Gravitation from Entanglement in Holographic CFTs

TL;DR

The paper demonstrates that the entanglement first law in holographic CFTs, for ball-shaped regions, is equivalent to the linearized gravitational equations in the AdS bulk. By leveraging the Iyer–Wald formalism and the Wald entropy functional, it extends the derivation from Einstein gravity to general higher curvature theories, deriving the holographic stress tensor from entanglement entropy and energy relations. It provides explicit computations in R^2 and R^4 gravity to illustrate how higher-derivative couplings modify the holographic dictionary, including an analysis of Fefferman–Graham expansions and additional FG modes. The work offers a conceptually transparent route from quantum information to bulk dynamics and suggests avenues for nonlinear and quantum generalizations of the holographic correspondence.

Abstract

Entanglement entropy obeys a 'first law', an exact quantum generalization of the ordinary first law of thermodynamics. In any CFT with a semiclassical holographic dual, this first law has an interpretation in the dual gravitational theory as a constraint on the spacetimes dual to CFT states. For small perturbations around the CFT vacuum state, we show that the set of such constraints for all ball-shaped spatial regions in the CFT is exactly equivalent to the requirement that the dual geometry satisfy the gravitational equations of motion, linearized about pure AdS. For theories with entanglement entropy computed by the Ryu-Takayanagi formula $S=A/(4G_N)$, we obtain the linearized Einstein equations. For theories in which the vacuum entanglement entropy for a ball is computed by more general Wald functionals, we obtain the linearized equations for the associated higher-curvature theories. Using the first law, we also derive the holographic dictionary for the stress tensor, given the holographic formula for entanglement entropy. This method provides a simple alternative to holographic renormalization for computing the stress tensor expectation value in arbitrary higher derivative gravitational theories.

Figures (3)

• Figure 1: Causal development $\mathcal{D}$ (left) of a ball-shaped region $B$ on a spatial slice of Minkowski space, showing the evolution generated by $H_B$. A conformal transformation maps $D$ to a hyperbolic cylinder $H^{d-1} \times$ time (right), taking $H_B$ to the ordinary Hamiltonian for the CFT on $H^{d-1}$.
• Figure 2: AdS-Rindler patch associated with a ball $B(R, x_0)$ on a spatial slice of the boundary. Solid blue paths indicate the boundary flow associated with $H_B$ and the conformal Killing vector $\zeta$. Dashed red paths indicate the action of the Killing vector $\xi$.
• Figure 3: Notation for regions in AdS$_{d+1}$, with radial coordinate $z$ and boundary space coordinate $\vec{x}$. $B(R,x_0)$ is the $(d-1)$-dimensional ball on the $z=0$ boundary of radius $R$ centered at $\vec{x}_0$ on the spatial slice at time $t_0$. $\tilde{B}$ is the $(d-1)$-dimensional hemispherical surface in AdS ending on $\partial B$, and $\Sigma$ is the enclosed $d$-dimensional spatial region.