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Positive symmetric Maxwell field-strength reduction

1 · Prerequisites

2 · Summary

This page studies the source-free Einstein–Maxwell Cauchy problem using the real two-form $F$ and its six electric/magnetic field-strength components. It fixes signature $(-, +, +, +)$ , the future-normal convention $K (X, Y) = g (\nabla_{X} n, Y)$ , orientation $ϵ_{123} = 1$ , and SI frame variables $e = E / c$ , $b = B$ ; no global potential $A$ is assumed.

Positive symmetric Maxwell field-strength reduction derives the six first-order evolution equations directly from $d F = 0$ and $d ⋆_{g} F = 0$ . Its spatial symbol is the symmetric curl block with eigenvalues $0, 0, \pm ∣ ξ ∣, \pm ∣ ξ ∣$ , positive time matrix, and the two Gauss constraints. Maxwell Gauss constraints propagate by form transport transports those constraints by the differential-form identities $d^{2} F = d^{2} ⋆_{g} F = 0$ , including the finite regularity needed by the high-Sobolev result.

Coupled reduced Einstein–Maxwell local evolution checks that this Maxwell block combines with the harmonic Einstein wave block without derivative coupling through the algebraic Maxwell stress. Einstein harmonic gauge propagation with Maxwell on shell recovers the unreduced Einstein equation when the initial harmonic vector vanishes, from the initial constraints, stress conservation and a local zero-data wave-energy argument. Local high-Sobolev and smooth electrovacuum gives the periodic $H^{s}$ result for $s \geq 5$ , its smooth-persistence and gauge-uniqueness claims, and a localized smooth existence clause on general initial slices. Local Sobolev stability and continuation for the reduced electrovacuum system records the endpoint continuation argument and $H^{s - 1}$ stability under fixed compact-state bounds.

Smooth local geometric existence and uniqueness for source-free Einstein–Maxwell data verifies the complete-data local Einstein–Maxwell model against the generic covariant local-development theorem. Causal and Cauchy-surface inputs for smooth electrovacuum developments records the actual causal and Cauchy-surface interfaces used by the local and global arguments. Source-free smooth maximal globally hyperbolic Einstein–Maxwell development applies the current smooth GHD quotient theorem only after those local obligations are checked. The original Sbierski result is vacuum-only; the Einstein–Maxwell conclusion is conditional on the locally verified model hypotheses. Maximality is within the smooth globally hyperbolic development class and supplies no geodesic-completeness result.

Conditional status. The CP01 item files for the electrovacuum model and data definitions, Maxwell stress identity, Gauss–Codazzi constraints, harmonic initialization, and periodic solver are present on disk, and they are listed in the batch-6 manifest, but remain draft with verification precheck pending. Their actual statements and proofs were inspected; the consumers use their exact interfaces conditionally and remain escalated until those draft proof statuses are resolved. The imported GR existence, Sobolev, local-development and maximal-development items used here are marked published, with precheck pending where recorded in their actual metadata. This is a mathematical Cauchy analysis; it reports no measurements or empirical inference.

Geodesic completeness here means that every maximal solution of ${\ddot{x}}^{a} + Γ^{a}_{b c} {\dot{x}}^{b} {\dot{x}}^{c} = 0$ has affine-parameter domain all of $R$ ; maximality forbids same-parameter extension to a larger interval. The development results make no such affine-domain assertion.

The classical Maxwell stress identity is proved for a $C^{1}$ metric and closed $C^{1}$ two-form, so the finite-Sobolev constraint recovery and endpoint restart do not invoke a smooth-only stress theorem.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck pendingOpen item page →

Maxwell stress divergence for classical fields

Statement

Let $M$ be a smooth Hausdorff, second-countable, oriented four-manifold, let $g$ be a time-oriented $C^{1}$ Lorentzian metric of signature $(-, +, +, +)$ , and let $F$ be a real $C^{1}$ two-form with $d F = 0$ . Use the connection and stress conventions of Electrovacuum tensor Cauchy system, interpreted here by their explicit coordinate formulas at this regularity. Fix $μ_{0} > 0$ and put

$T_{a b}^{E M} = μ_{0}^{- 1} (F_{a c} F_{b}^{c} - \frac{1}{4} g_{a b} F_{c d} F^{c d}) .$

The Christoffel formula defines a continuous metric-compatible torsion-free connection, $T^{E M}$ is $C^{1}$ , and the following continuous pointwise identity holds:

$\nabla_{a} (T^{E M})^{a}_{b} = μ_{0}^{- 1} F_{b c} \nabla_{a} F^{a c} .$

In particular, if a continuous vector field $J$ satisfies $\nabla_{a} F^{a b} = - μ_{0} J^{b}$ , then $\nabla^{a} T_{a b}^{E M} = - F_{b c} J^{c}$ . For $J = 0$ the stress is conserved. No Einstein equation, smoothness of $F$ beyond $C^{1}$ , or global potential/exactness hypothesis is needed.

At each point, for a future unit normal $n$ and a positively oriented orthonormal spatial basis $(e_{i})$ , set $e_{i} = F (e_{i}, n)$ and $b_{i} = \frac{1}{2} ϵ_{i}^{j k} F (e_{j}, e_{k})$ , with $ϵ_{123} = 1$ . The algebraic observer identities are

$T^{E M} (n, n) = \frac{∣ e ∣^{2} + ∣ b ∣^{2}}{2 μ_{0}}, T^{E M} (n, X) = - \frac{1}{μ_{0}} ⟨ e \times b, X ⟩$

for every spatial $X$ . These formulas have the same signs and normalization as the smooth electrovacuum convention. They assert no field existence or regularity threshold for a coupled PDE.

Facts & Assumptions

Given: The $C^{1}$ metric and closed two-form, positive $μ_{0}$ , and the continuous sourced-divergence hypothesis only when that conclusion is used.

[F1]

Electrovacuum tensor Cauchy system fixes $T^{E M} = μ_{0}^{- 1} (F_{a c} F_{b}^{c} - \frac{1}{4} g_{a b} F_{c d} F^{c d})$ , $F_{0 i} = - e_{i}$ , $F_{i j} = ϵ_{i j k} b_{k}$ , and the sourced sign $\nabla_{a} F^{a b} = - μ_{0} J^{b}$ . Its smooth-field domain is not used to infer the classical identity below.

Proof

Proof technique: differentiate the explicit tensor and contract the closedness identity; check the observer projections by finite algebra.

1.1givenF1algebra

In a chart the inverse matrix of $g$ is $C^{1}$ . Its Christoffel formula has continuous coefficients symmetric in its lower indices. Substitution into $\nabla_{a} g_{b c}$ cancels the first metric derivatives and gives zero, so raising indices commutes with the connection on $C^{1}$ tensors. The antisymmetrized connection terms also cancel, giving $\nabla_{a} F_{b c} + \nabla_{b} F_{c a} + \nabla_{c} F_{a b} = 0$ from $d F = 0$ . Multiplying by $F^{a c}$ and exchanging $a, c$ in the third term gives $2 F^{a c} \nabla_{a} F_{b c} = \frac{1}{2} \partial_{b} (F_{c d} F^{c d})$ . Every term is continuous and uses only the stated first derivatives.

2.1givenF1step 1.1algebra

Raise the first stress index and differentiate by the product rule: $\nabla_{a} (T^{E M})^{a}_{b} = μ_{0}^{- 1} [(\nabla_{a} F^{a c}) F_{b c} + F^{a c} \nabla_{a} F_{b c} - \frac{1}{4} \partial_{b} (F_{c d} F^{c d})]$ . The last two terms cancel by step 1.1. Inserting the sourced equation then gives $- F_{b c} J^{c}$ ; setting $J = 0$ gives conservation. Metric compatibility identifies this with $\nabla^{a} T_{a b}^{E M}$ . Thus the identity holds at $C^{1}$ without applying a smooth-only stress theorem.

3.1givenF1algebra∎

In the orthonormal basis, $F_{0 i} = - e_{i}$ and $F_{i j} = ϵ_{i j k} b_{k}$ , so $F_{a b} F^{a b} = 2 (∣ b ∣^{2} - ∣ e ∣^{2})$ . The $00$ component of $F_{a c} F_{b}^{c}$ is $∣ e ∣^{2}$ , and its $0 i$ component is $- e_{j} ϵ_{i j k} b_{k} = - (e \times b)_{i}$ . Substitution into the stress formula gives its two displayed observer projections. All calculations are at a single point, require no derivatives of the basis, and apply also when either field is zero.

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Positive symmetric Maxwell field-strength reduction

Statement

Let $U \subseteq R \times Σ$ be a coordinate neighborhood in an oriented smooth Lorentzian four-manifold of signature $(-, +, +, +)$ , foliated by spacelike $s$ -levels. Write $g = - α^{2} d s^{2} + h_{i j} (d x^{i} + β^{i} d s) (d x^{j} + β^{j} d s), α > 0, h = (h_{i j}) > 0,$ and let $n = α^{- 1} (\partial_{s} - β^{i} \partial_{i})$ be the future unit normal. On a smaller neighborhood choose a positively oriented orthonormal spatial frame $(e_{1}, e_{2}, e_{3})$ by Gram–Schmidt. Let $D$ be the Levi-Civita derivative of $h$ , define ${div}_{h} v : = D_{i} v^{i}$ and $({curl}_{h} v)_{i} : = ϵ_{i}^{j k} D_{j} v_{k}$ , and define the spatial cross product by $(u \times v)_{i} = ϵ_{i j k} u_{j} v_{k}$ in this frame. For a smooth real two-form $F$ , define the real spatial components $e_{i} : = F (e_{i}, n)$ and $b_{i} : = \frac{1}{2} ϵ_{i}^{j k} F (e_{j}, e_{k})$ , with $ϵ_{123} = 1$ ; thus $F_{0 i} = - e_{i}$ and $F_{i j} = ϵ_{i j k} b_{k}$ in the adapted tetrad. Assume the source-free model of draft definition Electrovacuum tensor Cauchy system, whose actual equations are $d F = 0$ and $d ⋆_{g} F = 0$ ; its draft status and manifest omission are recorded in [L1].

