The Noether charge entropy in anti-deSitter space and its field theory dual

TL;DR

This paper connects the Noether charge entropy of black branes in anti-deSitter space to local operators in the bulk, modifies Wald's entropy formula away from the horizon, and explores its dual field theory interpretation via AdS/CFT.

Contribution

It provides a modified expression for Wald's entropy away from the horizon and establishes its dual in the boundary field theory, considering theories asymptoting to Einstein gravity.

Findings

Modified Wald entropy includes an additional horizon-vanishing term.

The boundary entropy density relates to bulk operators via AdS/CFT.

Entropy variation reflects renormalization of gravitational couplings.

Abstract

We express the Noether charge entropy density of a black brane in anti-deSitter space in terms of local operators in the anti-deSitter space bulk. We find that Wald's expression for the Noether charge entropy needs to be modified away from the horizon by an additional term that vanishes on the horizon. We then determine the field theory dual of the Noether charge entropy for theories that asymptote to Einstein theory. We do so by calculating the value of the entropy density at the anti-deSitter space boundary and applying the standard rules of the AdS/CFT correspondence. We interpret the variation of the entropy density operator from the horizon to the boundary as due to the renormalization of the effective gravitational couplings as they flow from the ultra-violet to the infra-red. We discuss the cases of Einstein-Hilbert theory and f(R) theories in detail and make general comments about more complicated cases.