Linearized focusing and quantum entropy inequalities

Use natural units $c=\mathrm{\hslash }=k_B=1$, metric signature $(-,+,+,+)$, and the curvature convention $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z$ with ${\mathrm{Ric}}_{bd}=g^{ac}{R}_{abcd}$. Let $M$ be a fixed Hausdorff, second-countable, oriented smooth four-manifold. For each dimensionless $ϵ$ in an open interval about $0$, supply a smooth Lorentzian metric $g_{ϵ}$ on $M$ with the stated signature and a time orientation (a chosen future timelike cone). Each $(M,g_{ϵ})$ is globally hyperbolic: it is strongly causal, meaning every point has arbitrarily small neighborhoods that each causal curve meets in a connected parameter interval, and each causal diamond $J^{+}(p)\cap J^{-}(q)$ is compact. Here causal curves are piecewise $C^1$ with consistently future- or past-directed causal tangent, and $J^{\pm }(p)$ denote points reachable from or reaching $p$ by such curves.

For each $ϵ$, let $H_{ϵ}\subset M$ be a smooth embedded three-dimensional submanifold whose induced metric has a one-dimensional radical at every point. Its smooth nowhere-zero future null generator field $k_{ϵ}$ spans that radical and is affine, ${\mathrm{\nabla }}_{k_{ϵ}}^{g_{ϵ}}{k}_{ϵ}=0$. At $p\in H_{ϵ}$ define the screen space ${\mathcal{S}}_{ϵ,p}:=k_{ϵ}^{\perp }\mathrm{/}\mathbb{R}{k}_{ϵ}$, where $k_{ϵ}^{\perp }$ is taken in $T_{p}M$. Its positive screen metric is ${\bar{g}}_{ϵ}([X],[Y]):=g_{ϵ}(X,Y)$, its screen deformation form is $B_{ϵ}([X],[Y]):=g_{ϵ}({\mathrm{\nabla }}_X^{g_{ϵ}}{k}_{ϵ},Y),$ and its expansion and shear are ${\theta }_{ϵ}:={\mathrm{tr}}_{{\bar{g}}_{ϵ}}{B}_{ϵ}$ and ${\sigma }_{ϵ}:=B_{ϵ}-\frac{1}{2}{\theta }_{ϵ}{\bar{g}}_{ϵ}$, respectively. These are the supplied smooth null-screen conventions. The word “horizon” below denotes only $H_{ϵ}$; no causal- or event-horizon property is assumed.

Fix a nonempty compact smooth two-manifold $C$ and ${\lambda }_0\in \mathbb{R}$. For each $ϵ$, supply a smooth map ${\mathrm{\Phi }}_{ϵ}:[{\lambda }_0,\mathrm{\infty })\times C\to H_{ϵ}$ such that ${\mathrm{\partial }}_{\lambda }{\mathrm{\Phi }}_{ϵ}=k_{ϵ}\circ {\mathrm{\Phi }}_{ϵ}$, and every ${\mathrm{\Phi }}_{ϵ,\lambda }:C\to H_{ϵ}$ is an embedding with spacelike image. The family supplies a fixed identification of generator labels and affine normalization across $ϵ$. With $[\lambda ]=L$, this parameterization gives $[k_{ϵ}]=1$ in length coordinates.

Write $q_{ϵ,\lambda }:={\mathrm{\Phi }}_{ϵ,\lambda }^{\ast }{g}_{ϵ}$ for the positive-definite induced metric on $C$, and let ${\mu }_{ϵ,\lambda }$ be its Riemannian area density. On each finite parameter slab, the pulled-back metric, area density, screen deformation form, expansion, and shear are jointly $C^1$ in $(ϵ,\lambda )$ and smooth in $y$; the underlying metric and embedding family has the corresponding $ϵ$-regularity and spatial derivatives needed to define them. Call the reference geometry stationary in this model when $B_0=0$ along every generator; then ${\theta }_0={\sigma }_0=0$. Define $\delta \theta :={{\mathrm{\partial }}_{ϵ}{\theta }_{ϵ}\mid }_0$.

