Generalized Entropy of a Cut

Generalized entropy is the finite quantity obtained only after the gravitational entropy of a codimension-two cut and the quantum entropy on one chosen side are renormalized together. Neither summand is separately regulator independent. The combined functional is the object that can be varied to define quantum expansion, quantum extremality, and semiclassical horizon laws.

Required background. Quantum Information supplies density operators, von Neumann entropy, reduced algebras, and the distinction between a subsystem entropy and a thermodynamic state count used below.

Helpful background. Quantum Fields in Curved Spacetime supplies Hadamard states and point-splitting; Einstein Equations fixes the gravitational source convention; Black-Hole Geometry and Mechanics supplies codimension-two horizon cuts; and Semiclassical Gravity places the construction in the mean-metric approximation.

Units. This page restores $\hbar$ in entropy formulas while keeping $c=k_{\rm B}=1$. Hence $G\hbar$ has dimensions of area and the Einstein entropy is $A/(4G\hbar)$; setting $\hbar=1$ recovers the site-wide convention.

The renormalized cut functional

Let $\sigma$ be a smooth spacelike codimension-two surface that divides a Cauchy slice into an “outside” region $\Sigma_{\rm out}$ and its complement. Fix the quantum state, the algebra assigned to $\Sigma_{\rm out}$, the ultraviolet regulator $\epsilon$, and the same renormalization scheme used in the gravitational effective action. In four dimensions the regulated expression has the schematic form

$$S_{\rm gen}[\sigma] = S_{\rm grav}^{\rm bare}[\sigma;\epsilon] +S_{\rm out}^{\rm bare}[\Sigma_{\rm out};\epsilon] = S_{\rm grav}^{\rm ren}[\sigma] +S_{\rm out}^{\rm ren}[\Sigma_{\rm out}].$$

For Einstein gravity, the leading gravitational term is $A(\sigma)/(4G_{\rm ren}\hbar)$. Higher-curvature couplings contribute local surface terms fixed by the action. On a stationary Killing horizon these reduce to the appropriate Wald entropy, but a general nonstationary cut can also require extrinsic-curvature terms: substituting the stationary Wald formula on every cut is not sufficient. The distinction is explained in Bousso et al. 2016, Appendix A.2, PDF. Corners or gauge-field edge data require the same care as the entropy algebra. Thus a useful expansion is

$$S_{\rm grav}^{\rm ren} =\frac{A(\sigma)}{4G_{\rm ren}\hbar} +S_{R^2}^{\rm ren}[\sigma] +S_{\rm local}^{\rm ren}[\sigma],$$

where the last two symbols stand for local geometric terms fixed by the chosen effective action, not arbitrary additions made after the entropy is computed. With a regulator and a chosen factorization, the outside term starts from the von Neumann entropy $-\operatorname{Tr}(\rho_{\rm out}\log\rho_{\rm out})$. Continuum local QFT algebras generally have no reduced density matrix with a finite absolute von Neumann entropy. In that setting the notation denotes a specified regulated/subtracted entropy construction; relative entropy and suitable differences can be defined algebraically without pretending that the divergent absolute entropy is finite.

The leading area-law divergence of $S_{\rm out}^{\rm bare}$ renormalizes $1/G$; subleading curvature and logarithmic divergences renormalize higher-curvature couplings. This paired cancellation, including the role of conical and heat-kernel coefficients, is developed in Solodukhin 2011, §§ 2.6–2.9 and 3.1–3.12, PDF. The same logic is built into the generalized-entropy formulation used for quantum expansion in Bousso et al. 2016, §§ 3.1–3.2 and Appendix A, PDF.

Why the sum, not either term, is physical

Changing the cutoff shifts the two pieces oppositely. Schematically, with a leading divergence $c_2A/\epsilon^2$,

$$S_{\rm out}^{\rm bare}=\frac{c_2A}{\epsilon^2}+S_{\rm out}^{\rm finite}+\cdots, \qquad \frac{1}{4G_{\rm bare}\hbar} =\frac{1}{4G_{\rm ren}\hbar}-\frac{c_2}{\epsilon^2}+\cdots.$$

Substitution cancels the displayed divergence. A finite change of renormalization scheme can still move a local geometric functional between $S_{\rm grav}^{\rm ren}$ and $S_{\rm out}^{\rm ren}$, but it leaves $S_{\rm gen}$ unchanged when the couplings and entropy subtraction are translated consistently. It is therefore misleading to ask whether the area term or the matter term “really” carries a particular finite contribution without naming the scheme.

