AdS gravity entropy dictionary and quantum extremal surfaces

Let $\mathrm{\Sigma }$ be a locally compact Hausdorff space and $B\subseteq \mathrm{\Sigma }$ closed. Use ${\mathbb{Z}}_2=\{0,1\}$ with addition modulo two. A chain in $C_k^{\mathrm{l}\mathrm{f}}(\mathrm{\Sigma })$ is a formal sum of continuous singular $k$-simplex maps with mod-two coefficients such that every point of $\mathrm{\Sigma }$ has a neighborhood meeting only finitely many nonzero simplex images. Put $C_k^{\mathrm{l}\mathrm{f}}(\mathrm{\Sigma })=0$ for $k<0$. Let $C_k^{\mathrm{l}\mathrm{f}}(\mathrm{\Sigma };B)$ be the subgroup of these ambient chains whose simplex images are contained in $B$, and set $C_k^{\mathrm{l}\mathrm{f}}(\mathrm{\Sigma },B):=C_k^{\mathrm{l}\mathrm{f}}(\mathrm{\Sigma })\mathrm{/}{C}_k^{\mathrm{l}\mathrm{f}}(\mathrm{\Sigma };B)\mathrm{.}$ The face boundary, with all signs equal to plus in ${\mathbb{Z}}_2$, is well-defined and locally finite, preserves the $B$-supported subgroup, and induces ${\bar{\mathrm{\partial }}}^2=0$ on this quotient. Define $H_k^{\mathrm{l}\mathrm{f}}(\mathrm{\Sigma },B):=\ker {\bar{\mathrm{\partial }}}_k\mathrm{/}\mathrm{im}{\bar{\mathrm{\partial }}}_{k+1}\mathrm{.}$ For ambient locally finite chains $a,x$ with boundaries supported in $B$, the equality $[a]=[x]$ holds exactly when there is $r\in C_{k+1}^{\mathrm{l}\mathrm{f}}(\mathrm{\Sigma })$ with $\mathrm{\partial }r+a+x\in C_k^{\mathrm{l}\mathrm{f}}(\mathrm{\Sigma };B)$. Changing representatives by quotient boundaries or $B$-supported chains preserves this criterion. Finite ordinary chains are allowed; on compact $\mathrm{\Sigma }$ this is precisely the ordinary relative singular chain model.

Here is an explicit meaning of supplied fundamental-chain data. Let $Y\subseteq \mathrm{\Sigma }$ be an embedded $n$-manifold with boundary, $n\ge 1$. An ambient locally finite mod-two fundamental chain for $Y$ is a chain $c\in C_n^{\mathrm{l}\mathrm{f}}(\mathrm{\Sigma })$ whose simplex maps land in $Y$, whose boundary maps land in $\mathrm{\partial }Y$, and which satisfies the following local condition. For each $p\in Y\setminus \mathrm{\partial }Y$, retain the finitely many simplices of $c$ whose images contain $p$, obtaining an ordinary finite chain $c_p$ in $Y$. Its class in $H_n(Y,Y\setminus \{p\};{\mathbb{Z}}_2)$ is the unique nonzero element. The boundary of $c_p$ avoids $p$, so this class is defined. The local group is ${\mathbb{Z}}_2$, without an orientation choice. These are explicit conditions on supplied chains, not an assertion that arbitrary nonproper embeddings push intrinsic locally finite chains into $\mathrm{\Sigma }$.

A sufficient construction is as follows. Supply a pure $n$-dimensional ordered-vertex locally finite triangulation of $Y$, with boundary a subcomplex, whose realized simplex maps into $\mathrm{\Sigma }$ form a locally finite family. Proper inclusion of a locally finite triangulated $Y$ into $\mathrm{\Sigma }$ is sufficient for this ambient condition. Require that each interior codimension-one face has two incident top simplices and each boundary face has one. For every interior point $p$, supply a finite subcomplex neighborhood $N_p$ which is a compact $n$-manifold with boundary, with $p$ in its interior, such that the sum of its parametrized top simplices represents its compact mod-two relative fundamental class. A compact manifold ball or star with its verified relative fundamental chain supplies such a local certificate; no claim about an arbitrary triangulation's local stars is made. Then the sum of all top simplices, parametrized in the supplied vertex order, is an ambient locally finite fundamental chain for $Y$. In degree $n-1$, a supplied triangulated $n$-dimensional region with ambient top chain $r$ gives the displayed relative-chain criterion for boundary chains $a,x\in C_{n-1}^{\mathrm{l}\mathrm{f}}(\mathrm{\Sigma })$ when $\mathrm{\partial }r=a+x+b$ for some $b\in C_{n-1}^{\mathrm{l}\mathrm{f}}(\mathrm{\Sigma };B)$ and both $\mathrm{\partial }a$ and $\mathrm{\partial }x$ lie in $C_{n-2}^{\mathrm{l}\mathrm{f}}(\mathrm{\Sigma };B)$. These last conditions make $a$ and $x$ relative cycles, while the displayed decomposition supplies the relative boundary witness. No existence of a triangulation, a region, a chain certificate or an admissible candidate is asserted.

