The first law of AdS black holes in higher curvature gravity

TL;DR

This paper proves that the first law of black hole thermodynamics holds in a broad class of higher curvature gravity theories with AdS asymptotics, extending known results from Einstein gravity.

Contribution

It demonstrates the validity of the first law in higher curvature gravity theories with arbitrary Ricci scalar functions, using conformal and symplectic formalisms.

Findings

Conserved quantities remain unchanged under conformal transformation.

Mass, angular momentum, and entropy satisfy the first law in these theories.

The mass should be defined with respect to a nonrotating timelike Killing vector.

Abstract

We consider the first law of black hole thermodynamics in an asymptotically anti-de Sitter spacetime in the class of gravitational theories whose gravitational Lagrangian is an arbitrary function of the Ricci scalar. We first show that the conserved quantities in this class of gravitational theories constructed through conformal completion remain unchanged under the conformal transformation into the Einstein frame. We then prove that the mass and the angular momenta defined by these conserved quantities, along with the entropy defined by the Noether charge, satisfy the first law of black hole thermodynamics, not only in Einstein gravity but also in the higher curvature gravity within the class under consideration. We also point out that it is naturally understood in the symplectic formalism that the mass satisfying the first law should be necessarily defined associated with the timelike Killing vector nonrotating at infinity. Finally, a possible generalization into a wider class of gravitational theories is discussed.