Entanglement First Law and Conditional Linearized Field-Equation Inference

The entanglement first law can imply the linearized Einstein equation about vacuum AdS when it is combined with the RT dictionary, the boundary stress-tensor map, equality for every boundary ball, appropriate boundary conditions, and an injective geometric transform. The information identity alone contains none of those gravitational inputs, and the argument neither proves nonlinear dynamics nor runs automatically in reverse.

Required background. First Law of Entanglement and Quadratic Corrections supplies $\delta S=\delta\langle K\rangle$. The Ryu–Takayanagi Formula supplies the leading entropy/area dictionary.

Helpful background. The Semiclassical Einstein Equation distinguishes the target equation. Distributional Kernels and Distributions on Manifolds supplies the inversion domain, Semiclassical First Laws and Physical-Process Variations supplies the gravitational identity, and Modular Response Kernels and Stress-Tensor Insertions supplies the response language.

First application. Derive the linearized Einstein equation about vacuum AdS from first-law equality for all boundary balls under the standard holographic assumptions.

Ball modular energy

For a ball $B$ of radius $R$ centered at $\mathbf x_0$ in a CFT vacuum at $t=0$, the modular Hamiltonian is local:

$$K_B=2\pi\int_{\lvert\mathbf x-\mathbf x_0\rvert<R} d^{d-1}x\, \frac{R^2-\lvert\mathbf x-\mathbf x_0\rvert^2}{2R} T_{00}(\mathbf x).$$

For a nearby state, the exact information identity gives

$$\delta S_B=\delta\langle K_B\rangle.$$

Holography adds two independent maps: RT identifies $\delta S_B$ with the first variation of the area of the corresponding AdS–Rindler bifurcation surface, while holographic renormalization identifies $\delta\langle T_{\mu\nu}\rangle$ with the normalizable asymptotic metric coefficient.

From an Iyer–Wald identity to equations of motion

Let $\xi_B$ be the bulk Killing vector that vanishes on the RT surface. For a metric perturbation $h_{ab}$ about AdS, the covariant phase-space identity can be written schematically as

$$d\chi_B[h] =-2\,\xi_B^a\,\delta E_{ab}[h],\epsilon^b,$$

where $\delta E_{ab}=0$ is the linearized gravitational equation. Integrating over the wedge slice gives

$$\delta\langle K_B\rangle-\delta S_B =-2\int_{\Sigma_B}\xi_B^a\delta E_{ab}\epsilon^b.$$

The first law sets the left side to zero. If this holds for every translated, boosted, and resized ball, if the perturbation satisfies the appropriate asymptotic constraints, and if the resulting Radon-type transform is injective, then $\delta E_{ab}=0$ pointwise. This is the conditional derivation of Faulkner et al. 2014, §§2–4, building on the entanglement-equilibrium observation of Lashkari et al. 2014.

With bulk matter, the target is the linearized Einstein equation with the correctly normalized stress tensor. Higher-derivative entropy and equations must be varied together; inserting a Wald-like entropy while retaining Einstein’s equations is inconsistent.

Necessary assumptions

The inference uses:

• a CFT state with a controlled semiclassical AdS dual;
• the RT or corrected entropy dictionary;
• the boundary stress-tensor/metric normalization;
• all boundary balls, not one selected region;
• a fixed background and a perturbative state family;
• gauge and boundary conditions that make the integral transform injective;
• locality of the target bulk equations.

Remove any one item and the conclusion weakens. In particular, the first law is true in any differentiable family of density operators; most such systems do not imply Einstein gravity.

Adversarial controls and nonconverses

Use one ball. One weighted integral of $\delta E_{ab}$ can vanish while the integrand is nonzero. The family of all balls is what enables inversion.

Change the entropy functional only. Higher-curvature theories require matched changes to entropy, charges, and field equations. A mixed calculation can manufacture the Einstein tensor.

Reverse the implication. A metric satisfying the linearized Einstein equation does not by itself prove RT, a CFT dual, or the entanglement first law dictionary. Those were inputs to the forward argument.

Evidence ceiling

Under the standard holographic dictionary and inversion hypotheses, first-law equality for all vacuum balls implies the linearized bulk field equation about AdS. It does not derive the holographic dictionary, select a unique nonlinear action, establish finite-N locality, or show that entanglement alone produces gravity.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Faulkner, Thomas, Monica Guica, Thomas Hartman, Robert C. Myers, and Mark Van Raamsdonk. “Gravitation from Entanglement in Holographic CFTs.” Journal of High Energy Physics 2014, 051 (2014). DOI; arXiv:1312.7856.
• Lashkari, Nima, Michael B. McDermott, and Mark Van Raamsdonk. “Gravitational Dynamics from Entanglement ‘Thermodynamics’.” Journal of High Energy Physics 2014, 195 (2014). DOI; arXiv:1308.3716.