For each $ϵ$, supply a positive normalized state ${\omega }_{ϵ}$ on the selected free scalar-field algebra. Its two-point distribution $W_{ϵ}$ is a bisolution of the selected Klein–Gordon operator and has the supplied Hadamard wavefront relation ${\mathcal{C}}_{g_{ϵ}}^{+}$: pairs of future-null covectors related by parallel propagation along a future-directed null geodesic with common positive scale and opposite sign in the second slot, including the coincident-point case. Supply a state-independent Hadamard singular bidistribution $H_{\mu ,ϵ}$ such that $W_{ϵ}-H_{\mu ,ϵ}$ is smooth, and one fixed local stress prescription across the family, $T_{ab}^{\mathrm{r}\mathrm{e}\mathrm{n},ϵ}(x)={\lim }_{x^{\mathrm{\prime }}\to x}{D}_{ab}^{ϵ}(x,x^{\mathrm{\prime }})(W_{ϵ}-H_{\mu ,ϵ})(x,x^{\mathrm{\prime }})+C_{ab}[g_{ϵ};\mu ,\mathbf{c}](x),$ where $D_{ab}^{ϵ}$ is the supplied finite-order bidifferential operator and $C_{ab}$ the supplied state-independent local curvature correction with the supplier's fixed positive renormalization scale $\mu$ ($[\mu ]=L^{-1}$ in these natural units) and finite renormalization conditions $\mathbf{c}$. Thus the coincidence limit acts on a smooth difference, not on the singular distributions separately. The family $F_{ϵ}(\lambda ,y):=(T_{ab}^{\mathrm{r}\mathrm{e}\mathrm{n},ϵ}{k}_{ϵ}^a{k}_{ϵ}^b)({\mathrm{\Phi }}_{ϵ}(\lambda ,y))$ is jointly $C^1$ in $ϵ$ and continuous in $(\lambda ,y)$ on each finite parameter slab. No independent graviton-fluctuation source is included.

Restrict every model input to the admissible domain supplied with its chosen effective-model instance and to the regime below the supplier's positive energy cutoff $E_{\mathrm{c}\mathrm{u}\mathrm{t}}$. The supplier's SI cutoff is measured in joules; in the natural units fixed here, $[E_{\mathrm{c}\mathrm{u}\mathrm{t}}]=L^{-1}$. Neither the cutoff value nor its admissible input domain is chosen here. Keep any prescribed retarded data as supplied inputs, without imposing a collapse geometry or scattering history. Select the Einstein-Hilbert submodel by setting the supplier's curvature-squared couplings to zero. Write $G_N$ for the supplier's Newton's constant $G$ in the natural-unit convention $c=\mathrm{\hslash }=k_B=1$, so $G_N>0$ has dimension $L^2$. Define $\delta T_{kk}(\lambda ,y):={{\mathrm{\partial }}_{ϵ}{F}_{ϵ}(\lambda ,y)\mid }_0\mathrm{.}$ Then $[\lambda ]=L$, $[k_{ϵ}]=1$, $[\delta T_{kk}]=L^{-4}$, and $[\delta \theta ]=L^{-1}$. Assume $\delta T_{kk}$ is continuous, its tail is absolutely integrable on each generator, and ${\int }_{C}d{A}_0{\int }_{{\lambda }_0}^{\mathrm{\infty }}(v-{\lambda }_0)\mathrm{\mid }\delta T_{kk}(v,y)\mathrm{\mid }dv<\mathrm{\infty },$ where $d{A}_0:={\mu }_{0,{\lambda }_0}$ is the reference background area density. Adopt, only to first order in $ϵ$, the linearized focusing equation and future boundary condition ${\mathrm{\partial }}_{\lambda }\delta \theta (\lambda ,y)=-8\pi G_N\delta T_{kk}(\lambda ,y),{\lim }_{\lambda \to \mathrm{\infty }}\delta \theta (\lambda ,y)=0.$ Terms quadratic in expansion or shear are omitted only at this first perturbative order: both vanish in the stationary reference geometry and are of order $ϵ$ in the supplied family, so their squares are of order ${ϵ}^2$. This postulate is an adopted model equation; it neither asserts that the model describes nature nor identifies its null-stress moment with modular energy or entropy.

