Smarr Formula and an Extended First Law for Lovelock Gravity

TL;DR

This work derives a Smarr formula for static, asymptotically AdS black holes in Lovelock gravity by applying a scale perturbation to an extended first law that now includes variations of the dimensionful Lovelock couplings. Central to the construction are Killing-Lovelock potentials, which yield finite boundary contributions and define the thermodynamic potentials $\Psi^{(k)}$ conjugate to each coupling $b_k$; the resulting Smarr relation expresses the mass in terms of $T S$ and these Lovelock-driven contributions. The analysis resolves divergences inherent in AdS spacetimes and provides a geometric interpretation of the new potentials, including renormalized volume-type terms $\Theta^{(k)}$. The findings offer a framework to study Hawking-Page transitions and AdS/CFT aspects in general Lovelock theories, and set the stage for extensions to rotating black holes and more intricate phase structure.

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.