On the thermodynamics of periodic AdS black branes

TL;DR

This paper derives a simple formula for how the free energy of asymptotically AdS black branes changes with spatial period variations, clarifying thermodynamic properties of dual CFTs with broken translational symmetry.

Contribution

It provides a new, straightforward expression for free-energy variation in periodic AdS black branes, linking thermodynamics and boundary stress tensor relations.

Findings

Derived a simple free-energy variation formula.

Connected thermodynamics with boundary stress tensor relations.

Clarified properties of black branes with broken translational symmetry.

Abstract

We consider asymptotically AdS black brane solutions that are dual to CFTs with periodic dependence on the spatial directions, arising from either a spontaneous or an explicit breaking of translational symmetry. We derive a simple expression for the variation of the free-energy with respect to changing the periods. This explains some observations, based on numerics, that have arisen in the explicit construction of thermodynamically preferred black holes in the case that the spatial directions are infinite in extent and the symmetry is spontaneously broken. It also leads to simple Smarr-type relations involving the boundary stress tensor..