Semiclassical equation with fixed counterterms and state

Smooth null screen and subextremal Kerr–Newman parameter family

Conditional restricted classical area increase

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

Harvey Reall, Part III Black Holes

The Einstein-Hilbert specialization and the first-order truncation are part of the model. In particular, this is not a conclusion about nonzero curvature-squared couplings. The published restricted classical area theorem uses a separate null-energy premise; no such premise is adopted here.

Use natural units $c=\mathrm{\hslash }=k_B=1$, metric signature $(-,+,+,+)$, and $[\lambda ]=L$. Write $G_N$ for the Newton constant $G$ in the supplied semiclassical equation after setting $c=\mathrm{\hslash }=k_B=1$; then $G_N>0$ and $[G_N]=L^2$. Take a model instance of the adopted Einstein-Hilbert focusing postulate, with all its inputs restricted to the model-instance admissible domain below the supplied effective energy cutoff $E_{\mathrm{c}\mathrm{u}\mathrm{t}}$, whose dimension in these natural units is $L^{-1}$. The numerical cutoff and admissible domain are supplied with that instance.

Let $M$ be the common smooth four-manifold of the family of smooth time-oriented Lorentzian metrics $g_{ϵ}$ of signature $(-,+,+,+)$, and let $H_{ϵ}\subset M$ be smooth embedded null hypersurfaces whose induced metrics have one-dimensional radicals. Their smooth nowhere-zero future null generator fields $k_{ϵ}$ span those radicals and are affine, ${\mathrm{\nabla }}_{k_{ϵ}}^{g_{ϵ}}{k}_{ϵ}=0$. Use the postulate's global-hyperbolicity convention: strong causality, meaning that every point has arbitrarily small neighborhoods met by each causal curve in a connected parameter interval, and compact causal diamonds $J^{+}(p)\cap J^{-}(q)$. Let $C$ be a nonempty compact smooth two-manifold and ${\lambda }_0\in \mathbb{R}$. For dimensionless $ϵ$ in an open interval about $0$, take the supplied family ${\mathrm{\Phi }}_{ϵ}:[{\lambda }_0,\mathrm{\infty })\times C\to H_{ϵ},{\mathrm{\partial }}_{\lambda }{\mathrm{\Phi }}_{ϵ}=k_{ϵ}\circ {\mathrm{\Phi }}_{ϵ},$ with every ${\mathrm{\Phi }}_{ϵ,\lambda }:C\to H_{ϵ}$ an embedding with spacelike image and with fixed generator labels and affine normalization across $ϵ$. The metric and embedding families have the postulate's corresponding $ϵ$-regularity on their common ambient manifold. The pulled-back metric, area density, screen deformation form, expansion, and shear are jointly $C^1$ in $(ϵ,\lambda )$ and smooth in $y$ on each finite parameter slab, with the spatial regularity needed to define them.

At $p\in H_{ϵ}$, use the screen quotient ${\mathcal{S}}_{ϵ,p}=k_{ϵ}^{\perp }\mathrm{/}\mathbb{R}{k}_{ϵ}$ with positive metric ${\bar{g}}_{ϵ}([X],[Y])=g_{ϵ}(X,Y)$. Its deformation form and expansion are $B_{ϵ}([X],[Y])=g_{ϵ}({\mathrm{\nabla }}_X^{g_{ϵ}}{k}_{ϵ},Y),{\theta }_{ϵ}={\mathrm{tr}}_{{\bar{g}}_{ϵ}}{B}_{ϵ}\mathrm{.}$ Assume the reference geometry is stationary in this model in the specified sense $B_0=0$ along every generator, so ${\theta }_0$ and the shear ${\sigma }_0=B_0-\frac{1}{2}{\theta }_0{\bar{g}}_0$ vanish. Put $q_{ϵ,\lambda }:={\mathrm{\Phi }}_{ϵ,\lambda }^{\ast }{g}_{ϵ}$ for the positive-definite induced metric on $C$, and let ${\mu }_{ϵ,\lambda }$ be its Riemannian area density. Define $A_{ϵ}(\lambda ):={\int }_C{\mu }_{ϵ,\lambda },\delta A(\lambda ):={{\mathrm{\partial }}_{ϵ}{A}_{ϵ}(\lambda )\mid }_0,d{A}_0:={\mu }_{0,{\lambda }_0}\mathrm{.}$

