Generalised Smarr Formula and the Viscosity Bound for Einstein-Maxwell-Dilaton Black Holes

TL;DR

This paper explores the relationship between the shear viscosity to entropy ratio and a generalized Smarr relation in Einstein-Maxwell-Dilaton black holes, showing that the bound saturation correlates with a specific thermodynamic relation.

Contribution

It demonstrates that a generalized Smarr relation exists for these black holes and links it to the viscosity bound saturation, using two different derivation methods.

Findings

A generalized Smarr relation holds for Einstein-Maxwell-Dilaton black holes.

The Smarr relation is connected to the saturation of the viscosity to entropy bound.

Two methods independently derive the Smarr relation, confirming its robustness.

Abstract

We study the shear viscosity to entropy ratio $\eta/S$ in the boundary field theories dual to black hole backgrounds in theories of gravity coupled to a scalar field, and generalisations including a Maxwell field and non-minimal scalar couplings. Motivated by the observation in simple examples that the saturation of the $\eta/S\ge 1/(4\pi)$ bound is correlated with the existence of a generalised Smarr relation for the planar black-hole solutions, we investigate this in detail for the general black-hole solutions in these theories, focusing especially on the cases where the scalar field plays a non-trivial role and gives rise to an additional parameter in the space of solutions. We find that a generalised Smarr relation holds in all cases, and in fact it can be viewed as the bulk gravity dual of the statement of the saturation of the viscosity to entropy bound. We obtain the generalised Smarr relation, whose existence depends upon a scaling symmetry of the planar black-hole solutions, by two different but related methods, one based on integrating the first law of thermodynamics, and the other based on the construction of a conserved Noether charge.