The six evolution equations for $(e, b)$ have principal part $n (e) - {curl}_{h} b = terms with no derivatives of (e, b), n (b) + {curl}_{h} e = terms with no derivatives of (e, b) .$ For a spatial covector $ξ$ , let $C (ξ) v : = ξ^{♯} \times v$ in the oriented orthonormal frame. Set $θ^{0} : = α d s = - g (n, \cdot)$ and decompose any spacetime covector as $ζ = ζ (n) θ^{0} + ζ_{⊥}$ , where $ζ_{⊥} (n) = 0$ and $∣ ζ_{⊥} ∣_{h}$ is its spatial norm. The spatial principal block is the linear map on $R^{3} \oplus R^{3}$ defined by $S (ξ) (u, v) = (- C (ξ) v, C (ξ) u)$ . It is real symmetric, has eigenvalues $0, 0, ∣ ξ ∣, ∣ ξ ∣, - ∣ ξ ∣, - ∣ ξ ∣$ , and satisfies $∥ S (ξ) ∥ = ∣ ξ ∣$ . Therefore the principal symbol $P (ζ) = ζ (n) I_{6} + S (ζ_{⊥})$ is positive definite for every covector with $ζ (n) > ∣ ζ_{⊥} ∣_{h}$ . In the adapted coordinates its time matrix is $α^{- 1} I_{6}$ , and every coordinate spatial matrix is symmetric. Derivatives of the frame and metric enter only as smooth lower-order coefficients depending on $(g, g^{- 1}, \partial g, e, b)$ ; in a first-order metric state they are zero-order terms. The remaining two Maxwell equations are the constraints ${div}_{h} e = 0$ and ${div}_{h} b = 0$ .

Facts & Assumptions

Given: The oriented, time-oriented Lorentzian structure, a smooth spacelike foliation, a positive lapse and spatial metric, and a smooth two-form $F$ .

[F1]

Oriented Lorentzian fields, observers and coordinate conventions distinguishes the observer, adapted chart, and tetrad. In a positively oriented future tetrad with $E_{0} = n$ , its field convention is $F_{0 i} = - E_{i} / c$ and $F_{i j} = ϵ_{i j k} B_{k}$ ; here $e_{i} = E_{i} / c$ and $b_{i} = B_{i}$ are the mathematical components.

[F2]

Observer electric/magnetic reconstruction and invariants gives the pointwise reconstruction of the two-form from these six spatial components; Lorentzian exterior calculus and Hodge star defines $⋆_{g}$ by the oriented Lorentzian volume form and gives $(⋆_{g} F)_{a b} = \frac{1}{2} ({v o l}_{g})_{a b c d} F^{c d}$ .

[L1]

Draft supplier Electrovacuum tensor Cauchy system: its actual Definition fixes a smooth oriented, time-oriented Lorentzian four-manifold of signature $(-, +, +, +)$ , the source-free equations $d F = d ⋆_{g} F = 0$ , the SI constants, and the algebraic stress tensor. The item file is listed in the batch-6 manifest; its status is draft and its verification precheck is pending; this use remains conditional on that actual draft status.

Proof

1.1givenF1F2L1algebra

On a sufficiently small neighborhood the positive spatial metric makes each Gram–Schmidt denominator strictly positive. Hence the adapted orthonormal frame and its inverse coframe are smooth functions of $g$ ; differentiating the frame introduces only $g$ and $\partial g$ . In that frame the six components of $d F$ and $d ⋆_{g} F$ with one normal slot are exactly the six evolution equations. On a slice with induced metric $h$ and orientation inherited from the oriented tetrad, let ${v o l}_{h}$ be the corresponding Riemannian volume form. The all-spatial restrictions are $F ∣_{Σ} = ι_{b} {v o l}_{h}$ and $(⋆_{g} F) ∣_{Σ} = ι_{e} {v o l}_{h}$ : in the oriented orthonormal tetrad their components are $F_{i j} = ϵ_{i j k} b_{k}$ and $(⋆_{g} F)_{i j} = ϵ_{i j k} e_{k}$ . Here $(ι_{v} {v o l}_{h}) (X, Y) : = {v o l}_{h} (v, X, Y)$ . In oriented coordinates, ${v o l}_{h} = \sqrt{\det (h_{i j})} d x^{1} \land d x^{2} \land d x^{3}$ , so $d_{Σ} (ι_{v} {v o l}_{h}) = \partial_{i} (\sqrt{\det h} v^{i}) d x^{1} \land d x^{2} \land d x^{3} = (D_{i} v^{i}) {v o l}_{h}$ , using $Γ^{i}_{i k} = \partial_{k} \log \sqrt{\det h}$ from the Levi-Civita formula and determinant differentiation. Thus their all-spatial components vanish exactly when the two Gauss divergences vanish.

2.1step 1.1F1F2L1algebra

Freeze the tetrad at a point and use its timelike vector as the time direction. To make the Hodge-to-divergence conversion explicit, set $ε_{a b c d} : = ({v o l}_{g})_{a b c d}$ and raise its indices with $g^{- 1}$ . In a positively oriented chart $ε_{0123} = \sqrt{∣ \det g ∣}$ ; differentiating the determinant gives $Γ^{a}_{a c} = \partial_{c} \log \sqrt{∣ \det g ∣}$ , and the covariant derivative formula for a top form then gives $\nabla_{a} ε_{b c d e} = 0$ . The Hodge formula $(⋆_{g} F)_{c d} = \frac{1}{2} ε_{c d e f} F^{e f}$ , torsion-freeness and metric-compatibility, and $ε^{b a c d} ε_{c d e f} = - 2 (δ_{e}^{b} δ_{f}^{a} - δ_{f}^{b} δ_{e}^{a})$ give the exact identity $\nabla_{a} F^{a b} = \frac{1}{3!} ε^{b a c d} (d ⋆_{g} F)_{a c d} = \frac{1}{\sqrt{∣ \det g ∣}} \partial_{a} (\sqrt{∣ \det g ∣} F^{a b}) .$ For the first equality, expand $(d ⋆_{g} F)_{a c d} = 3 \nabla_{[a} (⋆_{g} F)_{c d]}$ and contract; the second follows from the same Christoffel trace identity and $Γ^{b}_{a c} F^{a c} = 0$ . Thus $d ⋆_{g} F = 0$ is equivalent to $\nabla_{a} F^{a b} = 0$ , and the $(0, i, j)$ components of $d F = 0$ , with $F_{0 i} = - e_{i}$ and $F_{i j} = ϵ_{i j k} b_{k}$ , give $n (b) + curl e = 0$ , while the spatial components of this divergence equation give $n (e) - curl b = 0$ . Hence the spatial symbol in the convention $n (U) + A^{j} e_{j} (U) = 0$ is $S (ξ)$ . Since $C (ξ)^{T} = - C (ξ)$ , its two off-diagonal blocks are transposes and $S (ξ)^{T} = S (ξ)$ . Also $∣ C (ξ) v ∣^{2} = ∣ ξ ∣^{2} ∣ v ∣^{2} - (ξ \cdot v)^{2}$ , so $∥ C (ξ) ∥ = ∣ ξ ∣$ and $∥ S (ξ) ∥ = ∣ ξ ∣$ .

3.1step 2.1algebra

The identity $C (ξ)^{2} v = ξ (ξ \cdot v) - ∣ ξ ∣^{2} v$ shows that $- C (ξ)^{2}$ has eigenvalues $0, ∣ ξ ∣^{2}, ∣ ξ ∣^{2}$ . Squaring $S (ξ)$ therefore gives eigenvalues $0, 0, ∣ ξ ∣^{2}, ∣ ξ ∣^{2}, ∣ ξ ∣^{2}, ∣ ξ ∣^{2}$ ; because $S$ is symmetric and has trace zero, its eigenvalues are $0, 0, ∣ ξ ∣, ∣ ξ ∣, - ∣ ξ ∣, - ∣ ξ ∣$ . For any $z \in R^{6}$ , $z^{T} P (ζ) z \geq (ζ (n) - ∣ ζ_{⊥} ∣_{h}) ∣ z ∣^{2}$ , proving positivity on the stated future covector cone. Taking $ζ = d s$ gives $d s (n) = α^{- 1}$ and $ζ_{⊥} = 0$ , hence the positive time matrix $α^{- 1} I_{6}$ . The shift contributes only scalar multiples of $I_{6}$ to spatial matrices, so those matrices remain symmetric.

4.1step 1.1step 3.1givenL1algebra∎

The frame-derivative, Hodge-coefficient, and commutator terms contain at most first derivatives of $g$ and no derivatives of $(e, b)$ ; they are smooth on the open state set where $g$ is Lorentzian, $α > 0$ , and $h > 0$ . This proves the claimed lower-order form and completes the conditional reduction proof.

