Restricted smooth horizon Einstein/NEC adoption

Use a smooth, oriented and time-oriented Lorentzian four-manifold $(M,g,N,o)$ of signature $(-,+,+,+)$, as in Thermodynamics-GR fields, currents, reference normalization and units. The manifold orientation is independent of the time orientation. On an open set $U\subseteq M$, let $F$ be a smooth real two-form and use the SI constants and length-coordinate convention of Oriented Lorentzian fields, observers and coordinate conventions. This adoption is parameterized by any complete explicitly supplied named smooth classical matter-model instance $\mathcal{M}=({\mathcal{C}}_m,{\mathcal{E}}_m,{\mathsf{J}}_m,{\mathsf{T}}_m)$ of Equation-based Einstein–Maxwell model. The model data are inputs, not unspecified functions to be inferred from the field equation. They specify the actual matter-field bundles and smooth configuration fields $\mathrm{\Psi }$ in ${\mathcal{C}}_m[g,F]$ on $U$, the exact equations and constraints ${\mathcal{E}}_m[g,F;\mathrm{\Psi }]=0$, all constitutive functions and parameters, and the explicit current and symmetric-stress maps ${\mathsf{J}}_m$ and ${\mathsf{T}}_m$. Any further domain, boundary or initial-data hypotheses of that selected model are retained. A specialization must give these actual equations and maps together with the model's name; providing only the symbols ${\mathcal{E}}_m,{\mathsf{J}}_m,{\mathsf{T}}_m$ or arbitrary smooth $j,T^m$ is not such a specification.

For the supplied model, take an admitted smooth configuration $\mathrm{\Psi }\in {\mathcal{C}}_m[g,F]$ satisfying all of its equations and constraints, and set $j:={\mathsf{J}}_m[g,F;\mathrm{\Psi }],T^m:={\mathsf{T}}_m[g,F;\mathrm{\Psi }]\mathrm{.}$ Thus $j$ is a smooth vector field and $T^m$ is a smooth symmetric covariant two-tensor on the same domain. The postulate and the restricted area deduction are uniform in these fully specified model data; they do not select one constitutive law or restrict the range to source-free matter or dust. The named zero-matter instance, when selected, is precisely ${\mathcal{M}}_0$ of the equation-based supplier. No matter action or circle-representation hypothesis is required for this model input.

Define $T_{ab}:=T_{ab}^{\mathrm{E}\mathrm{M}}+T_{ab}^m,T_{ab}^{\mathrm{E}\mathrm{M}}:=\frac{1}{{\mu }_0}(F_{ac}{F}_b{}^c-\frac{1}{4}{g}_{ab}{F}_{cd}{F}^{cd}),$ where raised indices use $g^{-1}$ and ${\mu }_0>0$. Define $G_{ab}:={\mathrm{Ric}}_{ab}-\frac{1}{2}R{g}_{ab}$ with the curvature convention of Oriented Lorentzian fields, observers and coordinate conventions. Adopt the smooth Maxwell equations $dF=0$ and ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$, together with the equation-based gravitational field law of Equation-based Einstein–Maxwell model on $U$: $G_{ab}+\mathrm{\Lambda }{g}_{ab}=\frac{8\pi G_N}{c^4}{T}_{ab},$ with constant $\mathrm{\Lambda }\in \mathbb{R}$, $G_N>0$, and $c>0$.

For a specified smooth embedded null hypersurface $H\subset U$ and a smooth nowhere-zero future-directed null generator field $k$ along $H$, adopt the null energy condition on this generator: $T_{ab}{k}^a{k}^b\ge 0\text{at every point of }H\mathrm{.}$ Here $H$, $k$, and the screen variables are defined by Smooth null screen and subextremal Kerr–Newman parameter family; the displayed inequality is a physical restriction on the selected classical stress and congruence. It is not a consequence of the field equation and is not asserted for every quantum state. For the restricted area result on this page, the chosen generator must also have the smooth future flow and persistent compact spacelike cuts stated in Null focusing and a restricted smooth future area theorem. This adoption neither constructs such a flow or horizon nor asserts that every event horizon has those regularity and completeness properties. No matter action, singular source, observational claim, or thermodynamic entropy is included.