Borel-Moore homology by locally finite chains

Relative singular homology

Relative fundamental class and boundary orientation

Fix an integer $d\ge 2$, a smooth $(d+1)$-manifold with boundary $\bar{M}$, its interior $M$, and a smooth Lorentzian metric $g$ on $M$ with signature $(-,+,\dots ,+)$. The model supplies a dimensionless smooth defining function $z:\bar{M}\to [0,\mathrm{\infty })$ with $z>0$ on $M$, $z=0$ and $dz\mathrm{\ne }0$ on $\mathcal{I}:=\mathrm{\partial }\bar{M}$, such that $\bar{g}:=z^{2}g$ extends smoothly to a nondegenerate Lorentzian metric of the same signature on $\bar{M}$ whose induced metric $\gamma :=\bar{g}{\mathrm{\mid }}_{T\mathcal{I}}$ is Lorentzian. Thus $\mathcal{I}$ is a timelike conformal boundary. For these signature conventions, a timelike vector $v$ satisfies $\bar{g}(v,v)<0$, a null vector is nonzero with $\bar{g}(v,v)=0$, and a causal vector is nonzero with $\bar{g}(v,v)\le 0$. A time orientation is a continuous choice, at every point, of one component of the timelike cone; causal vectors in the closure of that component are future-directed. The model supplies a time orientation on $(\bar{M},\bar{g})$, which induces one on $(\mathcal{I},\gamma )$. A submanifold is spacelike when the metric induced on each tangent space is positive definite.

A future-directed causal curve in $(\bar{M},\bar{g})$ or $(\mathcal{I},\gamma )$ is a locally Lipschitz map $\eta :J\to \bar{M}$ or $\eta :J\to \mathcal{I}$, respectively, from a real interval, with a nonzero future-directed tangent almost everywhere and $\bar{g}(\dot{\eta },\dot{\eta })\le 0$ or $\gamma (\dot{\eta },\dot{\eta })\le 0$. Extensions are geometric: a regular increasing reparameterization is an order-preserving locally bi-Lipschitz homeomorphism $h:J\to J_0$ onto an interval. A curve is inextendible if there are no such $h$, a larger interval $\tilde{J}$ containing $J_0$ as a proper subinterval, and a causal curve $\tilde{\eta }:\tilde{J}\to \bar{M}$ (or to $\mathcal{I}$ in the boundary case) with $\tilde{\eta }\circ h=\eta$. Thus a parametrization on all of $\mathbb{R}$ does not by itself make a curve inextendible. A boundary Cauchy slice ${\mathcal{C}}_{\mathrm{\partial }}$ is a smooth embedded spacelike hypersurface in $(\mathcal{I},\gamma )$ whose inverse image under every inextendible boundary causal curve consists of exactly one parameter value. A bulk Cauchy slice for it is a smooth embedded spacelike $d$-dimensional hypersurface $\mathrm{\Sigma }$ of $(\bar{M},\bar{g})$ with $\mathrm{\partial }\mathrm{\Sigma }={\mathcal{C}}_{\mathrm{\partial }}$ whose inverse image under every inextendible causal curve in $\bar{M}$ consists of exactly one parameter value. This is explicitly a boundary-inclusive convention: curves may meet or run along $\mathcal{I}$, and no ordinary Cauchy property for the AdS interior alone is imposed. Existence of these slices is supplied model data, not a theorem about every asymptotically AdS spacetime. For each candidate $X$, the model supplies such a slice ${\mathrm{\Sigma }}_X$ containing $X$; one common slice is allowed.

