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.