For each $ϵ$, the supplied state ${\omega }_{ϵ}$ is a positive normalized linear functional; its two-point distribution $W_{ϵ}$ is a bisolution of the selected Klein-Gordon operator with Hadamard wavefront relation ${\mathcal{C}}_{g_{ϵ}}^{+}$: its paired future-null covectors are related by parallel propagation along a future-directed null geodesic, with common positive scale and opposite sign in the second slot, including coincidence. Use the postulate's single fixed local stress prescription $T_{ab}^{\mathrm{r}\mathrm{e}\mathrm{n},ϵ}(x)={\lim }_{x^{\mathrm{\prime }}\to x}{D}_{ab}^{ϵ}(x,x^{\mathrm{\prime }})(W_{ϵ}-H_{\mu ,ϵ})(x,x^{\mathrm{\prime }})+C_{ab}[g_{ϵ};\mu ,\mathbf{c}](x),$ where $W_{ϵ}-H_{\mu ,ϵ}$ is smooth, $D_{ab}^{ϵ}$ is the supplied finite-order bidifferential operator, and $C_{ab}$ is the fixed state-independent local curvature correction. Define $\delta \theta (\lambda ,y):={{\mathrm{\partial }}_{ϵ}{\theta }_{ϵ}(\lambda ,y)\mid }_0,F_{ϵ}(\lambda ,y):=(T_{ab}^{\mathrm{r}\mathrm{e}\mathrm{n},ϵ}{k}_{ϵ}^a{k}_{ϵ}^b)({\mathrm{\Phi }}_{ϵ}(\lambda ,y)),\delta T_{kk}(\lambda ,y):={{\mathrm{\partial }}_{ϵ}{F}_{ϵ}(\lambda ,y)\mid }_0\mathrm{.}$ Assume $F_{ϵ}$ has the postulate's joint $C^1$ dependence in $ϵ$ and continuous dependence on $(\lambda ,y)$ on each finite parameter slab. Then $[k_{ϵ}]=1$, $[\delta T_{kk}]=L^{-4}$, and $[\delta \theta ]=L^{-1}$.

Assume $\delta T_{kk}$ is continuous, has an absolutely integrable tail on each generator, and satisfies the absolute first-moment bound ${\int }_{C}d{A}_0{\int }_{{\lambda }_0}^{\mathrm{\infty }}(v-{\lambda }_0)\mathrm{\mid }\delta T_{kk}(v,y)\mathrm{\mid }dv<\mathrm{\infty }\mathrm{.}$ The postulate adopts, only to first order in $ϵ$, the signed linearized focusing equation and future boundary condition ${\mathrm{\partial }}_{\lambda }\delta \theta (\lambda ,y)=-8\pi G_N\delta T_{kk}(\lambda ,y),{\lim }_{\lambda \to \mathrm{\infty }}\delta \theta (\lambda ,y)=0.$ Then the finite limit $\delta A_{\mathrm{\infty }}:={\lim }_{R\to \mathrm{\infty }}\delta A(R)$ exists and, for every $\lambda \ge {\lambda }_0$, $\delta A_{\mathrm{\infty }}-\delta A(\lambda )=8\pi G_N{\int }_{C}d{A}_0{\int }_{\lambda }^{\mathrm{\infty }}(v-\lambda )\delta T_{kk}(v,y)dv\mathrm{.}$ Consequently, if $M_{\lambda }:={\int }_{C}d{A}_0{\int }_{\lambda }^{\mathrm{\infty }}(v-\lambda )\delta T_{kk}(v,y)dv,$ then $\frac{\delta A_{\mathrm{\infty }}-\delta A(\lambda )}{4G_N}=2\pi M_{\lambda }\mathrm{.}$ Here $[\delta A]=L^2$ and $M_{\lambda }$ is dimensionless. The quotient by $4G_N$ is only a geometric normalization; it is not an entropy identification.

Linearized future horizon focusing model

Smooth null screen and subextremal Kerr–Newman parameter family

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

Harvey Reall, Part III Black Holes

The proof derives the cut-area variation locally and uses absolute integrability only for the future limit and Fubini step. The model does not identify $M_{\lambda }$ with a modular generator; such an identification would require a separate algebraic bridge. It does not infer monotonicity when $\delta T_{kk}$ changes sign.