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Entanglement Equilibrium and the Einstein Equation

Ted Jacobson

TL;DR

Jacobson demonstrates a deep link between the semiclassical Einstein equation and a maximal vacuum entanglement hypothesis (MVEH): in a small geodesic ball, the entanglement entropy is maximized at fixed volume when geometry and fields depart from maximal symmetry. By analyzing area deficits of geodesic balls and the entanglement structure of causal diamonds through a conformal Killing flow, the paper derives that Newton's constant is fixed by the universal entanglement density $\eta$, and the Einstein equation emerges as the condition for first-order entropy stationarity. For conformal fields, entropy stationarity is equivalent to the Einstein equation; for nonconformal fields a conjectured extra term $\delta X$ in the modular Hamiltonian modifies the relation but preserves the overall structure. The work provides an entanglement-based, microscopic perspective on gravity in the semiclassical regime, connecting horizon thermodynamics, conformal symmetry, and causal diamonds to spacetime dynamics.

Abstract

A link between the semiclassical Einstein equation and a maximal vacuum entanglement hypothesis is established. The hypothesis asserts that entanglement entropy in small geodesic balls is maximized at fixed volume in a locally maximally symmetric vacuum state of geometry and quantum fields. A qualitative argument suggests that the Einstein equation implies validity of the hypothesis. A more precise argument shows that, for first-order variations of the local vacuum state of conformal quantum fields, the vacuum entanglement is stationary if and only if the Einstein equation holds. For nonconformal fields, the same conclusion follows modulo a conjecture about the variation of entanglement entropy.

Entanglement Equilibrium and the Einstein Equation

TL;DR

Jacobson demonstrates a deep link between the semiclassical Einstein equation and a maximal vacuum entanglement hypothesis (MVEH): in a small geodesic ball, the entanglement entropy is maximized at fixed volume when geometry and fields depart from maximal symmetry. By analyzing area deficits of geodesic balls and the entanglement structure of causal diamonds through a conformal Killing flow, the paper derives that Newton's constant is fixed by the universal entanglement density , and the Einstein equation emerges as the condition for first-order entropy stationarity. For conformal fields, entropy stationarity is equivalent to the Einstein equation; for nonconformal fields a conjectured extra term in the modular Hamiltonian modifies the relation but preserves the overall structure. The work provides an entanglement-based, microscopic perspective on gravity in the semiclassical regime, connecting horizon thermodynamics, conformal symmetry, and causal diamonds to spacetime dynamics.

Abstract

A link between the semiclassical Einstein equation and a maximal vacuum entanglement hypothesis is established. The hypothesis asserts that entanglement entropy in small geodesic balls is maximized at fixed volume in a locally maximally symmetric vacuum state of geometry and quantum fields. A qualitative argument suggests that the Einstein equation implies validity of the hypothesis. A more precise argument shows that, for first-order variations of the local vacuum state of conformal quantum fields, the vacuum entanglement is stationary if and only if the Einstein equation holds. For nonconformal fields, the same conclusion follows modulo a conjecture about the variation of entanglement entropy.

Paper Structure

This paper contains 10 sections, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Area deficit and general relativity
  3. Causal diamond and conformal isometry
  4. Entanglement entropy of a diamond
  5. Equilibrium and the Einstein equation
  6. Discussion
  7. Area deficit calculation
  8. Conformal isometry of a flat causal diamond
  9. Entropy of de Sitter space
  10. First law of causal diamond mechanics

Figures (1)

  • Figure 1: Causal diamond, in a maximally symmetric spacetime, for a geodesic ball $\Sigma$ of radius $\ell$ with center $o$ and boundary $\partial \Sigma$. The dashed curves are flow lines of $\zeta$, the conformal Killing vector field, whose flow preserves the diamond and which vanishes at the top and bottom vertices and on $\partial\Sigma$. The vectors show $\zeta$ at four points of $\Sigma$.