RG flow and thermodynamics of causal horizons in higher-derivative AdS gravity

Abstract

In arXiv:1508.01343 [hep-th], one of the authors proposed that in AdS/CFT the gravity dual of the boundary c-theorem is the second law of thermodynamics satisfied by causal horizons in AdS and this was verified for Einstein gravity in the bulk. In this paper we verify this for higher derivative theories. We pick up theories for which an entropy expression satisfying the second law exists and show that the entropy density evaluated on the causal horizon in a RG flow geometry is a holographic c-function. We also prove that given a theory of gravity described by a local covariant action in the bulk a sufficient condition to ensure holographic c-theorem is that the second law of causal horizon thermodynamics be satisfied by the theory. This allows us to explicitly construct holographic c-function in a theory where there is curvature coupling between gravity and matter and standard null energy condition cannot be defined although second law is known to hold. Based on the duality between c-theorem and the second law of causal horizon thermodynamics proposed in arXiv:1508.01343 [hep-th] and the supporting calculations of this paper we conjecture that every Unitary higher derivative theory of gravity in AdS satisfies the second law of causal horizon thermodynamics. If this is not true then c-theorem will be violated in a unitary Lorentz invariant field theory.

Black hole singularity, generalized (holographic) c-theorem and entanglement negativity

“…In the presence of higher derivative terms instead of Bekenstein-Hawking entropy we have to use the entropy expression which satisfies the second law in the bulk and reduces to the Wald entropy when evaluated on a Killing horizon [44]. If we do this we will recover the a-charge at the asymptotic UV boundary as was shown in [12]. That means the a-function will start decreasing from a U V .…”

“…The holographic picture is based on the fact that the gravity dual of c-theorem is the second law of causal horizon thermodynamics in asymptotically AdS spaces [11,12]. In a nutshell, second law for causal horizons say that if we consider the future bulk light-cone of a boundary point then the expansion of the null geodesic generators of the light-cone is negative [17,18].…”

“…This is essentially holographic c-theorem [5][6][7][8][9][10] if we specialize to a domain-wall geometry. The bulk future light-cone interpolates between the UV-AdS and the IR-AdS and the monotonically decreasing Bekenstein-Hawking entropy density gives the holographic c-function [11,12].…”

“…It has curvature singularity hidden behind the black brane horizon and the geometry is not Lorentz invariant except near the AdS 5 boundary. The c-function we construct is just the Bekenstein-Hawking entropy density of a causal horizon in the black brane geometry [11,12]. The causal horizon originates at some point of the AdS boundary and terminates at the curvature singularity.…”

A holographic study of the a-theorem and RG flow in general quadratic curvature gravity

“…In [30] an a-function is introduced by using the Jacobson-Myers (JM) entropy functional. The non-increasing behavior of this function follows from the fact that the JM entropy functional satisfies the linearized second law of the causal horizon thermodynamics.…”