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Maxwell Gauss constraints propagate by form transport

Statement

Let $M$ be a smooth oriented four-manifold with a time-oriented $C^{1}$ Lorentzian metric $g$ of signature $(-, +, +, +)$ . Let a neighborhood of the initial slice $Σ_{0}$ have a $C^{1}$ spacelike foliation with a $C^{1}$ future unit normal field $n$ , transverse to $Σ_{0}$ . Here the normal's $C^{1}$ regularity is an explicit hypothesis; a merely $C^{1}$ defining time function alone need not provide it. Smooth coordinate time levels with a $C^{1}$ metric satisfy this hypothesis. Let $F$ be a $C^{1}$ real two-form on this neighborhood. Define its electric and magnetic spatial vectors by $e_{i} = F (e_{i}, n)$ and $b_{i} = \frac{1}{2} ϵ_{i}^{j k} F (e_{j}, e_{k})$ in a positively oriented orthonormal spatial frame. Suppose the six evolution equations are $ι_{n} d F = 0, ι_{n} d (⋆_{g} F) = 0,$ and suppose the initial Gauss constraints are $D_{i} e^{i} = 0$ and $D_{i} b^{i} = 0$ on $Σ_{0}$ , where $D$ is the Levi-Civita derivative of its induced $C^{1}$ Riemannian metric $h$ . These six equations are the normal-slot component equations of the field-strength reduction Positive symmetric Maxwell field-strength reduction; their displayed first-derivative formulation is used here directly, without assuming that supplier's smooth-field hypotheses. Then $d F = 0$ and $d (⋆_{g} F) = 0$ on a neighborhood of each point of $Σ_{0}$ . No Einstein equation, stress conservation law, smooth potential, or global exactness of $F$ is needed.

Facts & Assumptions

Given: The oriented $C^{1}$ Lorentzian metric and two-form, the explicit $C^{1}$ transverse normal, the two normal-contraction evolution identities, and the initial electric and magnetic divergence constraints.

[F1]

The displayed evolution equations are $ι_{n} d F = ι_{n} d (⋆_{g} F) = 0$ . A three-form in four dimensions has three independent normal-slot components, so these are six scalar first-order equations. The normal-slot equations and the all-spatial components are the pointwise decomposition derived in proof step 1.1 of Positive symmetric Maxwell field-strength reduction: in an oriented orthonormal tetrad, $F_{i j} = ϵ_{i j k} b_{k}$ and $(⋆_{g} F)_{i j} = ϵ_{i j k} e_{k}$ , while $d_{Σ} (ι_{v} {v o l}_{h}) = (D_{i} v^{i}) {v o l}_{h}$ . That computation uses only first derivatives of $F$ and the $C^{1}$ metric and normal, so it applies at the regularity stated here; it identifies the two all-spatial components with the initial Gauss divergences.

[F2]

Lorentzian exterior calculus and Hodge star defines the Hodge star from the oriented Lorentzian volume form and gives $d^{2} = 0$ ; The metric volume form, current flux through 3-faces and finite-chain charges fixes ${v o l}_{g} = \sqrt{∣ \det g ∣} d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ .

[L1]

Draft supplier Electrovacuum tensor Cauchy system fixes the source-free Maxwell equations as $d F = d ⋆_{g} F = 0$ on the oriented Lorentzian model. Its item file is listed in the batch-6 manifest; its status is draft and its verification precheck is pending; this use remains conditional on that actual draft status.

[F3]

A locally Lipschitz vector field has a unique local flow, the flow is $C^{1}$ in the initial point for a $C^{1}$ field, and the $C^{1}$ inverse function theorem applies when the flow map is transverse to $Σ_{0}$ (Picard-Lindelöf local existence and uniqueness for first-order systems, $C^{1}$ dependence of solutions on initial data, The Euclidean inverse function theorem).

Proof

1.1givenF1F2L1algebra

Put $U : = d F$ and $V : = d ⋆_{g} F$ . They are continuous three-forms. By [F1], $ι_{n} U = ι_{n} V = 0$ throughout the evolution. In a local frame with timelike vector $n$ , a three-form annihilated by $n$ has only its spatial volume component, so there are unique continuous scalars $q, r$ with $U = q ι_{n} {v o l}_{g}$ and $V = r ι_{n} {v o l}_{g}$ . On $Σ_{0}$ , the spatial components of $d F$ and $d ⋆_{g} F$ are respectively $(D_{i} b^{i}) {v o l}_{h}$ and, up to the fixed orientation sign, $(D_{i} e^{i}) {v o l}_{h}$ . Thus $q = r = 0$ on the initial slice.

2.1step 1.1F2algebra

Since $d U = d^{2} F = 0$ and $d V = d^{2} ⋆_{g} F = 0$ distributionally, the coordinate formula for exterior derivative gives $\partial_{μ} (\sqrt{∣ \det g ∣} q n^{μ}) = 0$ and the same equation for $r$ . The identity $\partial_{μ} (\sqrt{∣ \det g ∣} n^{μ}) = \sqrt{∣ \det g ∣} \nabla_{μ} n^{μ}$ follows from $Γ^{μ}_{μ ν} = \partial_{ν} \log \sqrt{∣ \det g ∣}$ , itself obtained by differentiating the determinant and using the Levi-Civita formula. Therefore $n (q) + q {div}_{g} n = 0$ and $n (r) + r {div}_{g} n = 0$ in the distributional sense.

3.1givenstep 1.1step 2.1F3algebra

The field $n$ is $C^{1}$ and transverse to $Σ_{0}$ by hypothesis. Its local flow map $Φ (τ, x)$ is $C^{1}$ and the differential at $τ = 0$ maps the time direction to $n$ and the slice directions identically, hence is invertible. By [F3], after shrinking, $(τ, x)$ are flow coordinates. Pullback preserves distributional closedness of these continuous three-forms under the $C^{1}$ flow chart: test against compactly supported smooth functions, use the change-of-variables formula, and approximate the resulting compactly supported $C^{1}$ test functions in their $C^{1}$ norm. In these coordinates $n = \partial_{τ}$ and ${v o l}_{g} = J (τ, x) d τ \land d x^{1} \land d x^{2} \land d x^{3}$ with continuous $J > 0$ . Thus $d U = 0$ reads $\partial_{τ} (J q) = 0$ distributionally, and $d V = 0$ reads $\partial_{τ} (J r) = 0$ . A continuous function whose distributional derivative in $τ$ vanishes is constant in $τ$ : convolving in $τ$ gives smooth functions with zero derivative, and uniform convergence on compact subcylinders passes that constancy to the original function; testing against spatial test functions first gives the same conclusion when both variables are present. Their initial values vanish by step 1.1, so $q = r = 0$ throughout the flow neighborhood.

4.1step 3.1L1algebra∎

The identities $U = q ι_{n} {v o l}_{g}$ and $V = r ι_{n} {v o l}_{g}$ now give $d F = 0$ and $d ⋆_{g} F = 0$ . The argument uses only the Maxwell evolution, its initial Gauss constraints, and $d^{2} = 0$ .

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Coupled reduced Einstein–Maxwell local evolution

Statement

Assume ${A C}_{ω}$ . Let $s \geq 5$ be an integer and let $U = (g_{a b}, p_{a b}, q_{a b i}, e_{i}, b_{i})$ be the finite-dimensional first-order state on the $2 π$ -periodic three-torus, with $p_{a b} = {\overset{ˉ}{\nabla}}_{0} g_{a b}$ and $q_{a b i} = {\overset{ˉ}{\nabla}}_{i} g_{a b}$ for a fixed smooth background connection $\overset{ˉ}{\nabla}$ . The component Sobolev spaces $H^{s}$ are those of Periodic high-Sobolev state spaces. Assume an open admissible state set $O$ on which $g$ is Lorentzian, $g^{00} < 0$ , the inverse spatial block $(g^{i j})$ is positive definite, the lapse is positive, and the chosen state variables and coefficients are smooth. Let $K ⋐ int K^{'} ⋐ O$ be compact sets and $R \geq 0$ . Assume the draft local symmetric-hyperbolic supplier Exact periodic positive symmetric-hyperbolic supplier with its stated integer- $s$ , coefficient, compact-state, choice and norm hypotheses. Then the harmonic-coordinate reduced Einstein equation with source $T_{a b} = μ_{0}^{- 1} (F_{a c} F_{b}^{c} - \frac{1}{4} g_{a b} F_{c d} F^{c d})$ , coupled to the six source-free Maxwell field-strength evolution equations, is a smooth quasilinear symmetric-hyperbolic system. For every initial state $U_{0} \in H^{s}$ taking values in $K$ and satisfying $∥ U_{0} ∥_{H^{s}} \leq R$ , its reduced system has a local solution with the regularity and compact-state bounds of Exact periodic positive symmetric-hyperbolic supplier. No Einstein or Maxwell constraint is assumed in this reduced-system statement.

Facts & Assumptions

Given: The fixed source-free Einstein–Maxwell equations, the positive symmetric Maxwell reduction, a smooth background connection, the admissible compact state sets, and the draft local solver interface.

[L1]

Draft definition Electrovacuum tensor Cauchy system fixes the curvature/sign convention, $κ > 0$ , $μ_{0} > 0$ , $Λ \in R$ , the trace-free Maxwell stress $T_{a b} = μ_{0}^{- 1} Q_{a b} (g, F)$ , and the equations $G_{a b} + Λ g_{a b} = κ T_{a b}$ , $d F = d ⋆_{g} F = 0$ . The item file is listed in batch 6; its status is draft and its verification precheck is pending; this use is conditional on the actual draft statement.

[L2]

Draft periodic-state definition Periodic high-Sobolev state spaces uses $T_{2 π}^{3} = R^{3} / (2 π Z)^{3}$ and the real componentwise Fourier norm $∥ u ∥_{H^{s}}^{2} = \sum_{k \in Z^{3}} (1 + ∣ k ∣^{2})^{s} ∣ \hat{u} (k) ∣^{2}$ , under ${A C}_{ω}$ . Its item file is listed in batch 6; its status is draft and its verification precheck is pending; this use remains conditional on that actual draft.

