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Conditional restricted classical area increase

Physical scope

A smooth classical Einstein–Maxwell solution with a specified future affine-complete null congruence, persistent compact spacelike cuts, and the adopted null energy condition on that congruence. The result is not a claim about arbitrary or nonsmooth event horizons, quantum states, or generalized entropy.

Statement

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 $M = (C_{m}, E_{m}, J_{m}, T_{m})$ 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 $Ψ \in C_{m} [g, F]$ satisfying $E_{m} [g, F; Ψ] = 0$ on $U$ , and use the same source maps $j = J_{m} [g, F; Ψ]$ and $T^{m} = T_{m} [g, F; Ψ]$ . 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 $\nabla_{k} k = 0$ . In the definitions of Smooth null screen and subextremal Kerr–Newman parameter family, $k$ is affine and its screen expansion is $θ = {tr}_{\overset{ˉ}{g}} B$ .

Assume ${A C}_{ω}$ as defined in The Axiom of Countable Choice ( ${A C}_{ω}$ ). Let $S_{0} \subset H$ be a compact smooth embedded spacelike two-dimensional submanifold transverse to $k$ , and let $λ_{0} \in R$ . Suppose a smooth map $Φ : [λ_{0}, \infty) \times S_{0} \to H$ satisfies $Φ (λ_{0}, x) = x$ and $\partial_{λ} Φ (λ, x) = k_{Φ (λ, x)}$ , with each $Φ_{λ} : = Φ (λ, \cdot)$ an embedding whose image $S_{λ}$ 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 $q_{λ} : = Φ_{λ}^{*} g$ on $S_{0}$ and the area $A_{H} (λ) : = \int_{S_{0}} μ_{q_{λ}} .$ Then $θ \geq 0$ on every future cut, $A_{H}^{'} (λ) = \int_{S_{0}} θ (Φ (λ, x)) μ_{q_{λ}} (x) = \int_{S_{λ}} θ d A \geq 0 (λ > λ_{0}),$ and $A_{H} (λ_{2}) \geq A_{H} (λ_{1})$ whenever $λ_{2} \geq λ_{1} \geq λ_{0}$ .

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.

Facts & Assumptions

Given: The fully specified named model $M$ and its admitted smooth configuration $Ψ$ , including all actual model equations, constraints and source maps; the equation-based Einstein–Maxwell field law and total-stress null energy condition adopted for that same configuration by Restricted smooth horizon Einstein/NEC adoption; a future affine null generator and its smooth flow with compact spacelike cuts for every $λ \geq λ_{0}$ ; and the stated ${A C}_{ω}$ assumption.

[F1]

Restricted smooth horizon Einstein/NEC adoption is parameterized by the complete named matter-model instance $M$ and an admitted configuration solving its actual $E_{m} = 0$ . Its sources are $j = J_{m} [g, F; Ψ]$ and $T^{m} = T_{m} [g, F; Ψ]$ . For this same configuration it adopts $G_{a b} + Λ g_{a b} = (8 π G_{N} / c^{4}) T_{a b}$ and $T_{a b} k^{a} k^{b} \geq 0$ on the specified smooth future null generator, with $T_{a b} = T_{a b}^{E M} + T_{a b}^{m}$ .

[F2]

Equation-based Einstein–Maxwell model defines $G_{a b} = {Ric}_{a b} - \frac{1}{2} R g_{a b}$ and $κ_{E} = 8 π G_{N} / c^{4} > 0$ in the equation-based model.

[F3]

Oriented Lorentzian fields, observers and coordinate conventions fixes $R (X, Y) Z = \nabla_{X} \nabla_{Y} Z - \nabla_{Y} \nabla_{X} Z - \nabla_{[X, Y]} Z$ and ${Ric}_{b d} = R^{a}_{b a d}$ , the same curvature and Ricci convention used by Null focusing and a restricted smooth future area theorem.

[F4]

Smooth null screen and subextremal Kerr–Newman parameter family defines a smooth null hypersurface, its smooth future null generator, the affine condition $\nabla_{k} k = 0$ , the positive screen metric on $k^{⊥} / R k$ , its deformation form $B$ , and the expansion $θ = {tr}_{\overset{ˉ}{g}} B$ .

[F5]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) states ${A C}_{ω}$ : each countable family of nonempty sets has a choice function.

[F6]

Null focusing and a restricted smooth future area theorem: under ${A C}_{ω}$ , a smooth future affine null flow from a compact spacelike cut, with every future cut a compact spacelike embedding and $Ric (k, k) \geq 0$ on the full half-line, has $θ \geq 0$ and satisfies the displayed derivative and monotonicity formulas. Its conditional proof is authored in the current mathematical carrier; its draft status and lack of independent approval do not remove its exact stated conditional interface.

[A1]

The named model's actual field variables, equations, constraints and source maps are fixed input, and $Ψ$ satisfies them on $U$ . The field equation and null energy inequality for its total stress hold at every point of the chosen generator flow; $G_{N} > 0$ and $c > 0$ .

[A2]

The map $Φ$ and all its compact spacelike embedded cuts exist for the whole future half-line $[λ_{0}, \infty)$ ; no finite-affine-parameter breakdown occurs on this specified domain.

[A3]

The Axiom of Countable Choice ${A C}_{ω}$ in [F5] is retained; no stronger choice principle is assumed.

Proof

1.1givenF1F2F3F6A1algebra

For the supplied model and admitted configuration in [F1] and [A1], use $T = T^{E M} + T_{m} [g, F; Ψ]$ from that same configuration. Set $κ_{E} : = 8 π G_{N} / c^{4} > 0$ and contract its adopted field equation from [F1]–[F2] with $k^{a} k^{b}$ . Since $k$ is null, $g_{a b} k^{a} k^{b} = 0$ , so both the cosmological term and the scalar-curvature term in $G_{a b} k^{a} k^{b}$ vanish. The resulting identity is ${Ric}_{a b} k^{a} k^{b} = κ_{E} T_{a b} k^{a} k^{b} \geq 0$ by [A1]. The curvature convention in [F3] identifies this Ricci contraction with the one in [F6]; hence the required null-convergence hypothesis holds along the full flow.

2.1givenF4F5F6A2A3step 1.1algebra∎

The geometric assumptions [F4], [A2], and [A3] match the screen, affine-flow, persistent-cut, and choice hypotheses of [F6]. Applying that mathematical theorem gives $θ \geq 0$ and the stated area derivative identity; its area density is positive, so $A_{H}^{'} (λ) \geq 0$ for $λ > λ_{0}$ . Its endpoint continuity and monotonicity conclusion give $A_{H} (λ_{2}) \geq A_{H} (λ_{1})$ for every $λ_{2} \geq λ_{1} \geq λ_{0}$ . This is a model deduction under the adopted postulates, not an inference from observation.

Remarks

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.

Conditional restricted classical area increase

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources