Nonlinear gravity from entanglement in conformal field theories

Thomas Faulkner,Eliot Hijano,Felix M Haehl,Mark Van Raamsdonk,Onkar Parrikar,Charles Rabideau

doi:10.1007/jhep08(2017)057

Abstract

In this paper, we demonstrate the emergence of nonlinear gravitational equations directly from the physics of a broad class of conformal field theories. We consider CFT excited states defined by adding sources for scalar primary or stress tensor operators to the Euclidean path integral defining the vacuum state. For these states, we show that up to second order in the sources, the entanglement entropy for all ball-shaped regions can always be represented geometrically (via the Ryu-Takayanagi formula) by an asymptotically AdS geometry. We show that such a geometry necessarily satisfies Einstein’s equations perturbatively up to second order, with a stress energy tensor arising from matter fields associated with the sourced primary operators. We make no assumptions about AdS/CFT duality, so our work serves as both a consistency check for the AdS/CFT correspondence and a direct demonstration that spacetime and gravitational physics can emerge from the description of entanglement in conformal field theories.

Highlights

Despite a great deal of evidence for the validity of the correspondence, our understanding of these questions is far from complete
In [8, 9], it was demonstrated that any spacetime geometry that correctly captures the entanglement entropy of a near-vacuum CFT state to first order in the perturbation must satisfy Einstein’s equations linearized about Anti de Sitter space
After motivating a class of CFT states with this property, we will show that with only a minimal assumption about the CFT, a spacetime which captures via the HRRT formula the entanglement entropy of such a state up to second order in perturbations around the vacuum state must satisfy Einstein’s equations perturbed to second order about AdS

Summary

Introduction

Despite a great deal of evidence for the validity of the correspondence, our understanding of these questions is far from complete. A surprising by-product of our analysis is that a spacetime satisfying Einstein’s equations and capturing the entanglement entropy for ball-shaped regions always exists as long as the CFT satisfies a single constraint relating the overall coefficient CT in the stress-tensor two-point function to an overall coefficient a∗ appearing in the vacuum entanglement entropy for a ball. This is a manifestation of the fact that to second order in the state deformation, the ball-entanglement entropies expressed in terms of CFT one-point functions are given by a nearly universal formula, depending only on the coefficients CT and a∗. If the ball entanglement entropies at this order are geometrical for certain CFTs, they must be geometrical for all CFTs with the same relation between these coefficients

Objectives

Results

Discussion

Conclusion