[L3]

Draft result Exact periodic positive symmetric-hyperbolic supplier: for $s \geq 5$ , smooth periodic symmetric matrices $A^{μ}$ and smooth $B$ on an open finite-dimensional state set, a time matrix uniformly positive on compact admissible neighborhoods, and bounded $H^{s}$ initial data valued in $K ⋐ int K'$ , there is a common $T > 0$ and a solution on $(- T, T)$ with $L^{\infty} H^{s} \cap C H^{s - 1}$ , $\partial_{s} U \in L^{\infty} H^{s - 1}$ and values in $K'$ . Its proof applies the current GR existence, uniqueness/stability and smooth-persistence suppliers. The draft file is listed in batch-6 inventory and its precheck is pending; this conclusion remains conditional.

[F1]

Positive symmetric Maxwell field-strength reduction gives the six-dimensional Maxwell principal block $S (ξ)$ , which is symmetric, has norm $∣ ξ ∣$ , and has positive time matrix $α^{- 1} I_{6}$ .

[L4]

Principal part of the reduced Einstein operator and its first-order symmetric hyperbolic reduction gives the reduced metric wave block with time matrix $diag (I, - g^{00} I, g^{i j} I)$ and states that a smooth algebraic stress map of additional first-order matter variables introduces no metric principal-derivative coupling.

[L5]

Draft initial-data interface Harmonic initial data relative to a fixed smooth background uses a fixed smooth positive reference $γ$ and sets $g_{i j} = h_{i j}$ , $g_{00} = - 1$ , $g_{0 i} = 0$ , $\partial_{0} g_{i j} = 2 K_{i j}$ , $\partial_{0} g_{00} = - 2 {tr}_{h} K$ , and $\partial_{0} g_{0 i} = h_{i j} h^{k ℓ} (Γ (h)^{j}_{k ℓ} - Γ (γ)^{j}_{k ℓ})$ . Its actual proof gives the required $H = 0$ and $H^{s}$ regularity. The item file is listed in batch 6 and its precheck is pending; this use remains conditional.

Proof

1.1givenL4F1algebra

In the admissible open state set, $g^{00} < 0$ and $(g^{i j}) > 0$ make the metric time matrix in [L4] positive definite. On a compact $K^{'}$ , its least eigenvalue is bounded below by a positive constant. The lapse condition gives the Maxwell time matrix $α^{- 1} I_{6} > 0$ , also uniformly on $K^{'}$ . Therefore the combined time matrix is the block sum of these two positive matrices.

1.2givenL1F1L4algebra

For a spatial covector $ξ$ , the metric wave spatial matrices are symmetric by [L4], while the Maxwell spatial matrices are symmetric by [F1]; the shift contributes a scalar identity block and preserves symmetry. The Einstein stress $Q (g, F)$ is algebraic in $(g, g^{- 1}, F)$ , so its components contain no derivatives of $U$ . The Maxwell tetrad and Hodge coefficients depend smoothly on $(g, g^{- 1})$ , and their derivatives multiply only the state components; because $p, q$ are included in $U$ , these are smooth zero-order terms. There are no off-diagonal derivative blocks between the metric and Maxwell equations.

2.1step 1.1step 1.2givenL2L3L5construct∎

Keep the arbitrary compact sets $K ⋐ int K^{'} ⋐ O$ and bound $R$ fixed as in the Statement. Steps 1.1–1.2 verify the coefficient hypotheses throughout $O$ , so they hold on a neighborhood of this $K^{'}$ , independently of any particular initial jet. Choose a compact time slab containing zero in its interior and lying inside the smooth background's time domain. On its product with the torus and $K^{'}$ , smoothness bounds each coefficient derivative and compactness gives a uniform positive lower bound for the combined time matrix. Every $U_{0}$ in the Statement belongs to the bounded class of [L2]; thus [L3] applies to these same $K, K^{'}, R$ and gives a common local interval and the asserted bounds for every such $U_{0}$ . The orthogonal harmonic jet in [L5] is included whenever it belongs to this bounded class, but no harmonic, derivative-compatibility, Einstein or Maxwell constraint is required for the reduced solver.

TheoremStatement: AI-adaptedProof: AI-adaptedprecheck pendingOpen item page →

Einstein harmonic gauge propagation with Maxwell on shell

Statement

Use the signature $(-, +, +, +)$ , orientation and constants of Electrovacuum tensor Cauchy system. Let $Σ_{0}$ be a smooth spacelike hypersurface in a neighborhood with a smooth spacelike foliation and future unit normal field $n$ . Fix a smooth background Lorentzian metric $\overset{ˉ}{g}$ there, with Levi-Civita connection $\overset{ˉ}{\nabla}$ . Let $g \in C^{3}$ be a Lorentzian metric and $F \in C^{2}$ a real two-form. Write $\nabla$ for the metric-selected connection of $g$ and $⋆_{g}$ for its oriented Lorentzian Hodge star. Let $h$ be the induced metric, set $K (X, Y) = g (\nabla_{X} n, Y)$ for tangent vectors, and regard the following frame components as spatial vectors $e, b$ on $Σ_{0}$ : $e_{i} : = F (e_{i}, n), b_{i} : = \frac{1}{2} ϵ_{i}^{j k} F (e_{j}, e_{k}),$ in a positively oriented orthonormal spatial frame $(e_{i})$ with $ϵ_{123} = 1$ . Define $H^{a} : = g^{b c} (Γ^{a}_{b c} [g] - {\overset{ˉ}{Γ}}^{a}_{b c}), H_{b} : = g_{b c} H^{c} .$ Suppose the harmonic-reduced Einstein equation ${Ric}_{a b} [g] - \nabla_{(a} H_{b)} = κ (T_{a b}^{E M} - \frac{1}{2} g^{c d} T_{c d}^{E M} g_{a b}) + Λ g_{a b}, T_{a b}^{E M} : = μ_{0}^{- 1} (F_{a c} F_{b}^{c} - \frac{1}{4} g_{a b} F_{c d} F^{c d}),$ and the six normal Maxwell evolution equations $ι_{n} d F = 0$ and $ι_{n} d (⋆_{g} F) = 0$ hold on that neighborhood. Assume $H ∣_{Σ_{0}} = 0$ . Let $h$ be the induced metric, use $K (X, Y) = g (\nabla_{X} n, Y)$ , and, in a positively oriented orthonormal spatial frame $(e_{i})$ , define $e_{i} : = F (e_{i}, n)$ and $b_{i} : = \frac{1}{2} ϵ_{i}^{j k} F (e_{j}, e_{k})$ with $ϵ_{123} = 1$ . Assume the initial constraints $D_{i} e^{i} = D_{i} b^{i} = 0, R (h) + ({tr}_{h} K)^{2} - ∣ K ∣_{h}^{2} = 2 Λ + \frac{κ}{μ_{0}} (∣ e ∣_{h}^{2} + ∣ b ∣_{h}^{2}), D^{j} K_{j i} - D_{i} {tr}_{h} K = - \frac{κ}{μ_{0}} (e \times_{h} b)_{i}$ hold on $Σ_{0}$ , where $D$ is the Levi-Civita derivative of $h$ , $D_{i} e^{i}$ and $D_{i} b^{i}$ are the spatial divergences, $R (h)$ is its scalar curvature, ${tr}_{h} K$ and $∣ K ∣_{h}$ are the induced trace and norm, and $e \times_{h} b$ is the oriented cross product. Then $H = 0$ on a neighborhood of each point of $Σ_{0}$ , and the reduced metric equation there is the full Einstein–Maxwell equation. In the periodic torus setting, Harmonic initial data relative to a fixed smooth background supplies one explicit initial jet satisfying $H ∣_{Σ_{0}} = 0$ ; that construction is not assumed for a general $Σ_{0}$ .

Facts & Assumptions

Given: The coupled reduced evolution, $H ∣_{Σ_{0}} = 0$ , and the Maxwell and Einstein initial constraints.

[L1]

Maxwell Gauss constraints propagate by form transport gives $d F = d ⋆_{g} F = 0$ locally for $C^{1}$ metric and field, an explicit $C^{1}$ transverse future unit normal, the normal-slot evolution equations $ι_{n} d F = ι_{n} d ⋆_{g} F = 0$ , and zero initial Gauss divergences. The smooth coordinate slices and $g \in C^{3}$ , $F \in C^{2}$ here meet those hypotheses.

[L2]

Maxwell stress divergence for classical fields proves the sourced identity $\nabla^{a} T_{a b} = - F_{b c} j^{c}$ for a $C^{1}$ Lorentzian metric and $C^{1}$ closed two-form with continuous $j$ satisfying $\nabla_{a} F^{a b} = - μ_{0} j^{b}$ . Thus its exact regularity hypotheses are met here. Here is the exact source-free conversion needed to apply it: with $ε_{a b c d} : = ({v o l}_{g})_{a b c d}$ and indices raised by $g^{- 1}$ , the Lorentzian Hodge formula gives $\nabla_{a} F^{a b} = \frac{1}{3!} ε^{b a c d} (d ⋆_{g} F)_{a c d} = \frac{1}{\sqrt{∣ \det g ∣}} \partial_{a} (\sqrt{∣ \det g ∣} F^{a b}) .$ The first equality follows by expanding $d ⋆_{g} F = 3 \nabla_{[a} (⋆_{g} F)_{c d]}$ , using metric-compatibility $\nabla_{a} ε_{b c d e} = 0$ , substituting $(⋆_{g} F)_{c d} = \frac{1}{2} ε_{c d e f} F^{e f}$ , and using $ε^{b a c d} ε_{c d e f} = - 2 (δ_{e}^{b} δ_{f}^{a} - δ_{f}^{b} δ_{e}^{a})$ ; the second uses $Γ^{a}_{a c} = \partial_{c} \log \sqrt{∣ \det g ∣}$ and $Γ^{b}_{a c} F^{a c} = 0$ . Thus $d ⋆_{g} F = 0$ gives $\nabla_{a} F^{a b} = 0$ , while $d F = 0$ is the supplier's other hypothesis, so the supplier applies with $j = 0$ and yields $\nabla^{a} T_{a b} = 0$ .