A boundary region $A$ is a relatively compact open subset of ${\mathcal{C}}_{\mathrm{\partial }}$ (its closure is compact) with smooth boundary $\mathrm{\partial }A$; write $\bar{A}$ for its closure in ${\mathcal{C}}_{\mathrm{\partial }}$. A candidate $X$ is a smooth embedded $(d-1)$-dimensional submanifold with boundary of $\bar{M}$, with $X\cap \mathcal{I}=\mathrm{\partial }X=\mathrm{\partial }A$, $X\setminus \mathrm{\partial }X\subset M$, $X\subseteq {\mathrm{\Sigma }}_X$, and positive-definite induced metric $\bar{g}{\mathrm{\mid }}_{TX}$ (equivalently, $g{\mathrm{\mid }}_{TX}$ is positive definite in the interior). Set $B_A:={\mathcal{C}}_{\mathrm{\partial }}\setminus A$, a closed subspace of ${\mathrm{\Sigma }}_X$ since ${\mathcal{C}}_{\mathrm{\partial }}=\mathrm{\partial }{\mathrm{\Sigma }}_X$ is closed and $A$ is open in it. Use the ambient locally finite mod-two relative chain model as follows. A chain in $C_k^{\mathrm{l}\mathrm{f}}({\mathrm{\Sigma }}_X)$ is a formal sum of continuous singular $k$-simplex maps into ${\mathrm{\Sigma }}_X$ such that every point has a neighborhood meeting only finitely many nonzero simplex images. The subgroup $C_k^{\mathrm{l}\mathrm{f}}({\mathrm{\Sigma }}_X;B_A)$ consists of chains whose simplex images all lie in $B_A$, and $C_k^{\mathrm{l}\mathrm{f}}({\mathrm{\Sigma }}_X,B_A)$ is the quotient with the induced face boundary; its homology is $H_k^{\mathrm{l}\mathrm{f}}({\mathrm{\Sigma }}_X,B_A;{\mathbb{Z}}_2)$. The mod-two coefficients mean $1+1=0$. The model supplies a finite fundamental chain $a_X$ for the compact manifold with boundary $\bar{A}$ and an ambient locally finite fundamental chain $x$ for $X$, with boundaries supported in $\mathrm{\partial }A$. Here fundamental means that, at every interior point of the represented manifold, the finite chain of simplices whose images contain that point has the unique nonzero mod-two local homology class. These are supplied chain data; local finiteness is required in the ambient slice, and is not inferred by pushing an intrinsic chain through an arbitrary nonproper inclusion. Define ${\mathcal{H}}_A$ by $[x]=[a_X]$ in $H_{d-1}^{\mathrm{l}\mathrm{f}}({\mathrm{\Sigma }}_X,B_A;{\mathbb{Z}}_2)$. Equivalently, the model supplies a witness chain $r_X\in C_d^{\mathrm{l}\mathrm{f}}({\mathrm{\Sigma }}_X)$ with $\mathrm{\partial }{r}_X+a_X+x\in C_{d-1}^{\mathrm{l}\mathrm{f}}({\mathrm{\Sigma }}_X;B_A)$. The specified bulk region $R_X\subseteq {\mathrm{\Sigma }}_X$ is part of the model data represented by this witness; when $R_X$ is an ordinary bulk region, its supplied ambient locally finite fundamental chain is used. A proper compatible locally finite triangulation with the finite local fundamental-class certificates described above is one sufficient construction by mod-two face cancellation, not a requirement on the full candidate class. The supplied class ${\mathcal{X}}_A$ (possibly empty) includes all this slice and chain data; no candidate, region, chain or triangulation existence theorem is asserted.

A named finite-regulator sequence for $X\in {\mathcal{X}}_A$ consists of labels $({\mathrm{\Lambda }}_n{)}_{n\ge 1}$, nonzero finite-dimensional complex inner-product spaces $H_{\mathrm{i}\mathrm{n},n}(X)$ and $H_{\mathrm{o}\mathrm{u}\mathrm{t},n}(X)$, a density operator ${\rho }_n(X)$ on their ordered tensor product, and a finite assigned geometric value $S_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v},n}[X]$ for each $n$. A density operator here is a self-adjoint endomorphism with nonnegative eigenvalues and trace one. The factor $H_{\mathrm{o}\mathrm{u}\mathrm{t},n}(X)$ represents the regulated bulk region $R_X$ and $H_{\mathrm{i}\mathrm{n},n}(X)$ its complementary factor; both factors and the state are supplied model data. Take $k_B>0$ in $\mathrm{J}{\mathrm{K}}^{-1}$. Define ${\rho }_{\mathrm{o}\mathrm{u}\mathrm{t},n}(X)$ by tracing the first, $H_{\mathrm{i}\mathrm{n},n}(X)$, factor out of the ordered tensor product, using the finite partial-trace convention in the finite-regulator outside-entropy definition: in orthonormal bases $(e_i)$ and $(f_a)$ of the two factors, the $(a,b)$ entry is ${\sum }_i({\rho }_n(X){)}_{(i,a),(i,b)}$. The finite matrix argument in that definition proves the resulting endomorphism is a basis-independent density operator. With natural logarithms and $0\log 0:=0$, set

