Regulated generalized-entropy model

Fix a smooth, oriented, time-oriented Lorentzian four-manifold $(M,g,N)$ of signature $(-,+,+,+)$, its metric-selected connection, and one smooth embedded null hypersurface $\mathcal{H}\subset M$. Use the null-hypersurface convention in the mathematical definition of the smooth null screen family. Supply a smooth, nonzero, future null generator field $k$ tangent to $\mathcal{H}$ with ${\mathrm{\nabla }}_{k}k=0$. Choose a nonempty compact smooth embedded spacelike two-surface $\mathrm{\Sigma }\subset \mathcal{H}$ transverse to $k$, meaning $T_p\mathrm{\Sigma }\oplus \mathbb{R}{k}_p=T_p\mathcal{H}$ at every $p\in \mathrm{\Sigma }$. If $i:\mathrm{\Sigma }\hookrightarrow M$ is inclusion, the induced metric $h=i^{\ast }g{\mathrm{\mid }}_{T\mathrm{\Sigma }}$ is positive definite. In any local surface chart $(y^1,y^2)$, define its area density by ${\mu }_h=\sqrt{\det (h_{AB})}\mathrm{\mid }d{y}^{1}d{y}^2\mathrm{\mid }$ and set $A_{\mathrm{\Sigma }}={\int }_{\mathrm{\Sigma }}{\mu }_h$. Admissible model data require $0<A_{\mathrm{\Sigma }}<\mathrm{\infty }$.

A named finite-regulator sequence consists of labels $({\mathrm{\Lambda }}_n{)}_{n\ge 1}$, nonzero finite-dimensional complex inner-product spaces $H_{\mathrm{i}\mathrm{n},n}$ and $H_{\mathrm{o}\mathrm{u}\mathrm{t},n}$, and a density operator ${\rho }_n$ on the ordered tensor product $H_{\mathrm{i}\mathrm{n},n}\otimes H_{\mathrm{o}\mathrm{u}\mathrm{t},n}$ for each $n$. The physical Boltzmann constant $k_B>0$ has units $\mathrm{J}{\mathrm{K}}^{-1}$, while $c>0$ and $\mathrm{\hslash }>0$ have units $\mathrm{m}{\mathrm{s}}^{-1}$ and $\mathrm{J}\mathrm{s}$, respectively. Define

$S_{\mathrm{o}\mathrm{u}\mathrm{t},n}:=S_{\mathrm{o}\mathrm{u}\mathrm{t}}({\rho }_n;H_{\mathrm{i}\mathrm{n},n},H_{\mathrm{o}\mathrm{u}\mathrm{t},n}),S_{\mathrm{g}\mathrm{e}\mathrm{n},n}[\mathrm{\Sigma }]:=S_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v},n}[\mathrm{\Sigma }]+S_{\mathrm{o}\mathrm{u}\mathrm{t},n}\mathrm{.}$

Here $S_{\mathrm{o}\mathrm{u}\mathrm{t},n}$ is the finite spectral entropy of the partial-trace state specified by the finite-regulator outside-entropy definition, using natural logarithms and $0\log 0:=0$. The assigned geometric value $S_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v},n}[\mathrm{\Sigma }]$ is finite and has units $\mathrm{J}{\mathrm{K}}^{-1}$; its terms, including every chosen local counterterm and finite normalization condition, are fixed as part of the declared scheme. Any dimensional regulator label has its units recorded with that scheme.

On the optional area-term branch, one allowed summand of $S_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v},n}[\mathrm{\Sigma }]$ is

$\frac{k_B{c}^3{A}_{\mathrm{\Sigma }}}{4G_{{\mathrm{\Lambda }}_n}\mathrm{\hslash }},$

where each $G_{{\mathrm{\Lambda }}_n}>0$ is specified model data with the units of Newton's constant, ${\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$. The relation of $G_{{\mathrm{\Lambda }}_n}$ to the reference $G_N$ is also declared model data. The renormalized generalized entropy $S_{\mathrm{g}\mathrm{e}\mathrm{n}}^{\mathrm{r}\mathrm{e}\mathrm{n}}[\mathrm{\Sigma }]$ is defined exactly when the sequence $(S_{\mathrm{g}\mathrm{e}\mathrm{n},n}[\mathrm{\Sigma }]{)}_{n\ge 1}$ converges to a finite real value in the declared scheme; its value is that limit. If the sequence has no finite limit, this definition assigns no renormalized value. The limit belongs only to the named sequence and scheme.

With the physical $k_B$, both terms in each sum have units $\mathrm{J}{\mathrm{K}}^{-1}$, and $A_{\mathrm{\Sigma }}$ is in ${\mathrm{m}}^2$.

Thermodynamics-GR fields, currents, reference normalization and units

Smooth null screen and subextremal Kerr–Newman parameter family

Finite-regulator outside density and entropy

Finite subtraction ambiguity

Conditional area entropy coefficient with free offset

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

H. S. Reall, Part III Black Holes, University of Cambridge lecture notes (2024)

Aron C. Wall, A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices

The finite-matrix outside-entropy definition establishes that each $S_{\mathrm{o}\mathrm{u}\mathrm{t},n}$ is a finite real number; it does not establish convergence as $n\to \mathrm{\infty }$. The finite-part ambiguity result shows why the subtraction and its finite normalization condition must be fixed: changing a subtraction by a constant changes the resulting finite limit by that constant.

The conditional area-calibration theorem gives the coefficient $k_B{c}^3\mathrm{/}(4G_N\mathrm{\hslash })$ only on its selected strict subextremal stationary Kerr–Newman family, with its explicit ${\mathrm{A}\mathrm{C}}_{\omega }$ premise, adopted reversible-heat identification, and reduced-temperature model. It does not calibrate a regulator-dependent $G_{{\mathrm{\Lambda }}_n}$ on an arbitrary background. In this definition any such $G_{{\mathrm{\Lambda }}_n}$ is specified model input; when it is identified with $G_N$, the theorem's full hypotheses still apply.

Here ${\mathrm{A}\mathrm{C}}_{\omega }$ means that every sequence $(X_n{)}_{n\ge 1}$ of nonempty sets has a choice function $f$ with $f(n)\in X_n$, as in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). This axiom is part of the conditional calibration interface only; this finite-regulator definition selects no such sequence of objects.

No state evolution, Hawking emission, change of the classical metric caused by matter, or generalized-second-law inequality is part of this definition.