[L3]

The proof of draft supplier Gauss-Codazzi and Maxwell initial constraints uses the pointwise Gauss–Codazzi identities $G (n, n) = \frac{1}{2} (R (h) + ({tr}_{h} K)^{2} - ∣ K ∣_{h}^{2})$ and $G (n, V) = D^{j} K_{j i} V^{i} - D_{V} {tr}_{h} K$ for tangent $V$ , as well as the Maxwell stress projections. These geometric identities depend only on the metric's second jet and hold for the $C^{3}$ metric here; combining them with the stated initial constraints makes $E (n, n) = E (n, V) = 0$ . The EM constraint supplier's item is listed in batch 6; its status is draft and its precheck is pending, so the Maxwell constraint and stress conventions remain conditional on that actual draft proof.

[F1]

The Einstein residual through the gauge vector, and the subsidiary wave equation satisfied by the gauge vector: on a solution of the harmonic-reduced Einstein equation with constant $κ, Λ$ and $\nabla^{a} T_{a b} = 0$ , the unreduced residual is $E_{a b} = \frac{1}{2} (\nabla_{a} H_{b} + \nabla_{b} H_{a} - g_{a b} \nabla_{c} H^{c})$ and $H$ satisfies $□_{g} H_{b} + {Ric}_{b a} H^{a} = 0$ .

[F2]

Principal part of the reduced Einstein operator and its first-order symmetric hyperbolic reduction gives a symmetric first-order reduction of the wave operator, with positive definite time matrix on an admissible spacelike-slice chart and symmetric spatial matrices.

Proof

1.1givenL1L2F1

By [L1], the reduced Maxwell evolution and its initial divergence constraints give $d F = d ⋆_{g} F = 0$ . The coordinate/Hodge identity in [L2] therefore supplies the divergence hypothesis for the stress lemma with $j = 0$ , giving $\nabla^{a} T_{a b} = 0$ before using the unreduced Einstein equations. The Bianchi identity and the divergence identity for a trace-adjusted symmetric covector derivative are local identities in the third jet of $g$ and second jet of $H$ . At each point these jets are exactly realized by smooth Taylor polynomials in a chart, whose metric remains Lorentzian nearby. Hence the unrestricted coordinate identities underlying [F1] transfer at $g \in C^{3}$ , $H \in C^{2}$ ; substituting the actual reduced equation and conserved stress gives its homogeneous wave equation. This transfers a differential identity only, not a smooth solution.

2.1givenL3F1step 1.1algebra

Since $H = 0$ as a field on $Σ_{0}$ , its covariant derivative in every tangential direction vanishes there. The normal-projection identities in [L3] involve only the second jet of $g$ and the algebraic stress, so the same finite-jet argument in step 1.1 applies. The stated Einstein constraints therefore give $E (n, n) = E (n, e_{i}) = 0$ for the unreduced residual. The residual formula of [F1], evaluated in the orthonormal frame $(n, e_{i})$ , gives $E (n, n) = \frac{1}{2} (\nabla_{n} H^{♭}) (n)$ and $E (n, e_{i}) = \frac{1}{2} (\nabla_{n} H^{♭}) (e_{i})$ : the tangential derivatives vanish, and $\nabla_{c} H^{c} = - (\nabla_{n} H^{♭}) (n)$ . Hence $\nabla_{n} H^{♭} = 0$ and, by nondegeneracy of $g$ , $\nabla_{n} H = 0$ initially.

3.1step 1.1step 2.1F1F2algebra

Fix a point of $Σ_{0}$ and choose local coordinates $(s, x^{1}, x^{2}, x^{3})$ with spacelike $s$ -levels, $s = 0$ on $Σ_{0}$ , and $\partial_{s}$ normal to the slice at that point. Such coordinates follow by a linear change of the spatial coordinates by multiples of $s$ in any slice chart. At that point $g^{00} < 0$ and $(g^{i j}) > 0$ ; these strict inequalities persist on a smaller cylinder. Set $P_{b} : = \partial_{s} H_{b}$ and $Q_{b i} : = \partial_{i} H_{b}$ . The wave equation in [F1] has scalar principal part $g^{c d} \partial_{c} \partial_{d} H_{b}$ ; connection and Ricci terms are lower order. In the ordering $W = (H_{b}, P_{b}, Q_{b i})$ , its first-order reduction consists of $\partial_{s} H_{b} - P_{b} = 0$ , the wave equation solved for $\partial_{s} P_{b}$ , and the compatibility equations $\partial_{s} Q_{b i} - \partial_{i} P_{b} = 0$ . Multiplying the wave equation by $- 1$ and the compatibility equations by $g^{j i}$ gives time matrix $A^{0} = diag (I, - g^{00} I, (g^{i j}) I)$ and, for each spatial index $k$ , the only nonzero spatial blocks $A_{P P}^{k} = - 2 g^{0 k} I$ , $A_{P Q_{j}}^{k} = - g^{k j} I$ , and $A_{Q_{i} P}^{k} = - g^{i k} I$ . Thus $A^{0}$ is positive definite on a sufficiently small cylinder with spacelike $s$ -levels, and each $A^{k}$ is symmetric; all remaining coefficients and their first derivatives are bounded there by $g \in C^{3}$ and $H \in C^{2}$ . Choose a compact coordinate ball about the chosen point and shrink the time interval so that the shrinking balls $B_{s}$ of radius $r - λ s$ remain in this cylinder. If $A^{0} ⪰ a I$ and $M : = \sup_{∣ ν ∣ = 1} ∥ ν_{i} A^{i} ∥$ , choose $λ > M / a$ . For $s \geq 0$ set $E (s) = \int_{B_{s}} W^{T} A^{0} W d x$ . The symmetric system's product rule and spatial integration by parts give $E^{'} (s) \leq C E (s) - \int_{\partial B_{s}} W^{T} (λ A^{0} + ν_{i} A^{i}) W d S$ . The boundary matrix is positive semidefinite because $λ a > M$ , so $E^{'} \leq C E$ . The zero initial values established in steps 1.1 and 2.1 and Gronwall force $E = 0$ in the shrinking cone. Reversing time proves the same on the past cone. Thus $H = 0$ in a neighborhood of the chosen point, and the point was arbitrary.

4.1step 3.1step 1.1L2L3F1algebra∎

When $H = 0$ , the harmonic-reduced Einstein equation differs from the unreduced equation by terms linear in $\nabla H$ , which now vanish. Step 1.1 already gives both Maxwell equations, so the full coupled Einstein–Maxwell equations hold near the initial surface. The stress identity is applied at its explicit classical regularity, with no smoothness improvement inferred.

TheoremStatement: AI-adaptedProof: AI-adaptedprecheck pendingOpen item page →

Local high-Sobolev and smooth electrovacuum

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ . Let $s \geq 5$ be an integer, $κ > 0$ , $μ_{0} > 0$ , and $Λ \in R$ . Fix $L_{0} > 0$ in metres and put $Σ = T_{2 π L_{0}}^{3}$ with length coordinates $X^{i}$ . On $Σ$ , let $h \in H^{s + 1}$ be a symmetric Riemannian metric with $h \geq c_{0} I$ for some $c_{0} > 0$ , let $K \in H^{s}$ be a symmetric covariant two-tensor, and let $e, b \in H^{s}$ be spatial vector fields. Sobolev membership means that the length-frame arrays $h_{i j}, K_{i j}$ and oriented orthonormal arrays $e_{A}, b_{A}$ , composed with $D_{L_{0}} : T_{2 π}^{3} \to Σ$ , $[θ] \mapsto [L_{0} θ]$ , belong to the real componentwise Fourier spaces of Periodic high-Sobolev state spaces; these are not tensor pullbacks. All spatial derivatives in the geometric equations are $\partial_{X^{i}} = L_{0}^{- 1} \partial_{θ^{i}}$ . Use the length coordinate $t = x^{0} = c t_{S I}$ , so $g, h$ are dimensionless, $K$ has units $m^{- 1}$ , and $e = E / c$ and $b = B$ both have units tesla in an oriented orthonormal frame; $κ μ_{0}^{- 1} ∣ e ∣^{2}$ has units $m^{- 2}$ . Use the sign convention $K (X, Y) = g (\nabla_{X} n, Y)$ for the future unit normal. Assume $D_{i} e^{i} = 0, D_{i} b^{i} = 0, R (h) + ({tr}_{h} K)^{2} - ∣ K ∣_{h}^{2} = 2 Λ + \frac{κ}{μ_{0}} (∣ e ∣_{h}^{2} + ∣ b ∣_{h}^{2}), D^{j} K_{j i} - D_{i} {tr}_{h} K = - \frac{κ}{μ_{0}} (e \times_{h} b)_{i},$ where $D$ is the Levi-Civita derivative of $h$ . Then there are $T > 0$ and a solution of the full source-free Einstein–Maxwell system on $(- T, T) \times Σ$ with $g \in L^{\infty} H^{s + 1} \cap C^{0} H^{s}, \partial_{t} g \in L^{\infty} H^{s} \cap C^{0} H^{s - 1}, F \in L^{\infty} H^{s} \cap C^{0} H^{s - 1} .$ The initial two-form is reconstructed by $F (n, e_{i}) = - e_{i}$ and $F (e_{i}, e_{j}) = ϵ_{i j k} b_{k}$ ; no independent Maxwell normal derivative is prescribed. The metric remains Lorentzian with spacelike $t$ -levels in a fixed compact admissible state neighborhood. Smooth data give a smooth solution on the same base-order interval. For a fixed harmonic background, on any common local interval with the stated compact-state and uniform $H^{s}$ bounds, the harmonic-reduced first-order state is unique and continuously dependent on its initial state in $C^{0} H^{s - 1}$ . In addition, on any smooth oriented three-manifold, smooth constrained data with these same equations have a smooth full development in a neighborhood of each initial point. For finite-Sobolev data these equations mean their classical pointwise form at the $C^{3}$ metric and $C^{2}$ field regularity proved below; smoothness is asserted only for smooth initial data. No endpoint regularity threshold, strong $C^{0} H^{s}$ stability, global continuation, or geodesic completeness is asserted. Here an affinely parametrized metric geodesic solves ${\ddot{x}}^{a} + Γ^{a}_{b c} {\dot{x}}^{b} {\dot{x}}^{c} = 0$ on an interval; a maximal such solution admits no extension to a larger interval with the same affine parameter. Geodesic completeness means every maximal affine solution has parameter domain all of $R$ .

