Smarr Formula for Lovelock Black Holes: a Lagrangian approach

TL;DR

This work presents a general procedure to derive Smarr formulas for black holes in any diffeomorphism-invariant theory by leveraging Wald's Noether-charge formalism, recasting the Smarr identity as a surface term plus a bulk contribution. Applied to Lovelock gravity, it yields a mass–parameter relation $M=\sigma\mu\Omega_{D-2}$ for static black holes and a generalized Smarr formula $(1-\gamma/\sigma)M\doteq TS-\hat{W}$, where the static work term $\hat{W}$ reduces to a boundary integral. The analysis uncovers topological contributions to entropy and the Smarr balance in even dimensions, showing these can be interpreted as artefacts of the formalism that can be absorbed by a modified Noether charge, with implications for the physical meaning of $S^{top}$. The results highlight the utility of Wald's approach for higher-curvature theories while underscoring limitations in rotating cases and the interpretation of topological terms, pointing to future work on extending the framework and refining the entropy prescription.

Abstract

We argue that the Smarr Formula for black holes can be expressed in terms of a Noether charge surface integral plus a suitable volume integral, for any gravitational theory. The integrals can be constructed as an application of Wald's formalism. We apply this formalism to compute the mass and the Smarr Formula for static Lovelock black holes. Finally, we propose a new prescription for Wald's entropy in the case of Lovelock black holes, which takes into account topological contributions to the entropy functional.