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Universality of Gravity from Entanglement

Brian Swingle, Mark Van Raamsdonk

TL;DR

The paper investigates how gravity with matter emerges universally from entanglement in holographic CFTs by extending the entanglement first law to subleading $1/N$ corrections. Using the holographic entanglement entropy formula that includes bulk entanglement, it derives a local bulk constraint that the linearized Einstein equations are sourced by the bulk stress-energy tensor, thereby recovering Newtonian gravity in appropriate limits. It argues that gravity’s universality follows from the fact that all bulk degrees of freedom contribute to entanglement, and discusses the implications for non-linear gravity and potential CFT-based routes to the holographic dictionary via entanglement renormalization. The work provides a concrete mechanism linking quantum information structure in the boundary theory to classical spacetime dynamics in the bulk, reinforcing the central role of entanglement in the emergence of gravity.

Abstract

The entanglement "first law" in conformal field theories relates the entanglement entropy for a ball-shaped region to an integral over the same region involving the expectation value of the CFT stress-energy tensor, for infinitesimal perturbations to the CFT vacuum state. In recent work, this was exploited at leading order in $N$ in the context of large N holographic CFTs to show that any geometry dual to a perturbed CFT state must satisfy Einstein's equations linearized about pure AdS. In this note, we investigate the implications of the leading 1/N correction to the exact CFT result. We show that these corrections give rise to the source term for the gravitational equations: for semiclassical bulk states, the expectation value of the bulk stress-energy tensor appears as a source in the linearized equations. In particular, the CFT first law leads to Newton's Law of gravitation and the fact that all sources of stress-energy source the gravitational field. In our derivation, this universality of gravity comes directly from the universality of entanglement (the fact that all degrees of freedom in a subsystem contribute to entanglement entropy).

Universality of Gravity from Entanglement

TL;DR

The paper investigates how gravity with matter emerges universally from entanglement in holographic CFTs by extending the entanglement first law to subleading corrections. Using the holographic entanglement entropy formula that includes bulk entanglement, it derives a local bulk constraint that the linearized Einstein equations are sourced by the bulk stress-energy tensor, thereby recovering Newtonian gravity in appropriate limits. It argues that gravity’s universality follows from the fact that all bulk degrees of freedom contribute to entanglement, and discusses the implications for non-linear gravity and potential CFT-based routes to the holographic dictionary via entanglement renormalization. The work provides a concrete mechanism linking quantum information structure in the boundary theory to classical spacetime dynamics in the bulk, reinforcing the central role of entanglement in the emergence of gravity.

Abstract

The entanglement "first law" in conformal field theories relates the entanglement entropy for a ball-shaped region to an integral over the same region involving the expectation value of the CFT stress-energy tensor, for infinitesimal perturbations to the CFT vacuum state. In recent work, this was exploited at leading order in in the context of large N holographic CFTs to show that any geometry dual to a perturbed CFT state must satisfy Einstein's equations linearized about pure AdS. In this note, we investigate the implications of the leading 1/N correction to the exact CFT result. We show that these corrections give rise to the source term for the gravitational equations: for semiclassical bulk states, the expectation value of the bulk stress-energy tensor appears as a source in the linearized equations. In particular, the CFT first law leads to Newton's Law of gravitation and the fact that all sources of stress-energy source the gravitational field. In our derivation, this universality of gravity comes directly from the universality of entanglement (the fact that all degrees of freedom in a subsystem contribute to entanglement entropy).

Paper Structure

This paper contains 21 sections, 33 equations, 2 figures.

Table of Contents

  1. Overview
  2. Basic machinery
  3. The entanglement "first law" for CFTs
  4. Entanglement in holographic CFTs
  5. Argument for linearized equations coupled to matter
  6. Review of the leading order results
  7. Bulk matter from $1/N$ corrections to holographic entanglement entropy.
  8. Corrections to $\delta S_B^{\text{grav}}$
  9. Corrections to $\delta E_B^{\text{grav}}$
  10. The main argument
  11. Discussion
  12. Domain of validity of the result
  13. Origin of the universality of gravity
  14. From the linearized equations to the non-linear equations
  15. Holographic entanglement dictionary from entanglement renormalization
  16. ...and 6 more sections

Figures (2)

  • Figure 1: Bulk setup for deriving local equations from the entanglement first law. Dashed curves show the flow generated by the Killing vector $\xi$.
  • Figure 2: Panel (1) shows the geometrical setup we are considering. Panel (2) represents the effect of a conformal transformation (whose inverse is analogous to stereographic projection) which maps the infinite line to a sphere and the region $A$ to half the sphere. Panel (3) represents the effect of yet another conformal transformation where this time we send one of the intersections of $A$ and $\tilde{A}$ to infinity.