First law and anisotropic Cardy formula for three-dimensional Lifshitz black holes

TL;DR

This paper confirms that the entropy of three-dimensional Lifshitz black holes can be derived from an anisotropic Cardy formula, using new solutions and a nonminimal scalar field in massive gravity, validating the formula's general applicability.

Contribution

It introduces new Lifshitz black hole and soliton solutions with dynamical exponent z=3 and verifies the anisotropic Cardy formula for their entropy.

Findings

Confirmed the anisotropic Cardy formula for multiple Lifshitz black holes.

Computed black hole and soliton masses using a quasilocal conserved charge method.

Validated the first law of thermodynamics and Smarr relation for these solutions.

Abstract

The aim of this paper is to confirm in new concrete examples that the semiclassical entropy of a three-dimensional Lifshitz black hole can be recovered through an anisotropic generalization of the Cardy formula derived from the growth of the number of states of a boundary non-relativistic field theory. The role of the ground state in the bulk is played by the corresponding Lifshitz soliton obtained by a double Wick rotation. In order to achieve this task, we consider a scalar field nonminimally coupled to new massive gravity for which we study different classes of Lifshitz black holes as well as their respective solitons, including new solutions for a dynamical exponent z=3. The masses of the black holes and solitons are computed using the quasilocal formulation of conserved charges recently proposed by Gim, Kim, Kulkarni and Yi and based on the off-shell extension of the ADT formalism. We confirm the anisotropic Cardy formula for each of these examples, providing a stronger base for its general validity. Consistently, the first law of thermodynamics together with a Smarr formula are also verified.