Canonical Energy and Second-Order Relative Entropy

Second-order boundary relative entropy is a positive information metric. In a holographic code sector it matches the canonical energy of the corresponding bulk perturbation, provided gauge, extremal-surface displacement, counterterms, and edge contributions are handled consistently. This relation is a perturbative stability test, not a proof of nonlinear gravitational dynamics.

Required background. Boundary Relative Entropy and Bulk Modular Data supplies the leading relation. Entanglement First Law and Conditional Linearized Field-Equation Inference supplies the first-order cancellation.

Helpful background. Quantum Fisher Information in QFT supplies the boundary Hessian. Presymplectic Systems and the Covariant Phase-Space Ambiguity Map supplies the bulk form. Linear Response and Semiclassical Stability fixes the stability interpretation, and State Perturbations and Relative-Entropy Susceptibility fixes the state-family expansion.

First application. Compute the second variation for a small AdS perturbation sourced by a boundary deformation and match it to canonical energy in the wedge.

Relative-entropy Hessian

For a smooth normalized family $\rho(\lambda)$ with reference $\rho(0)=\sigma$,

$$S(\rho(\lambda)\Vert\sigma) =\frac{\lambda^2}{2}\, \mathcal I_\sigma(\delta\rho,\delta\rho) +O(\lambda^3).$$

The linear term vanishes by the entanglement first law. $\mathcal I_\sigma$ is positive on physical state variations, modulo directions that do not change the restricted state. The precise monotone metric depends on how a multiparameter family and operator ordering are defined; for relative entropy it is the Bogoliubov–Kubo–Mori Hessian.

Bulk canonical energy

Let $\omega(g;\delta_1g,\delta_2g)$ be the renormalized covariant symplectic current and $\xi_B$ the AdS–Rindler Killing field for a ball. The canonical energy of a linearized perturbation $h$ on the wedge slice is

$$\mathcal E_B(h,h) =\int_{\Sigma_B} \omega\!\left(g;h,\mathcal L_{\xi_B}h\right) +\mathcal E_{\mathrm{matter}}+\mathcal E_{\mathrm{edge}}.$$

In Hollands–Wald gauge, the coordinate location and Killing behavior at the extremal surface are fixed so that surface variations are represented consistently. Then the holographic second-variation identity gives

$$\mathcal I_\sigma(\delta\rho_B,\delta\rho_B) =\mathcal E_B(h,h)$$

at the declared semiclassical order Lashkari and Van Raamsdonk 2016, §§2–4.

The equality includes boundary counterterms required to make the symplectic form finite. Adding an exact form to the presymplectic potential can shift surface terms, so a convention-independent comparison must transform the charges and entropy term together.

Euclidean-source application

Prepare a one-parameter family by a small Euclidean boundary source $\lambda J$ for an operator $O$. The first-order bulk field is obtained from the Euclidean bulk-to-boundary propagator. Continue it to the Lorentzian wedge, evaluate $\omega(\phi,\mathcal L_{\xi_B}\phi)$, and integrate over $\Sigma_B$.

On the boundary, the same quadratic form is an integrated two-point function with modular ordering. Equality checks the source normalization, continuation, and counterterms. Positivity then constrains perturbations that lie in the physical code sector.

Stability meaning and limitations

Positive canonical energy is a necessary stability condition in many stationary gravitational problems. A negative direction signals that the assumed background or boundary conditions cannot represent a local information-theoretic minimum. The converse is weaker: positivity in the tested wedge and perturbation class does not prove nonlinear or global stability.

Gauge directions should have zero norm after all boundary charges are fixed. If a purported pure gauge perturbation carries nonzero canonical energy, the gauge transformation is not trivial at the boundary or a surface term has been omitted.

Adversarial controls

Move the extremal surface without the gauge term. The omitted displacement changes the quadratic area variation and spoils the match.

Use an unrenormalized symplectic current. Divergent boundary flux cannot equal a finite boundary information metric.

Promote positivity globally. A positive quadratic form near one state says nothing by itself about distant states, topology-changing branches, or finite-amplitude evolution.

Evidence ceiling

The second-order identity matches the boundary relative-entropy Hessian to renormalized bulk canonical energy for a controlled state family and wedge. It supports perturbative stability in that sector. It does not supply higher-order equations, prove a unique bulk action, or establish stability outside the tested perturbations.

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

• Hollands, Stefan, and Robert M. Wald. “Stability of Black Holes and Black Branes.” Communications in Mathematical Physics 321, 629–680 (2013). DOI; arXiv:1201.0463.
• Lashkari, Nima, and Mark Van Raamsdonk. “Canonical Energy Is Quantum Fisher Information.” Journal of High Energy Physics 2016, 153 (2016). DOI; arXiv:1508.00897.