Facts & Assumptions

Given: Countable Choice, the constrained data and constants in the Statement, and a fixed smooth comparison metric $\overset{ˉ}{g} = - d t^{2} + γ$ with Levi-Civita connection $\overset{ˉ}{\nabla}$ .

[L1]

Draft definition Electrovacuum tensor Cauchy system fixes the source-free system $G + Λ g = κ μ_{0}^{- 1} Q (g, F)$ , $d F = d ⋆_{g} F = 0$ , signature $(-, +, +, +)$ and $Q_{a b} = F_{a c} F_{b}^{c} - \frac{1}{4} g_{a b} F_{c d} F^{c d}$ . Its item file is listed in batch 6; its status is draft and its precheck is pending; this use is conditional on that draft.

[L2]

Draft periodic-state definition Periodic high-Sobolev state spaces fixes the componentwise real Fourier $H^{s} (T_{2 π}^{3})$ norm, the metric/first-derivative/electric/magnetic state variables, and ${A C}_{ω}$ . Its item file is listed in batch 6; its status is draft and its precheck is pending; this use is conditional on that draft.

[L3]

Draft harmonic initialization Harmonic initial data relative to a fixed smooth background chooses the initial metric jet relative to a fixed smooth positive background, with $g_{i j} = h_{i j}$ , $g_{00} = - 1$ , $g_{0 i} = 0$ , $\partial_{0} g_{i j} = 2 K_{i j}$ and the stated remaining derivatives so $H = 0$ . Its item file is listed in batch 6; its status is draft and its precheck is pending; this use is conditional on that draft.

[F1]

Coupled reduced Einstein–Maxwell local evolution verifies the combined metric-wave and field-strength principal blocks. Its actual proof applies Exact periodic positive symmetric-hyperbolic supplier, whose draft file is listed in batch 6 but has precheck pending; the reduced local existence conclusion is therefore conditional.

[L5]

Draft supplier Exact periodic positive symmetric-hyperbolic supplier: for integer $s \geq 5$ , smooth periodic symmetric coefficient matrices and lower-order terms on an open state set, a uniformly positive time matrix on compact admissible neighborhoods, and bounded $H^{s}$ initial data with values in such a neighborhood, there is a common local solution with $L^{\infty} H^{s} \cap C H^{s - 1}$ bounds, $\partial_{t} U \in L^{\infty} H^{s - 1}$ and compact-state control. Its draft item file is listed in batch-6 inventory, with status draft and verification precheck pending; its stated argument is conditional, not accepted.

[F2]

Maxwell Gauss constraints propagate by form transport recovers both full Maxwell equations from $ι_{n} d F = ι_{n} d ⋆_{g} F = 0$ and zero initial Gauss divergences when the metric, field and transverse future unit normal are $C^{1}$ . The smooth coordinate time levels and the metric/field regularity established in proof 4.1 supply this explicit normal regularity.

[L4]

Maxwell stress divergence for classical fields proves the sourced identity $\nabla^{a} T_{a b} = - F_{b c} j^{c}$ for a $C^{1}$ Lorentzian metric and $C^{1}$ closed two-form with continuous $j$ satisfying $\nabla_{a} F^{a b} = - μ_{0} j^{b}$ . Thus its exact regularity hypotheses are met here. The exact source-free conversion is, with $ε_{a b c d} : = ({v o l}_{g})_{a b c d}$ and indices raised by $g^{- 1}$ , $\nabla_{a} F^{a b} = \frac{1}{3!} ε^{b a c d} (d ⋆_{g} F)_{a c d} = \frac{1}{\sqrt{∣ \det g ∣}} \partial_{a} (\sqrt{∣ \det g ∣} F^{a b}) .$ The first equality follows by expanding $d ⋆_{g} F = 3 \nabla_{[a} (⋆_{g} F)_{c d]}$ , using metric-compatibility $\nabla_{a} ε_{b c d e} = 0$ , $(⋆_{g} F)_{c d} = \frac{1}{2} ε_{c d e f} F^{e f}$ and $ε^{b a c d} ε_{c d e f} = - 2 (δ_{e}^{b} δ_{f}^{a} - δ_{f}^{b} δ_{e}^{a})$ ; the second follows from $Γ^{a}_{a c} = \partial_{c} \log \sqrt{∣ \det g ∣}$ and $Γ^{b}_{a c} F^{a c} = 0$ . Hence $d ⋆_{g} F = 0$ implies the divergence equation with $j = 0$ , and the model's $d F = 0$ supplies the other hypothesis.

[F3]

Einstein harmonic gauge propagation with Maxwell on shell propagates the harmonic condition and recovers the full Einstein equation when the initial constraints and Maxwell stress conservation hold.

[F4]

Local existence of Sobolev and classical solutions of a periodic symmetric hyperbolic system, uniform on bounded H^s data supplies the bounded-set common existence time and $L^{\infty} H^{s} \cap C H^{s - 1}$ state regularity under AC $_{ω}$ . The inspected carrier is marked published, but its verification precheck remains pending.

[F5]

Smooth data give smooth solutions of a periodic symmetric hyperbolic system on the existence interval gives smooth persistence on the same interval for smooth data. The inspected carrier is marked published, but its verification precheck remains pending.

[F6]

Uniqueness of classical solutions and continuous dependence in H^{s-1} for periodic symmetric hyperbolic systems gives uniqueness and continuous dependence in $H^{s - 1}$ for classical periodic solutions with the stated common compact-state and $H^{s}$ bounds. Its actual statement includes AC $_{ω}$ and a common existence-time hypothesis; the inspected carrier is marked published, but its verification precheck remains pending.

[F7]

Smooth composition, difference, coefficient product and commutator estimates on bounded H^s data gives composition and product bounds on compact state sets; The periodic Sobolev space H^k of the three-torus is a Banach algebra for k at least four gives the $H^{q}$ algebra property for $q \geq 4$ ; Periodic Sobolev embedding: H^s on the three-torus controls C^m when s minus m exceeds three halves gives $H^{q} ↪ C^{m}$ when $q - m > 3 / 2$ .

Proof

1.1givenL1L2L3F7

The constraints in the Statement are the normal projections and Maxwell Gauss constraints of the model [L1] under the stated $K, e, b$ conventions. Initialization [L3] gives $H = 0$ and the compatible derivative variables $p = {\overset{ˉ}{\nabla}}_{0} g$ , $q_{i} = {\overset{ˉ}{\nabla}}_{i} g$ on the initial slice. The initial state lies in $H^{s}$ : $h \in H^{s + 1}$ gives $q_{i} \in H^{s}$ , $K, e, b \in H^{s}$ , and the harmonic normal derivatives are in $H^{s}$ by the smooth composition estimates [F7] on the compact range of the uniformly positive $h$ . Compactness of $T^{3}$ and $h \geq c_{0} I$ place the initial state in a compact subset of the open admissible set.

1.2L2F7algebra

(Endpoint product and composition estimates.) Put $⟨ k ⟩ = (1 + ∣ k ∣^{2})^{1 / 2}$ . For integer $m \in {2, 3}$ , the inequality $⟨ k ⟩^{m} \leq 2^{m - 1} (⟨ ℓ ⟩^{m} + ⟨ k - ℓ ⟩^{m})$ and the convolution formula for Fourier coefficients give $∥ u v ∥_{H^{m}} \leq C_{m} (∥ ⟨ k ⟩^{m} \hat{u} ∥_{ℓ^{2}} ∥ \hat{v} ∥_{ℓ^{1}} + ∥ \hat{u} ∥_{ℓ^{1}} ∥ ⟨ k ⟩^{m} \hat{v} ∥_{ℓ^{2}}) .$ Cauchy–Schwarz gives $∥ \hat{u} ∥_{ℓ^{1}} \leq (\sum_{k} ⟨ k ⟩^{- 2 m})^{1 / 2} ∥ u ∥_{H^{m}}$ , and the weight sum is finite in dimension three for $m \geq 2$ ; the analogous bound holds for $v$ . Young's convolution inequality (or the triangle inequality in $ℓ^{2}$ followed by summation in $ℓ^{1}$ ) therefore proves the algebra estimate in both $H^{2}$ and $H^{3}$ . For a smooth finite-dimensional map $G$ on a neighborhood of a compact value set, Fourier approximation and the chain rule through order three show that $G (Z) \in H^{3}$ when $Z \in H^{3}$ : the terms are bounded-coefficient multiples of $D^{α} Z$ , products $D Z D^{2} Z$ , or products of at most three first derivatives; $H^{3} ↪ C^{1}$ places first derivatives in $L^{\infty}$ and the highest derivatives in $L^{2}$ . The same expansions show continuity under strong $H^{3}$ convergence, since $H^{3} ↪ C^{1}$ gives uniform convergence of the coefficients and first derivatives while higher derivatives converge in $L^{2}$ . This also covers smooth dependence on $(t, x)$ on a compact coefficient-time slab. Together with the supplied estimates at orders $H^{k}$ for $k \geq 4$ in [F7], these are the endpoint estimates needed below.

