Smarr Formula and an Extended First Law for Lovelock Gravity

TL;DR

This paper derives a generalized Smarr formula and an extended first law for static AdS black holes in Lovelock gravity, incorporating the effects of dimensionful Lovelock couplings and their variations.

Contribution

It introduces an extended first law that accounts for Lovelock coupling variations and provides a geometric derivation of the Smarr formula in Lovelock gravity.

Findings

Derived a Smarr formula relating mass and geometric quantities.

Extended the first law to include Lovelock coupling variations.

Provided geometric expressions for thermodynamic potentials.

Abstract

We study properties of static, asymptotically AdS black holes in Lovelock gravity. Our main result is a Smarr formula that gives the mass in terms of geometrical quantities together with the parameters of the Lovelock theory. As in Einstein gravity, the Smarr formula follows from applying the first law to an infinitesimal change in the overall length scale. However, because the Lovelock couplings are dimensionful, we must first prove an extension of the first law that includes their variations. Key ingredients in this construction are the Killing-Lovelock potentials associated with each of the the higher curvature Lovelock interactions. Geometric expressions are obtained for the new thermodynamic potentials conjugate to variation of the Lovelock couplings.