$S_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k},n}[R_X]:=-k_B\mathrm{tr}\text{\hspace{0.17em}⁣}({\rho }_{\mathrm{o}\mathrm{u}\mathrm{t},n}(X)\log {\rho }_{\mathrm{o}\mathrm{u}\mathrm{t},n}(X)),S_{\mathrm{g}\mathrm{e}\mathrm{n},n}[X]:=S_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v},n}[X]+S_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k},n}[R_X]\mathrm{.}$

Require each $S_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v},n}[X]$ to have units $\mathrm{J}{\mathrm{K}}^{-1}$; its geometric terms, counterterms, and finite normalization are fixed model data in one declared regulator and renormalization scheme. On the optional Einstein–Hilbert area branch, the same fixed scheme also specifies, for each label ${\mathrm{\Lambda }}_n$ and candidate, a dimensionless Borel function $0\le {\chi }_{{\mathrm{\Lambda }}_n,X}\le 1$ with compact support $K_{n,X}$ in $X^{\circ }:=X\setminus \mathrm{\partial }X$. Put $h_X:=g{\mathrm{\mid }}_{T{X}^{\circ }}$, a smooth positive-definite induced metric. Its density is ${\mu }_{h_X}=\sqrt{\det (h_{ab})}\mathrm{\mid }d{y}^1\cdots d{y}^{d-1}\mathrm{\mid }$ in a surface chart. The determinant transformation $h_y=J^T{h}_{x}J$ gives the absolute-Jacobian density law. Under Countable Choice, meaning that every countable family of nonempty sets has a choice function, the density determines the corresponding compact-finite Borel measure ${\mathrm{vol}}_{h_X}$, independent of charts and orientation (compact-finite means finite on compact sets). Define the finite regulator-specific physical area by

${\mathrm{Area}}_{{\mathrm{\Lambda }}_n}[X]:={\int }_{X^{\circ }}{\chi }_{{\mathrm{\Lambda }}_n,X}d{\mathrm{vol}}_{h_X},0\le {\mathrm{Area}}_{{\mathrm{\Lambda }}_n}[X]\le {\mathrm{vol}}_{h_X}(K_{n,X})<\mathrm{\infty }\mathrm{.}$

The bound follows directly from $0\le \chi \le 1$ and compact finiteness. Smooth cutoffs give the usual intrinsic integral $\int \chi {\mu }_{h_X}$; sharp compact Borel cutoffs $\chi ={\mathbf{1}}_K$ are included. The measure domain here is the Borel domain, with no implicit completion. The geometric area has units ${\mathrm{m}}^{\mathrm{d}-1}$ in the model's length convention. Countable Choice is carried precisely for this measure/integration interface, not for the chain algebra. With supplied $G_{{\mathrm{\Lambda }}_n}>0$ of units ${\mathrm{m}}^{\mathrm{d}}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, this branch sets

$S_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v},n}[X]=\frac{k_B{c}^3{\mathrm{Area}}_{{\mathrm{\Lambda }}_n}[X]}{4G_{{\mathrm{\Lambda }}_n}\mathrm{\hslash }}+S_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r},n}[X],$

where $c$ has units $\mathrm{m}{\mathrm{s}}^{-1}$, $\mathrm{\hslash }$ has units $\mathrm{J}\mathrm{s}$, and the finite $S_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r},n}$ includes the scheme's specified counterterms and finite normalization. The cutoff functions, their relation to ${\mathrm{\Lambda }}_n$, and every subtraction are fixed by that named scheme; no unregulated physical-metric area of a boundary-anchored surface is substituted. General assigned finite $S_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v},n}$ data outside this optional area branch remain allowed.