Let $(M,g,N,o)$ be the smooth, oriented, time-oriented Lorentzian four-manifold of Oriented Lorentzian fields, observers and coordinate conventions, and let $U\subseteq M$ be open. For any complete explicitly supplied named smooth classical matter-model instance $\mathcal{M}=({\mathcal{C}}_m,{\mathcal{E}}_m,{\mathsf{J}}_m,{\mathsf{T}}_m)$ of Restricted smooth horizon Einstein/NEC adoption, retain its specified matter-field bundles, configuration fields, exact equations and constraints, constitutive inputs, regularity, and additional domain or data hypotheses. Take a smooth admitted configuration $\mathrm{\Psi }\in {\mathcal{C}}_m[g,F]$ satisfying ${\mathcal{E}}_m[g,F;\mathrm{\Psi }]=0$ on $U$, and use the same source maps $j={\mathsf{J}}_m[g,F;\mathrm{\Psi }]$ and $T^m={\mathsf{T}}_m[g,F;\mathrm{\Psi }]$. Assume the equation-based Einstein–Maxwell equations and restricted total-stress null energy condition of that postulate for this model and configuration. The claim is uniform over these supplied named models, without requiring a matter action or choosing a particular matter closure. Let $H\subset U$ be a smooth embedded null hypersurface with a smooth nowhere-zero future null generator field $k$ satisfying ${\mathrm{\nabla }}_{k}k=0$. In the definitions of Smooth null screen and subextremal Kerr–Newman parameter family, $k$ is affine and its screen expansion is $\theta ={\mathrm{tr}}_{\bar{g}}B$.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ as defined in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Let $S_0\subset H$ be a compact smooth embedded spacelike two-dimensional submanifold transverse to $k$, and let ${\lambda }_0\in \mathbb{R}$. Suppose a smooth map $\mathrm{\Phi }:[{\lambda }_0,\mathrm{\infty })\times S_0\to H$ satisfies $\mathrm{\Phi }({\lambda }_0,x)=x$ and ${\mathrm{\partial }}_{\lambda }\mathrm{\Phi }(\lambda ,x)=k_{\mathrm{\Phi }(\lambda ,x)}$, with each ${\mathrm{\Phi }}_{\lambda }:=\mathrm{\Phi }(\lambda ,\cdot )$ an embedding whose image $S_{\lambda }$ is compact and spacelike. Thus the same affine parameter runs for every future value and the cuts persist without a caustic or endpoint on this domain. Define the induced Riemannian metric $q_{\lambda }:={\mathrm{\Phi }}_{\lambda }^{\ast }g$ on $S_0$ and the area $A_H(\lambda ):={\int }_{S_0}{\mu }_{q_{\lambda }}\mathrm{.}$ Then $\theta \ge 0$ on every future cut, $A_H^{\mathrm{\prime }}(\lambda )={\int }_{S_0}\theta (\mathrm{\Phi }(\lambda ,x)){\mu }_{q_{\lambda }}(x)={\int }_{S_{\lambda }}\theta dA\ge 0(\lambda >{\lambda }_0),$ and $A_H({\lambda }_2)\ge A_H({\lambda }_1)$ whenever ${\lambda }_2\ge {\lambda }_1\ge {\lambda }_0$.

This is a conditional deduction for the stated smooth classical model and future-complete congruence. It makes no assertion about an arbitrary event horizon, a rough null hypersurface, quantum energy conditions, or a horizon entropy.

The Raychaudhuri and area-variation arguments are contained in the exact conditional mathematical supplier [F6]. The physical step here is the null contraction of the adopted Einstein equation. A different matter model, a failure of the stated energy condition, or a generator that breaks down before the full future half-line is outside this conclusion.

