Entropic Counterpart of Perturbative Einstein Equation

TL;DR

This paper derives a diffeomorphism invariant formulation of linearized Einstein equations using holographic entanglement entropy, analyzing small fluctuations around various black hole backgrounds in AdS space.

Contribution

It introduces a novel entropic reformulation of linearized Einstein equations in holography, connecting boundary entanglement changes to bulk gravitational dynamics.

Findings

Reformulation of Einstein equations in terms of entanglement entropy

Constraints on entanglement entropy from linearized gravity

Entanglement entropy of boosted subsystems encodes metric components

Abstract

Entanglement entropy in a field theory, with a holographic dual, may be viewed as a quantity which encodes the diffeomorphism invariant bulk gravity dynamics. This, in particular, indicates that the bulk Einstein equations would imply some constraints for the boundary entanglement entropy. In this paper we focus on the change in entanglement entropy, for small but arbitrary fluctuations about a given state, and analyze the constraints imposed on it by the perturbative Einstein equations, linearized about the corresponding bulk state. Specifically, we consider linear fluctuations about BTZ black hole in 3 dimension, pure AdS and AdS Schwarzschild black holes in 4 dimensions and obtain a diffeomorphism invariant reformulation of linearized Einstein equation in terms of holographic entanglement entropy. We will also show that entanglement entropy for boosted subsystems provides the information about all the components of the metric with a time index.