Assign $S_{\mathrm{g}\mathrm{e}\mathrm{n}}^{\mathrm{r}\mathrm{e}\mathrm{n}}[X]$ exactly when the combined sequence $(S_{\mathrm{g}\mathrm{e}\mathrm{n},n}[X]{)}_{n\ge 1}$ converges to a finite real number in that scheme, and then set it equal to that limit. No separate convergence of the two summands is required. If the combined sequence lacks a finite limit, no renormalized value is assigned.

A smooth one-parameter candidate family means a jointly smooth map $F:(-{ϵ}_0,{ϵ}_0)\times S\to \bar{M}$, where ${ϵ}_0>0$ and $S$ is a fixed smooth $(d-1)$-manifold with boundary, such that each $F_{ϵ}:=F(ϵ,\cdot )$ is an embedding with image $X_{ϵ}\in {\mathcal{X}}_A$, $F_{ϵ}(\mathrm{\partial }S)=\mathrm{\partial }A$ for the same fixed boundary region $A$, and $X_0=X$. Every member comes with its own supplied slice and chains satisfying $[x_{ϵ}]=[a_{X_{ϵ}}]$ in $H_{d-1}^{\mathrm{l}\mathrm{f}}({\mathrm{\Sigma }}_{X_{ϵ}},B_A;{\mathbb{Z}}_2)$. In this definition, preserving the homology condition means precisely this memberwise relative-chain criterion for the same $A$; it does not compare classes in different groups or assume an identification between the slices. The initial velocity is $V_{F_0(s)}:={\mathrm{\partial }}_{ϵ}F(0,s)$. Require it to be smooth and orthogonal to $T_{p}X$ at every interior point $p=F_0(s)$, meaning $g(V_p,w)=0$ for every $w\in T_{p}X$, and to extend smoothly by zero at the fixed anchor. When it exists, define this specified family's first variation by ${\delta }_V{S}_{\mathrm{g}\mathrm{e}\mathrm{n}}^{\mathrm{r}\mathrm{e}\mathrm{n}}[X]:={\frac{d}{dϵ}{S}_{\mathrm{g}\mathrm{e}\mathrm{n}}^{\mathrm{r}\mathrm{e}\mathrm{n}}[X_{ϵ}]\mid }_{ϵ=0}$. Call $X$ quantum stationary when this derivative exists and is zero for every such allowed family. No family existence or identification of different homology groups is asserted. The term records the quantum bulk-entropy contribution to the functional; it does not assert a universal semiclassical or microscopic interpretation.

Specified regulated generalized entropy

Ambient locally finite mod-two relative chains for candidate regions

Riemannian volume density

Riemannian volume is the radon measure of the riemannian density

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Ahmed Almheiri et al., The entropy of Hawking radiation

The current def-tgr-regulated-generalized-entropy supplier is restricted to one specified compact section of a smooth null hypersurface. This definition supplies a separate candidate-surface model for the AdS page; it borrows the fixed-scheme and entropy-unit convention only and does not claim that the null-screen domain already covers these AdS surfaces.

Fix an effective bulk model on a smooth Lorentzian spacetime $(M,g)$ of dimension $d+1$, with $d\ge 2$ and signature $(-,+,\dots ,+)$, together with a time orientation, meaning a continuous choice of one component of the timelike cone at each point. Here AdS means anti-de Sitter. In this item, asymptotically locally AdS means that $M$ is the interior of a smooth manifold with boundary $\bar{M}$; write $\mathcal{I}:=\mathrm{\partial }\bar{M}$. There is a dimensionless smooth defining function $z$ with $z>0$ on $M$, $z=0$ and $dz\mathrm{\ne }0$ on $\mathcal{I}$, such that $\bar{g}=z^{2}g$ extends smoothly to a nondegenerate Lorentzian metric of the same signature on $\bar{M}$ whose induced boundary metric $\gamma :=\bar{g}{\mathrm{\mid }}_{T\mathcal{I}}$ is Lorentzian. The chosen time orientation extends continuously to $(\bar{M},\bar{g})$ and induces one on $(\mathcal{I},\gamma )$. The model fixes its AdS length $L>0$ in metres, the asymptotic AdS boundary condition, bulk fields, semiclassical approximation, and conformal-boundary data; semiclassical here means that the bulk geometry is treated classically while the bulk entropy is computed from regulated quantum-state data on that geometry. No bulk field equation is inferred here.

