Lifshitz black brane thermodynamics in the presence of a nonlinear electromagnetic field

TL;DR

This paper explores the thermodynamics of Lifshitz black branes influenced by nonlinear electromagnetic fields, deriving key relations and a generalized Smarr formula in asymptotic Lifshitz spacetimes.

Contribution

It introduces a method to analyze Lifshitz black brane thermodynamics with nonlinear electromagnetic fields, including conserved quantities and a generalized Smarr relation.

Findings

Derived conserved quantity relating horizon and infinity parameters.

Calculated energy density using generalized counterterm method.

Established a generalized Smarr formula for Lifshitz black branes.

Abstract

In this paper, we investigate the thermodynamics of Lifshitz black branes in the presence of a nonlinear massless electromagnetic field. We begin by introducing the appropriate action in grand-canonical and canonical ensembles for nonlinear electromagnetic field. The condition on the parameters of the metric for having black brane solutions will be presented. Since the field equations cannot be solved for an arbitrary value of the critical exponent $z$, we obtain a conserved quantity along the $r$ coordinate that enables us to relate the parameters of the metric at the horizon and at infinity. Then, we calculate the energy density of the Lifshitz black brane through the use of the counterterm method generalized for the asymptotic Lifshitz spacetimes. Finally, we present a relation between the energy density and the thermodynamical quantities, electric potential, charge density, temperature and entropy density. This relation is the generalization of Smarr formula for anti-de Sitter black branes.