Cardy formula for charged black holes with anisotropic scaling

TL;DR

This paper extends the Lifshitz Cardy formula to charged black holes with anisotropic scaling, relating their thermodynamic properties and validating the formula across various black hole solutions with different scaling behaviors.

Contribution

The paper introduces a generalized Cardy formula for charged Lifshitz black holes, incorporating charge, dynamical exponent, and ground state energy, and tests it across multiple scenarios.

Findings

Validates the extended Cardy formula for various Lifshitz black holes.

Shows the formula's consistency with known Smarr relations.

Extends the approach to hyperscaling violation and charged AdS black holes.

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.