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

Abstract

We write down the most general membrane equations dual to black holes for a general class of gravity theories, up to sub-leading order in 1/D in large D limit. We derive a “minimal” entropy current which satisfies a local form of second law from these membrane equations. We find that consistency with second law requires the membrane equations to satisfy certain constraints. We find additional constraints on the membrane equations from the existence of membrane solutions dual to stationary black holes. Finally we observe a tension between second law and matching with Wald entropy for dual stationary black hole configurations, for the minimal entropy current. We propose a simple modification of the membrane entropy current so that it satisfies second law and also the stationary membrane entropy matches the Wald entropy.

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

“…37 Thus, the results in this section are validation of the proof provided above. 37 It must be noted that there is an overall negative sign in the equations of motion used in the calculations above ((i.e.) Evv) when compared to the expressions used in [22].…”

“…To see this, we must compare the expressions written above for J v and J i as listed in table 7 with section 3.1 in [22]. 37 Thus, the results in this section are validation of the proof provided above. 37 It must be noted that there is an overall negative sign in the equations of motion used in the calculations above ((i.e.)…”

“…Before we proceed, let us also mention that in the literature, the concept of an entropy current in the context of dynamical black holes has already been introduced; see [33][34][35][36][37]. Although they share the common primary goal in terms of constructing a local entropy current, the methodology and the working principles that we are using in our paper are very different from theirs.…”

“…For example, in [33] the context was to interpret the field equations of gravity as an equation of state, the primary motivation behind [34,35] has mainly been the celebrated fluid gravity correspondence [38]. On the other hand, a certain duality between JHEP09(2021)169 membrane dynamics and black holes called the membrane-gravity duality had been the driving principle for [36,37].…”

Large D black holes in an environment

“…The broader goal of this project is to understand the properties of the membranes dual to black hole solutions in the most generic situation, e.g., in [15] it was established that the general structures of the dual membrane equations for black holes in arbitrary higher derivative theories of gravity are constrained by the second law of thermodynamics. By studying black holes in presence of generic matter fields we hope to uncover constraints on matter stress tensors (beyond the null energy conditions).…”