# Nonlinear Gravity from Entanglement in Conformal Field Theories

## TL;DR

This paper shows that nonlinear gravitational equations can emerge from the entanglement structure of conformal field theories, linking quantum entanglement with spacetime geometry without relying on AdS/CFT assumptions.

## Contribution

It demonstrates that Einstein's equations can be derived perturbatively from entanglement entropy in CFT states, establishing a direct connection between entanglement and gravity.

## Key findings

- Entanglement entropy corresponds to asymptotically AdS geometries for sourced CFT states.
- Perturbative Einstein's equations are satisfied up to second order in sources.
- Spacetime and gravity can emerge from entanglement in conformal field theories.

## 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.