2.1step 1.1F1L5F4algebra

Compose the length-frame state components with $D_{L_{0}}$ . Replacing each $\partial_{X^{i}}$ by $L_{0}^{- 1} \partial_{θ^{i}}$ multiplies its symmetric spatial matrix by the fixed positive scalar $L_{0}^{- 1}$ and preserves smoothness and the positive time matrix. By [F1], the metric and Maxwell principal blocks are symmetric with positive time matrices and have no derivative coupling. At the initial orthogonal state $g^{00} < 0$ , $(g^{i j}) > 0$ and $α > 0$ ; these strict inequalities persist on a compact admissible state neighborhood, so the complete coefficient family meets the positivity and compact-state assumptions of [L5]. Applying that draft supplier conditionally gives the reduced solution and its $L^{\infty} H^{s} \cap C H^{s - 1}$ and $\partial_{t} U \in L^{\infty} H^{s - 1}$ bounds. The actual inspected generic existence theorem [F4] states the same periodic coefficient interface and is marked published with precheck pending; it is independent current support, not acceptance of the CP01 draft supplier.

3.1step 1.1step 2.1L3F1algebra

The first-order metric equations impose ${\overset{ˉ}{\nabla}}_{0} g = p$ and ${\overset{ˉ}{\nabla}}_{0} q_{i} - {\overset{ˉ}{\nabla}}_{i} p = C_{i}$ , where $C_{i} = [{\overset{ˉ}{\nabla}}_{0}, {\overset{ˉ}{\nabla}}_{i}] g$ is the fixed background-curvature commutator. For $Z_{i} = q_{i} - {\overset{ˉ}{\nabla}}_{i} g$ , subtract the commutator identity for $g$ from the equation for $q_{i}$ . The derivative-slot connection terms cancel and give a homogeneous linear ODE along each coordinate time line for $Z_{i}$ ; its initial value is zero by [L3], so uniqueness for a linear ODE gives $Z_{i} = 0$ . Thus $q_{i} = {\overset{ˉ}{\nabla}}_{i} g$ throughout the interval.

4.1step 2.1step 3.1step 1.2L2F2F3F7algebra

(Third-order spacetime regularity.) The Fourier norm identity $∥ g ∥_{H^{s + 1}}^{2} ≍ ∥ g ∥_{H^{s}}^{2} + \sum_{i} ∥ \partial_{i} g ∥_{H^{s}}^{2}$ , together with step 3.1 and the smooth background coefficients, gives $g \in L^{\infty} H^{s + 1}$ . The same identity at one lower order and the $C H^{s - 1}$ continuity of the first-order state give $g \in C H^{s}$ . The tetrad reconstruction of $F$ is a smooth algebraic function of $(g, e, b)$ on the compact admissible state set; [F7] gives $F \in L^{\infty} H^{s} \cap C H^{s - 1}$ . Solve the reduced system as $U_{t} = R (t, x, U, \partial_{i} U)$ with $R$ smooth on the compact admissible range. For every $s \geq 5$ , $U \in C H^{s - 1}$ and $\partial_{i} U \in C H^{s - 2}$ ; the composition estimates of [F7] at orders at least four and step 1.2 at orders two and three give $U_{t} \in C H^{s - 2}$ . Differentiating this identity in time, justified by Fourier approximation and the same chain/product estimates, gives $U_{t t} = \partial_{t} R (t, x, U, \partial U) + D_{U} R (t, x, U, \partial U) U_{t} + D_{\partial U} R (t, x, U, \partial U) \partial U_{t} .$ Its coefficient maps lie in $C H^{s - 2}$ , while $\partial U_{t} \in C H^{s - 3}$ ; the algebra estimates of [F7] for orders at least four and of step 1.2 for orders two and three imply $U_{t t} \in C H^{s - 3}$ . At the base order $s = 5$ , these spaces are $H^{3}$ and $H^{2}$ , and for $s = 6$ the lower product order is $H^{3}$ , also covered by step 1.2. The identities $p = {\overset{ˉ}{\nabla}}_{0} g$ and $q_{i} = {\overset{ˉ}{\nabla}}_{i} g$ then give all third spatial derivatives of $g$ from two spatial derivatives of $q$ , the $t^{2} x$ derivatives from one spatial derivative of $p_{t}$ , and $t^{3}$ derivatives from $p_{t t}$ ; the other derivatives of total order at most three follow directly from $p, q \in C H^{4}$ and $p_{t} \in C H^{3}$ . Since $H^{2} (T^{3}) ↪ C^{0}$ , all these derivatives are continuous. The same smooth composition argument gives $F \in C^{2}$ from $(g, e, b)$ , so the classical regularity required by [F2] and [F3] holds.

5.1step 2.1step 4.1F2F3F5F6L3L4

Maxwell propagation [F2] now gives $d F = d ⋆_{g} F = 0$ . The exact Hodge-to-divergence identity in [L4] converts the second equation to $\nabla_{a} F^{a b} = 0$ , so the stress supplier applies with $j = 0$ and gives $\nabla^{a} T_{a b} = 0$ . Thus [F3] applies with the initial Einstein constraints and $H = 0$ from [L3]; hence the reduced metric equation is the full Einstein equation. Smooth persistence [F5] applies to smooth data on the same interval. Finally, [F6] applies to any two reduced solutions on a common interval with these uniform compact-state and $H^{s}$ bounds and yields both uniqueness and the claimed $C H^{s - 1}$ continuous-dependence estimate.

6.1step 1.1step 3.1F1F2F3algebra∎

(Local smooth data on an arbitrary slice.) Fix a point of a smooth initial three-manifold and a relatively compact slice chart. Choose local product coordinates $(x^{0}, x^{i})$ with $x^{0} = 0$ on the slice and $\partial_{0} = n$ there. Choose a smooth positive spatial background $γ$ in this chart and put $Q^{i} : = h^{j k} (Γ (h)^{i}_{j k} - Γ (γ)^{i}_{j k})$ . Prescribe the local metric jet by $g_{i j} = h_{i j}$ , $g_{00} = - 1$ , $g_{0 i} = 0$ , $\partial_{0} g_{i j} = 2 K_{i j}$ , $\partial_{0} g_{00} = - 2 {tr}_{h} K$ , and $\partial_{0} g_{0 i} = h_{i j} Q^{j}$ , relative to the fixed product background $\overset{ˉ}{g} = - (d x^{0})^{2} + γ_{i j} (x) d x^{i} d x^{j}$ . The Christoffel formula gives $H^{0} = - {tr}_{h} K + {tr}_{h} K = 0$ and $H^{i} = - Q^{i} + h^{j k} (Γ (h)^{i}_{j k} - Γ (γ)^{i}_{j k}) = 0$ on the slice. These smooth local data, including the auxiliary derivative variables, take values after shrinking in a convex finite-dimensional neighborhood whose closure lies in the open symmetric-hyperbolic state set. Extend $γ$ to a smooth positive metric on a periodic three-torus, agreeing on a smaller ball, by interpolating with a flat metric through the convex cone of positive matrices. Extend the state by a smooth cutoff to a constant admissible state; convexity keeps all values in that neighborhood. Extend the coefficient matrices by the same spatial cutoff, interpolating $A^{0}$ to the identity and each symmetric $A^{i}$ and the lower-order term to zero. The convex cone of positive matrices preserves a positive time matrix, and the spatial matrices remain symmetric. The conditional local solver in [F1] therefore gives a smooth reduced solution on a torus cylinder, which satisfies the original reduced equations wherever the cutoff is one. The derivative-definition relations remain valid along the time lines there, by the same homogeneous commutator ODE as step 3.1. The Maxwell divergence residuals vanish near the chosen point by the local flow argument of [F2]. The Einstein constraints and $H = 0$ hold on the initial ball; the local cone-energy proof of [F3] therefore gives $H = 0$ in a smaller domain of dependence. On that neighborhood the reduced solution solves the full source-free system. This proves the additional local existence clause without imposing constraints on the auxiliary extension outside the slice ball.

PropositionStatement: AI-adaptedProof: AI-adaptedprecheck pendingOpen item page →

Local Sobolev stability and continuation for the reduced electrovacuum system

Statement

Assume ${A C}_{ω}$ and the conditional local existence theorem Local high-Sobolev and smooth electrovacuum. Let $s \geq 5$ , let a harmonic-reduced source-free Einstein–Maxwell solution on $[t_{0}, T)$ have first-order state $U = (g, p, q, e, b) \in L^{\infty} H^{s} \cap C H^{s - 1}$ with $\partial_{t} U \in L^{\infty} H^{s - 1}$ , and suppose its values stay in a fixed compact admissible state set and $\sup_{t < T} ∥ U (t) ∥_{H^{s}} < \infty$ . If $T$ is finite and lies inside the smooth coefficient-time domain, the reduced solution extends through $T$ to a larger interval. If the original state has $p = {\overset{ˉ}{\nabla}}_{0} g$ , $q_{i} = {\overset{ˉ}{\nabla}}_{i} g$ , vanishing harmonic vector $H$ , and satisfies the full source-free Einstein–Maxwell equations, the extension has these same properties and extends the full development. The same conclusion holds at a finite left endpoint. On every common interval where two reduced solutions have the same fixed harmonic background, compact-state range and uniform $H^{s}$ bound, their states are unique and satisfy an $H^{s - 1}$ continuous-dependence estimate.

Facts & Assumptions

Given: Countable Choice, the real componentwise Fourier $H^{s} (T_{2 π}^{3})$ state and the bounded-state reduced Einstein–Maxwell solution in the Statement.

[F1]

Local high-Sobolev and smooth electrovacuum gives the full solution regularity, derivative-variable identities, propagation of the Einstein and Maxwell constraints, harmonic gauge, and the Maxwell transport, stress and Einstein subsidiary identities used below. The endpoint is restarted with [L1] in the same reduced state, rather than reinitializing the special lapse/shift jet of that theorem. Its use remains conditional on the CP01 draft carriers recorded there, each of which has precheck pending and is listed in the batch-6 manifest inventory.

