Towards the Cardy formula for hyperscaling violation black holes

Abstract

The aim of this paper is to propose a generalized Cardy formula in the case of three-dimensional hyperscaling violation black holes. We first note that for the hyperscaling violation metrics, the scaling of the entropy in term of the temperature (defined as the effective spatial dimensionality divided by the dynamical exponent) depends explicitly on the gravity theory. Starting from this observation, we first explore the case of quadratic curvature gravity theory for which we derive four classes of asymptotically hyperscaling violation black holes. For each solution, we compute their masses as well as those of their soliton counterparts obtained through a double Wick rotation. Assuming that the partition function has a certain invariance involving the effective spatial dimensionality, a generalized Cardy formula is derived. This latter is shown to correctly reproduce the entropy where the ground state is identified with the soliton. Comparing our formula with the one derived in the standard Einstein gravity case with source, we stress the role played by the effective spatial dimensionality. From this observation, we speculate the general form of the Cardy formula in the case of hyperscaling violation metric for an arbitrary value of the effective spatial dimensionality. Finally, we test the viability of this formula in the case of cubic gravity theory.

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

“…[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].…”

“…In the hyperscaling case, one of the difficulty is to correctly identify the functional dependence of the effective spatial dimension. For example, for Einstein gravity with scalar field source, the effective spatial dimension is d eff = D − 2 − θ, while for higher-order gravity theories, this dependence may be different as shown below, see also [13,14].…”

Cardy formula for charged black holes with anisotropic scaling

“…This task can be interesting by itself in the hyperscaling violation case, since as shown in [29], the spatial effective dimensionality d eff may vary in function of the order of the gravity theories involved in the action.…”