Choose a smooth embedded hypersurface ${\mathcal{C}}_{\mathrm{\partial }}\subset \mathcal{I}$ whose induced metric is positive definite and which meets every inextendible future-directed boundary causal curve at exactly one parameter value. A causal curve in $(\bar{M},\bar{g})$ or $(\mathcal{I},\gamma )$ is a locally Lipschitz map from a real interval with a nonzero future-directed tangent almost everywhere and nonpositive squared tangent norm. Inextendibility is tested after an order-preserving locally bi-Lipschitz reparameterization $h:J\to J_0$: a curve $\eta :J\to M_0$ is inextendible exactly when there are no such $h$, larger interval $\tilde{J}⊋J_0$, and causal curve $\tilde{\eta }:\tilde{J}\to M_0$ with $\tilde{\eta }\circ h=\eta$, where $M_0$ is the bulk or boundary spacetime in question. Each bulk candidate slice ${\mathrm{\Sigma }}_X$ is a smooth embedded $d$-dimensional hypersurface in $(\bar{M},\bar{g})$ with positive-definite induced metric and $\mathrm{\partial }{\mathrm{\Sigma }}_X={\mathcal{C}}_{\mathrm{\partial }}$, and every inextendible causal curve in $\bar{M}$ meets it at exactly one parameter value. This is a boundary-inclusive convention: curves may meet or run along $\mathcal{I}$. The existence of these slices is supplied model data; no interior-only Cauchy existence is inferred. Choose a relatively compact open region $A\subset {\mathcal{C}}_{\mathrm{\partial }}$ (its closure is compact) with smooth boundary $\mathrm{\partial }A$. The regulated boundary Hilbert space and its finite-dimensional subspace ${\mathcal{C}}_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}$ are model data; regulated code subspace means this specified finite-dimensional subspace, and every boundary density matrix in the chosen state family is supported on it. A boundary density matrix ${\rho }_{\mathrm{\partial }}$ is a positive trace-class endomorphism of trace one on the regulated boundary Hilbert space, with range in ${\mathcal{C}}_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}$. For such a matrix ${\rho }_{\mathrm{\partial }}$, define its exact finite-cutoff entropy by $S_{\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}}({\rho }_{\mathrm{\partial }}):=-k_B\mathrm{tr}({\rho }_{\mathrm{\partial }}\log {\rho }_{\mathrm{\partial }})$, using natural logarithms and $0\log 0:=0$. An admissible bulk candidate $X$ is a smooth embedded spacelike $(d-1)$-dimensional submanifold with boundary of $\bar{M}$, with $X\cap \mathcal{I}=\mathrm{\partial }X=\mathrm{\partial }A$, $X\setminus \mathrm{\partial }X\subset M$, and $X\subseteq {\mathrm{\Sigma }}_X$; its induced metric is positive definite and its supplied region $R_X\subseteq {\mathrm{\Sigma }}_X$ and chains satisfy $[x]=[a_X]$ in the ambient locally finite mod-two relative homology group defined by the candidate-surface model. A quantum-stationary candidate (also called a quantum extremal surface) is an admissible $X$ at which the first variation of the scheme-fixed generalized entropy exists and vanishes for every family $F$ specified in the preceding candidate-surface definition: $F$ is jointly smooth, each $F_{ϵ}$ is an admissible embedding with the same anchor and memberwise relative-chain homology condition, and its initial velocity $V$ satisfies $g(V,w)=0$ for every interior tangent vector $w\in TX$ and extends smoothly by zero at the anchor. Let ${\mathcal{Q}}_A$ be the prescribed class of such candidates for the chosen boundary state. Use SI units: $k_B$ is in $\mathrm{J}{\mathrm{K}}^{-1}$, $c$ is in $\mathrm{m}{\mathrm{s}}^{-1}$, and $\mathrm{\hslash }$ is in $\mathrm{J}\mathrm{s}$. Write $G_{{\mathrm{\Lambda }}_n}>0$ for the model-supplied Newton coupling at regulator label ${\mathrm{\Lambda }}_n$, in ${\mathrm{m}}^{\mathrm{d}}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ for this $(d+1)$-dimensional bulk. When the chosen Einstein–Hilbert branch uses the area term, use the same label-dependent physical-metric area ${\mathrm{Area}}_{{\mathrm{\Lambda }}_n}[X]={\int }_{X^{\circ }}{\chi }_{{\mathrm{\Lambda }}_n,X}d{\mathrm{vol}}_{g{\mathrm{\mid }}_{T{X}^{\circ }}}$ defined by the candidate-surface model, with its explicit Countable Choice integration premise and its fixed finite cutoff data. This term is $k_B{c}^3{\mathrm{Area}}_{{\mathrm{\Lambda }}_n}[X]\mathrm{/}(4G_{{\mathrm{\Lambda }}_n}\mathrm{\hslash })$, and all additional finite terms remain in the prescribed $S_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r},n}$. The cutoff area is in ${\mathrm{m}}^{\mathrm{d}-1}$, so $[c^3{\mathrm{Area}}_{{\mathrm{\Lambda }}_n}[X]]={\mathrm{m}}^{\mathrm{d}+2}{\mathrm{s}}^{-3}=[G_{{\mathrm{\Lambda }}_n}\mathrm{\hslash }]$ and the factor multiplying $k_B$ is dimensionless. The metric, Borel cutoff, label dependence and subtraction convention are thus fixed before the combined renormalized limit is tested; an unregulated area at the conformal boundary is not asserted finite. For $d=3$ these are the usual four-dimensional SI units. Adopt

