Linearized Einstein's Equation around pure BTZ from Entanglement Thermodynamics

TL;DR

This paper demonstrates that Einstein's equations around a BTZ black hole in AdS3 can be derived from the first law of entanglement entropy in the dual thermal CFT2, extending previous vacuum state results.

Contribution

It extends the derivation of Einstein's equations from entanglement thermodynamics to thermal states dual to BTZ black holes in AdS3/CFT2.

Findings

Einstein's equations follow from the first law of entanglement in a thermal state.

The modular Hamiltonian for the thermal CFT2 state is utilized in the derivation.

The approach applies linearized perturbations around the BTZ black hole geometry.

Abstract

It is known that the linearized Einstein's equation around the pure $AdS$ can be obtained from the constraint $ \Delta S = \Delta\left< H \right> $, known as the first law of entanglement, on the boundary $CFT$. The corresponding dual state in the boundary $CFT$ is the vacuum state around which the linear perturbation is taken. In this paper we revisit this question, in the context of $ {AdS}_3/{CFT}_2 $, with the state of the boundary ${CFT}_2$ as a thermal state. The corresponding dual geometry is a planar BTZ black hole. By considering the linearized perturbation around this black brane we show that Einstein's equation follows from the first law of entanglement. The modular Hamiltonian in a thermal state of the ${CFT}_2$ that we have used has been recently found in arXiv:1608.01283 [cond-mat.stat-mech].