An entropy current for dynamical black holes in four-derivative theories of gravity

Abstract

We propose an entropy current for dynamical black holes in a theory with arbitrary four derivative corrections to Einstein's gravity, linearized around a stationary black hole. The Einstein-Gauss-Bonnet theory is a special case of the class of theories that we consider. Within our approximation, our construction allows us to write down a completely local version of the second law of black hole thermodynamics, in the presence of the higher derivative corrections considered here. This ultra-local, stronger form of the second law is a generalization of a weaker form, applicable to the total entropy, integrated over a compact 'time-slice' of the horizon, a proof of which has been recently presented in [1]. We also provide a general algorithm to construct the entropy current for the four derivative theories, which may be straightforwardly generalized to arbitrary higher derivative corrections to Einstein's gravity. This algorithm highlights the possible ambiguities in defining the entropy current.

An entropy current and the second law in higher derivative theories of gravity

“…2 The context and the backdrop of our current work. Our analysis in this paper is significantly motivated and based on the important results that were reported recently in [14] and subsequently in a follow-up paper [22]. Therefore, to better understand the context of this current paper, it is useful and informative that we discuss them a priori.…”

“…Although we would have nothing to say about those situations within this paper's premise, see [19][20][21] where a similar problem has been studied with exciting conclusions. 3 For a detailed discussion the reader is requested to see section 2 of [22] where a comprehensive review of [14] is given.…”

“…As a check of its consistency, this entropy function defined locally in time should reduce to the known expression of Wald entropy for stationary configurations once the dynamics are switched off. However, in [22], a subtle technical issue of this construction regarding its implementation was pointed out. Actually, this constructive algorithm of [14] stumbles upon a road-block at the very leading order in amplitude expansion.…”

“…The main goal of the analysis carried out in [22] was to perform a brute force calculation focussing on only four derivative theories of gravity to check if the arguments presented in [14] are indeed true. By explicitly working out the "time-time" component of the equations of motion for those theories, it was shown that there are terms, in that zero boost-weight sector of it, which cannot be cast in the form required to identify them as the Wald entropy density.…”

General theory of large D membranes consistent with second law of thermodynamics

“…The authors of this paper used the physical process version of first law to arrive at this correction. The authors of [18] then showed that the requirement of physical process first law can be relaxed by correctly capturing the presence of a spacial entropy current on the horizon along with the entropy density derived in [16,17]. In a different context the authors of [19] found a correction to the Wald entropy which satisfies the second law for non-linear but spherical dynamics of black holes.…”