Fix $G_N,c,{\mu }_0>0$ and ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$. Let $m_{\ast }>0$ be the selected SI mass parameter, $J\in \mathbb{R}$ the selected angular-momentum parameter, and $Q_e\in \mathbb{R}$ the selected electric charge parameter, with zero magnetic flux. Define the geometric length parameters and field normalization by $m:=\frac{G_N{m}_{\ast }}{c^2},a:=\frac{J}{m_{\ast }c},q:=\sqrt{\frac{G_N}{4\pi {\epsilon }_0{c}^4}}{Q}_e,\alpha :=\sqrt{\frac{4\pi G_N}{{\mu }_0{c}^4}}\mathrm{.}$ Assume the strict subextremal condition $m^2>a^2+q^2$. Set $s:=\sqrt{m^2-a^2-q^2},r_{+}:=m+s,D:=r_{+}^2+a^2,A_H:=4\pi D\mathrm{.}$ Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ as defined in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), the choice convention inherited by the selected geometric flux parametrization; no full Axiom of Choice is used. In the selected smooth ingoing KN model, the outer Killing horizon has generator $\chi :={\mathrm{\partial }}_v+\frac{a}{D}\mathrm{\Psi },$ where $v$ is a length coordinate and $\mathrm{\Psi }=X{\mathrm{\partial }}_Y-Y{\mathrm{\partial }}_X$ is the smooth rotation field of the unit two-sphere. Normalize time by $t=v\mathrm{/}c$, and define ${\mathrm{\Omega }}_{\mathrm{S}\mathrm{I}}:=\frac{ca}{D},{\kappa }_{\mathrm{a}\mathrm{c}\mathrm{c}}:=\frac{c^{2}s}{D},{\mathrm{\Phi }}_{\mathrm{S}\mathrm{I}}:=\frac{cq{r}_{+}}{\alpha D}\mathrm{.}$ Thus $c\chi ={\mathrm{\partial }}_t+{\mathrm{\Omega }}_{\mathrm{S}\mathrm{I}}\mathrm{\Psi }$; ${\kappa }_{\mathrm{a}\mathrm{c}\mathrm{c}}$ is the acceleration-unit surface gravity; and ${\mathrm{\Phi }}_{\mathrm{S}\mathrm{I}}$ is the voltage in the displayed gauge, constant on the horizon cross-section and vanishing at spatial infinity. The quantities ${\mathrm{\Omega }}_{\mathrm{S}\mathrm{I}}$, ${\kappa }_{\mathrm{a}\mathrm{c}\mathrm{c}}$, and ${\mathrm{\Phi }}_{\mathrm{S}\mathrm{I}}$ are constant on each fixed cross-section because the selected horizon formulas depend only on the family parameters. Let $E_{\mathrm{p}\mathrm{a}\mathrm{r}}:=m_{\ast }{c}^2$ be the energy-valued parameter derived from the selected mass input. For ordinary differentials on the strict physical parameter domain, with $G_N,c,{\mu }_0$ fixed, the exact family identities are $d{E}_{\mathrm{p}\mathrm{a}\mathrm{r}}=\frac{{\kappa }_{\mathrm{a}\mathrm{c}\mathrm{c}}{c}^2}{8\pi G_N}d{A}_H+{\mathrm{\Omega }}_{\mathrm{S}\mathrm{I}}dJ+{\mathrm{\Phi }}_{\mathrm{S}\mathrm{I}}d{Q}_e,$ and $E_{\mathrm{p}\mathrm{a}\mathrm{r}}=\frac{{\kappa }_{\mathrm{a}\mathrm{c}\mathrm{c}}{c}^2}{4\pi G_N}{A}_H+2{\mathrm{\Omega }}_{\mathrm{S}\mathrm{I}}J+{\mathrm{\Phi }}_{\mathrm{S}\mathrm{I}}{Q}_e\mathrm{.}$ The quantities have units $[E_{\mathrm{p}\mathrm{a}\mathrm{r}}]=\mathrm{J}$, $[A_H]={\mathrm{m}}^2$, $[{\kappa }_{\mathrm{a}\mathrm{c}\mathrm{c}}]=\mathrm{m}{\mathrm{s}}^{-2}$, $[{\mathrm{\Omega }}_{\mathrm{S}\mathrm{I}}]={\mathrm{s}}^{-1}$, and $[{\mathrm{\Phi }}_{\mathrm{S}\mathrm{I}}]=\mathrm{V}$. The joint Schwarzschild point $J=Q_e=0$ is included by the separate exact Schwarzschild horizon-data supplier; the generic physical horizon theorem itself states a nonzero-rotation-or-charge hypothesis.

These identities concern the selected stationary family and its stated normalizations. They are not a dynamical-process first law, and $E_{\mathrm{p}\mathrm{a}\mathrm{r}}$ is not identified here with ADM or Komar gravitational energy. No operational temperature, physical entropy, Hawking calibration, or extremal-boundary claim follows from this calculation.

The explicit zero-spin, zero-charge case uses [F6] because the generic physical horizon interface [L1] assumes $a^2+q^2>0$. The strict family excludes the extremal boundary, where $s=0$; no positive-temperature division or thermodynamic interpretation is made.