Lifshitz-like Magnetic Black Branes: Third Law of Thermodynamics and the Null Energy Condition

TL;DR

This paper develops a method to solve Einstein-dilaton-Maxwell models analytically, applies it to three models, and investigates the relationship between the null energy condition and the third law of thermodynamics.

Contribution

It introduces a potential reconstruction approach for solving these models and analyzes the correlation between NEC and the third law across different setups.

Findings

Models I and II show no correlation between NEC and the third law.

Model III exhibits that NEC validity implies the third law validity.

Explicit blackening functions enable detailed third law analysis.

Abstract

We develop a procedure to solve Einstein-dilaton-Maxwell models in quadratures using the potential reconstruction approach. We then apply this procedure to three distinct models, examining both the null energy condition (NEC) and the validity of the third law of thermodynamics in each case. The explicit knowledge of the blackening function -- as opposed to relying solely on numerical data -- allows us to discuss the validity of the third law in detail. The three models considered are: (I) a 5D model with two Maxwell fields, featuring anisotropy specified by a Gaussian function and a Lifshitz function; (II) the same 5D model as in (I), but with anisotropy parametrized by two Lifshitz parameters; and (III) a 6D model with one 2-form and one 3-form field, with the metric parametrized by two Lifshitz parameters. We show that for models I and II the parameter regions, where both the NEC and the third law are satisfied, exhibit no correlation between the two conditions. In contrast, for model III the validity of the NEC implies the validity of the third law.