A Second Law for higher curvature gravity

Abstract

The Second Law of black hole thermodynamics is shown to hold for arbitrarily complicated theories of higher curvature gravity, so long as we allow only linearized perturbations to stationary black holes. Some ambiguities in Wald's Noether charge method are resolved. The increasing quantity turns out to be the same as the holographic entanglement entropy calculated by Dong. It is suggested that only the linearization of the higher-curvature Second Law is important, when consistently truncating a UV-complete quantum gravity theory.You've just invented a new theory of gravity. Like Einstein's, your theory is generally covariant and formulated in terms of a metric g ab . But the action is more complicated; it is an arbitrary function of the Riemann curvature tensor, perhaps some scalars φ, and their derivatives:(1) where L g is the exciting gravitational piece, while L m is a boring minimally coupled matter sector obeying the null energy condition T ab k a k b ≥ 0 (NEC) (k a being null). Such "higher curvature" corrections are known to occur with small coefficients due to quantum and/or stringy corrections.The equation of motion from varying the metric isTo make up for all the excitement in the gravitational sector, you decide the matter sector should be an ordinary field theory minimally coupled to g ab , obeying the null energy condition T ab k a k b ≥ 0 (NEC) for k a null. A lesser mind would ask whether this theory is in agreement with observation, or perhaps whether the vacuum is even stable. But not you! You are concerned with a far deeper question: do black holes in your theory still obey the Second Law of horizon thermodynamics?In GR, the NEC (plus a version of cosmic censorship) implies that the area of any future event horizon H is always increasing [1]. So Bekenstein [2] postulated that black holes have entropy proportional to their area; Hawking radiation [3] showed that it was more than just an analogy, and that in turn had all kinds of ramifications [4]! Does this only make sense for the Einstein-Hilbert action, or is it true more broadly? We shall see that there is indeed a Second Law in all such theories of higher curvature gravity theories, provided that you only consider linearized perturbations δg ab , δφ of the gravitational fields (possibly sourced by a first order perturbation to δT ab ) evaluated on a stationary black hole background (or more precisely, a bifurcate Killing horizon 1 ). This has previously been done for f (Lovelock) gravity [5], and quadratic curvature gravity [6].Pick a gauge so that u and v are null coordinates which increase as one moves spacelike away from the horizon, so that u = 0 is the future horizon H, v is an affine parameter along the null generators of the horizon, v = 0 is the past horizon, i, j indices point in the D − 2 transverse directions, the metric obeysand the Killing symmetry acts like a standard Lorentz boost on the null coordinates:The Killing weight n of a tensor (with all indices lowered) is given by the number of v-indices minus the number of ...

Deriving covariant holographic entanglement

“…As discussed in [39,40] the holographic entanglement entropy functionals serve as a good starting point to examine the second law for higher derivative black hole entropy. The discussion thus far has been confined to the linear response regime of small amplitude fluctuations away from equilibrium.…”

“…These solutions are completely real in the Rindler wedges, but, for even q, they might present some imaginary phases in the Milne wedges. 40 Once one has a solution, after regularizing the bulk properly, one should be able to compute the action. This procedure was carried out explicitly in d = 2 from the Euclidean perspective in [38,44], although the explicit evaluation of this action is far from being trivial because (among other things) of the regularization of the bulk.…”

“…39 We are going to assume that at finite q there are no relevant contributions from bulk singularities (if any). 40 Using the coordinates in (3.11), the asymptotic boundary tells us that the Milne wedges are reached by r → i −1 r, τ → τ + i π 2…”

Entropy increases at linear order in scalar-hairy Lovelock gravity

“…Using this property, by analogy with the JM entropy of the purely Lovelock gravity, we introduce a new formula of the entropy after adding the scalar field and showed that this JM entropy increases in Vaidya-like black hole solution for the scalar-hairy Lovelock gravity under first-order approximation. Moreover, we showed that different from the entropy in F (Riemann) gravity obtained in [29], here the difference between the JM entropy and Wald entropy contains the correction from the scalar field.…”

“…If we restrict attention to linearized metric perturbations to stationary black holes, it has been shown that the Jacobson-Myers (JM) entropy of the Lovelock gravity and the holographic entropy of quadratic curvature gravity obey the second law [25][26][27][28]. More generally, Wall gave a general method to evaluate the corrected entropy which satisfies the linearized second law and showed that it takes the form [29]…”