Regulated generalized entropy for AdS candidate surfaces

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.