State dependence is different from scheme dependence. Holding the effective action fixed while changing a Hadamard state changes $S_{\rm out}^{\rm ren}$ nonlocally and may change the backreacted geometry and area. A regulator change, by contrast, is a relabeling compensated by local couplings. The cut and side must also be specified: $S_{\rm gen}$ is a functional of $\sigma$, the state, and the chosen outside algebra, not a number belonging to the spacetime alone.

A finite example: one entangled pair

To isolate the state dependence, hold the geometry and its counterterms fixed and consider two regulated modes on opposite sides of the cut in the state

$$|\psi\rangle=\sqrt p\,|0\rangle_{\rm in}|0\rangle_{\rm out} +\sqrt{1-p}\,|1\rangle_{\rm in}|1\rangle_{\rm out}, \qquad 0\le p\le1.$$

Tracing out the inside gives eigenvalues $p$ and $1-p$, hence an additional outside entropy $-p\ln p-(1-p)\ln(1-p)$. It is zero for a product state and $\ln2$ for $p=1/2$. These are finite differences in a regulated two-mode example; they are added to the common renormalized field-theory contribution. The calculation shows what the matter term measures, without assigning the ultraviolet area divergence to a collection of physical pairs. Changing the pair can also change its energy; holding the mean geometry fixed is justified here only to the perturbative order at which that backreaction is neglected.

Checks and range of validity

Independent check: stationary horizon. For a classical stationary black hole with quantum fields in a regular equilibrium state, the leading gravitational limit $G\hbar\to0$ at fixed geometry suppresses the matter variation relative to $A/(4G\hbar)$. The generalized entropy then reduces to the Bekenstein–Hawking entropy at leading order. Dimensional analysis also works: with $\hbar$ restored, $A/(G\hbar)$ is dimensionless; setting $\hbar=1$ gives the usual $A/G$ form.

Independent check: finite renormalization. Add a finite local curvature counterterm $\Delta I=\alpha\int\sqrt{-g}\,R^2$. Its conical variation shifts $S_{\rm grav}$ by a local functional of $\sigma$; the definition of the renormalized exterior entropy shifts by the opposite convention-dependent piece. A calculation that changes only one side fails this check.

The construction assumes a semiclassical state and a smooth cut on scales above the cutoff. It does not derive fixed-background Hawking radiation, and it does not by itself assign a fine-grained entropy to collected radiation. Replica saddles, islands, Page curves, and recovery statements require additional boundary-region and saddle-selection data and belong to Black-Hole Information.

Exercise

Suppose a regulator change shifts the outside entropy by $\Delta S_{\rm out}=\lambda\int_\sigma\sqrt{\gamma}\,R[\gamma]$. What must happen to the gravitational surface entropy, and what observable conclusion follows?

Solution

The renormalized gravitational couplings must be redefined so that $\Delta S_{\rm grav}=-\lambda\int_\sigma\sqrt{\gamma}\,R[\gamma]$. The two shifts then cancel in $S_{\rm gen}$. Consequently, the value and shape variation of the combined generalized entropy are scheme independent at the retained order, although the values assigned to its two pieces are not. For a closed two-dimensional cut of fixed topology, Gauss–Bonnet gives $\int_\sigma\sqrt\gamma\,R[\gamma]=4\pi\chi(\sigma)$, so this particular counterterm is constant under smooth shape deformations: cancellation of its value does not provide a nontrivial test of the shape derivative.

References

• Bousso, R., Fisher, Z., Leichenauer, S., and Wall, A. C. (2016). “A Quantum Focussing Conjecture.” Physical Review D 93, 064044. Open PDF.
• Solodukhin, S. N. (2011). “Entanglement Entropy of Black Holes.” Living Reviews in Relativity 14, 8. Open PDF.