …”

A Cardy-like formula for rotating black holes with planar horizon

“…In the standard AdS situation, the "effective" spatial dimension d eff = D − 2 and the dynamical exponent z = 1. Under these hypothesis, formulas for the asymptotic growth of the number of states have been obtained in the non-rotating case in [12][13][14], and their spinning generalizations in the isotropic case z = 1 were found in [21]. In the present work, we extend this formula for a generic dynamical exponent z.…”

“…[11][12][13][14][16][17][18]. The matching between the gravitational entropy and the Cardy formula (2.2) is perfectly consolidated for three-dimensional black holes that are asymptotically AdS (even in the weaker sense).…”

“…On the other hand, in the non-rotating case with anisotropy, i.e. J = 0 with z = 1, the formula (2.4) reproduces the Lifshitz Cardy formula in three dimensions with d eff = 1, see [11], and also the generic formula for hyperscaling violation metric [12][13][14].…”

“…namely the Lifshitz spacetimes [11] or the hyperscaling violation geometries [12][13][14]. The interest on anisotropic spacetimes has considerably grown up this last decade essentially due to the will of extending the ideas underlying the gauge/gravity duality to strongly coupled field theories with an anisotropic scaling symmetry [15].…”