The first law of entanglement entropy in AdS black hole backgrounds

TL;DR

This paper investigates the first law of entanglement entropy in CFTs dual to AdS black hole backgrounds, demonstrating its validity in various dimensions and boundary conditions using the AdS/CFT correspondence.

Contribution

It establishes the first law of entanglement entropy in odd-dimensional AdS black hole backgrounds and explores boundary conditions affecting energy flux in the dual CFT.

Findings

First law holds in (2+1)D CFT with metric perturbations.

First law is valid in (4+1)D CFT under Neumann boundary conditions.

Constructed asymptotic geometry of 6D AdS black droplet showing negative energy at infinity.

Abstract

The first law for entanglement entropy in CFT in an odd-dimensional asymptotically AdS black hole is studied by using the AdS/CFT duality. The entropy of CFT considered here is due to the entanglement between two subsystems separated by the horizon of the AdS black hole, which itself is realized as the conformal boundary of a black droplet in even-dimensional global AdS bulk spacetime. In $(2+1)$-dimensional CFT, the first law is shown to be always satisfied by analyzing a class of metric perturbations of the exact solution of a $4$-dimensional black droplet. In $(4+1)$-dimensions, the first law for CFT is shown to hold under the Neumann boundary condition at a certain bulk hypersurface anchored to the conformal boundary of the boundary AdS black hole. From the boundary view point, this Neumann condition yields there being no energy flux across the boundary of the boundary AdS black hole. Furthermore, the asymptotic geometry of a $6$-dimensional small AdS black droplet is constructed as the gravity dual of our $(4+1)$-dimensional CFT, which exhibits a negative energy near the spatial infinity, as expected from vacuum polarization.