$S_{\mathrm{\partial }}^{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}(A):={\min }_{X\in {\mathcal{Q}}_A}{S}_{\mathrm{g}\mathrm{e}\mathrm{n}}^{\mathrm{r}\mathrm{e}\mathrm{n}}[X]$

as the model's semiclassical boundary-entropy prediction only when ${\mathcal{Q}}_A$ is nonempty, the values are finite, and the minimum is attained. It does not identify this prediction with $S_{\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}}({\rho }_{\mathrm{\partial }})$. A path-integral saddle here is a stationary configuration of the chosen gravitational action; saddle dominance means that its contribution controls the specified path-integral approximation over competing configurations. Candidate existence, finite convergence, and dominance of the selected saddle are separate premises. The minimum rule is an adopted physical premise, not a consequence of the finite code entropy identity or the stationary Noether first law. It does not assert that the model describes nature, a universal RT/QES identity, or a continuum factorization of quantum fields.

Sector entropy and relative entropy identities

Regulated generalized entropy for AdS candidate surfaces

Specified regulated generalized entropy

Stationary first law from the covariant boundary identity

Daniel Harlow, Jerusalem Lectures on Black Holes and Quantum Information

Ahmed Almheiri et al., The entropy of Hawking radiation

The finite-sector entropy proposition supplies an exact finite matrix identity in nats; multiplying by $k_B$ converts that entropy convention to physical units, but it does not identify a sector term with geometric area. The Noether first-law supplier gives an Einstein–Hilbert area coefficient only under its stationary hypotheses. Neither result derives this adopted dictionary for a general asymptotically AdS model.

Let $I\subset \mathbb{R}$ be an open interval of boundary time, measured in seconds, and let $t_0\in I$. Suppose the prescribed candidate class for the adopted entropy dictionary consists, near $t_0$, of exactly two admissible surfaces $X_1(t)$ and $X_2(t)$. Let their finite generalized entropies be $S_1,S_2\in C^1(I,\mathbb{R})$, with values in $\mathrm{J}{\mathrm{K}}^{-1}$; here $C^1$ means continuously differentiable. Assume $S_1(t_0)=S_2(t_0)$, that $S_1(t)\mathrm{\ne }{S}_2(t)$ for all sufficiently close $t\mathrm{\ne }{t}_0$, and that $S_1^{\mathrm{\prime }}(t_0)\mathrm{\ne }{S}_2^{\mathrm{\prime }}(t_0)$. Then the adopted semiclassical boundary-entropy prediction $S_{\mathrm{\partial }}^{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}(t)=\min \{S_1(t),S_2(t)\}$ is continuous at $t_0$ and is not differentiable there. This conclusion concerns the selected two-candidate minimum. The hypotheses do not identify it with an exact finite-cutoff microscopic entropy, which need not have this cusp.

Restricted semiclassical entropy dictionary

Ahmed Almheiri et al., The entropy of Hawking radiation

For the symmetric branches $S_1=S_0+a(t-t_0)$ and $S_2=S_0-a(t-t_0)$, with $a>0$ measured in $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{s}}^{-1}$, the plot below shows the crossing and the lower envelope. It is illustrative only; the proof above establishes the result for all functions satisfying the stated hypotheses.