[L1]

Draft supplier Exact periodic positive symmetric-hyperbolic supplier: for integer $s \geq 5$ , smooth periodic symmetric coefficient matrices with uniformly positive time matrix on a compact admissible state neighborhood, and initial $H^{s}$ data with values in that neighborhood and bounded norm, there is a common local solution with $L^{\infty} H^{s} \cap C H^{s - 1}$ bounds and $\partial_{t} U \in L^{\infty} H^{s - 1}$ . Its item file is listed in batch 6; its status is draft and its verification precheck is pending; this input remains conditional.

[F2]

Uniqueness of classical solutions and continuous dependence in H^{s-1} for periodic symmetric hyperbolic systems proves uniqueness and $H^{s - 1}$ continuous dependence for classical periodic symmetric-hyperbolic solutions with common compact-state and $H^{s}$ bounds. Its inspected statement assumes ${A C}_{ω}$ and a common existence interval; its published carrier has precheck pending.

[F3]

Local existence of Sobolev and classical solutions of a periodic symmetric hyperbolic system, uniform on bounded H^s data gives a common existence interval and $\partial_{t} U \in L^{\infty} H^{s - 1}$ for the bounded data class of the preceding supplier. Its published carrier has verification precheck pending.

[F4]

Periodic high-Sobolev state spaces fixes $∥ u ∥_{H^{s}}^{2} = \sum_{k \in Z^{3}} (1 + ∣ k ∣^{2})^{s} ∣ \hat{u} (k) ∣^{2}$ componentwise. Smooth composition, difference, coefficient product and commutator estimates on bounded H^s data and The periodic Sobolev space H^k of the three-torus is a Banach algebra for k at least four give the coefficient/product bounds on compact state sets, and Periodic Sobolev embedding: H^s on the three-torus controls C^m when s minus m exceeds three halves gives $H^{q} ↪ C^{0}$ when $q > 3 / 2$ .

[F5]

Coupled reduced Einstein–Maxwell local evolution verifies the concrete reduced coefficient family on the open admissible state set: its metric time block is $diag (I, - g^{00} I, g^{i j} I)$ , its Maxwell time block is $α^{- 1} I_{6}$ , every spatial block is symmetric, and all coefficients and lower-order terms are smooth functions of the first-order state and the fixed smooth background. This verification applies to arbitrary admissible reduced states, including endpoint states with nonzero shift; it requires no constraints or special harmonic initial jet.

Proof

1.1givenL1F3F4F5algebra

By [F5], the reduced equation has the form $A^{0} (t, x, U) \partial_{t} U + A^{i} (t, x, U) \partial_{i} U + B (t, x, U) = 0$ in the sense of Periodic first-order symmetric hyperbolic systems: coefficients, admissible values, data classes and solutions. Its time matrix is $diag (I, - g^{00} I, g^{i j} I, α^{- 1} I_{6})$ , positive definite throughout the admissible open set. Every spatial matrix is symmetric: the metric wave blocks and the Maxwell curl blocks are symmetric, while the shift adds a scalar identity to the latter. The algebraic stress and the frame derivatives, expressed using $p, q$ , create only smooth zero-order terms, so there is no off-diagonal derivative coupling. Enclose the fixed compact state range in the interior of a larger compact subset of this open set; on that subset and a compact coefficient-time slab containing $T$ in its interior, smoothness and strict positivity give bounded coefficient derivatives and a uniform positive time-matrix lower bound. These verify the concrete hypotheses required by [L1] and [F3] at the restart. Since $∥ U ∥_{H^{s}}$ is uniformly bounded and $H^{s - 1}$ is an algebra for $s \geq 5$ , the equation and [F4] also give $∥ \partial_{t} U ∥_{H^{s - 1}} \leq C_{R}$ near $T$ .

2.1step 1.1F3F4algebra

For each finite Fourier projection $P_{N}$ , every coefficient of $P_{N} U$ is absolutely continuous and its derivative is the corresponding coefficient of $P_{N} \partial_{t} U$ . Thus $∥ P_{N} U (t) - P_{N} U (t^{'}) ∥_{H^{s - 1}} \leq C_{R} ∣ t - t^{'} ∣$ . Letting $N \to \infty$ and using Fourier convergence in $H^{s - 1}$ gives the same estimate for $U (t) - U (t^{'})$ . Hence $U (t)$ has a strong $H^{s - 1}$ limit $U_{T}$ as $t ↑ T$ .

3.1step 2.1F4algebra

For each finite frequency set ${∣ k ∣ \leq N}$ , strong convergence of its coefficients gives $\sum_{∣ k ∣ \leq N} (1 + ∣ k ∣^{2})^{s} ∣ {\hat{U}}_{T} (k) ∣^{2} = \lim_{t ↑ T} \sum_{∣ k ∣ \leq N} (1 + ∣ k ∣^{2})^{s} ∣ \hat{U} (t, k) ∣^{2} \leq R^{2},$ where $R = \sup_{t < T} ∥ U (t) ∥_{H^{s}}$ . Taking the supremum over $N$ in the Fourier definition [F4] proves $U_{T} \in H^{s}$ and $∥ U_{T} ∥_{H^{s}} \leq R$ . By the embedding in [F4], $U (t) \to U_{T}$ uniformly in space. Since each $U (t, x)$ lies in the fixed compact state set, the closedness of that set gives $U_{T} (x)$ in the same set for every $x$ .

4.1step 3.1F1F4algebra

For the full-development conclusion, assume the original solution has the compatibility, harmonic and full-equation properties stipulated in the Statement. The derivative identities from [F1] then pass to $U_{T}$ : $q_{T} = {\overset{ˉ}{\nabla}}_{i} g_{T}$ because these relations are continuous in the strong $H^{s - 1}$ limit. Since $q_{T} \in H^{s}$ , the Fourier norm identity gives $g_{T} \in H^{s + 1}$ ; likewise $p_{T}, e_{T}, b_{T} \in H^{s}$ . At the endpoint order $H^{3}$ needed when $s = 5$ , the Fourier convolution estimate $⟨ k ⟩^{3} \leq 4 (⟨ ℓ ⟩^{3} + ⟨ k - ℓ ⟩^{3})$ and $∥ \hat{u} ∥_{ℓ^{1}} \leq (\sum_{k} ⟨ k ⟩^{- 6})^{1 / 2} ∥ u ∥_{H^{3}}$ prove $∥ u v ∥_{H^{3}} \leq C ∥ u ∥_{H^{3}} ∥ v ∥_{H^{3}}$ ; the same argument with exponent $2$ proves the $H^{2}$ product estimate. The chain rule through order three, $H^{3} ↪ C^{1}$ , and Fourier approximation give continuity of smooth composition in $H^{3}$ on compact value sets: first derivatives converge uniformly, higher derivatives converge in $L^{2}$ , and every chain-rule product therefore converges in $L^{2}$ . In particular, matrix inversion is continuous in $H^{3}$ on a compact set of positive-definite metrics. For $s \geq 6$ the supplied algebra/composition estimates at $H^{s - 2} \geq H^{4}$ apply directly, and for $s = 5$ these local $H^{3}$ estimates cover the curvature and connection products. The constraints involve at most two spatial derivatives of $h_{T}$ and one derivative of $K_{T}, e_{T}, b_{T}$ , which converge in $H^{s - 2}$ from the first-order state and its derivative identities; each inverse-metric factor and each product in the constraint expressions is thus continuous at the endpoint. The harmonic vector is a smooth function of $(g_{T}, p_{T}, q_{T})$ , so $H_{T} = 0$ as the limit of $H (t) = 0$ . Therefore $U_{T}$ determines constrained data of precisely the regularity required by [F1].

5.1step 3.1step 4.1L1F1F3F5algebra

Restart the same harmonic-reduced coefficient system of [F5] at time $T$ with the actual complete state $U_{T}$ , keeping the fixed background and coordinates unchanged. The coefficients are smooth on the same time-state domain, $T$ is interior to its time interval, and step 3.1 supplies an $H^{s}$ datum with compact admissible range and bounded norm. Applying [L1] (equivalently [F3] after translating time) gives a reduced solution $V$ on $(T - ε, T + ε)$ with $V (T) = U_{T}$ . This gives a reduced extension candidate for every state in the Statement, without reinitialization to lapse one, shift zero, or the special metric jet of the original initial slice. For the full-development conclusion, assume now the additional properties used in step 4.1. The reduction's equations propagate $q_{i} - {\overset{ˉ}{\nabla}}_{i} g = 0$ by the homogeneous background-commutator ODE, so $V$ has the required metric derivative identities. It has the same $H^{s}$ bounds and compact-state control as the local solver class; the spacetime-regularity argument in proof step 4.1 of [F1], which uses only those bounds, the derivative identities and the fixed smooth coefficients, applies to $V$ after translating time and yields $g_{V} \in C^{3}$ and $F_{V} \in C^{2}$ . Its zero initial Maxwell divergence residuals from step 4.1 propagate by the normal-form transport argument of [F1]; the coordinate time foliation is smooth and its normal is $C^{1}$ at this regularity. Thus both full Maxwell equations hold, and the explicit classical stress identity carried by [F1] gives conservation at the restarted $C^{3}$ metric/ $C^{2}$ field regularity; no smooth-only stress theorem is applied. The harmonic residual identity from [F1] gives the homogeneous wave equation for $H$ . Initially $H = 0$ by step 4.1, so its tangential derivatives vanish. In any orthonormal frame with future unit time vector $n$ , the Einstein residual is $E_{a b} = \frac{1}{2} (\nabla_{a} H_{b} + \nabla_{b} H_{a} - g_{a b} \nabla_{c} H^{c})$ , hence $E (n, n) = \frac{1}{2} \nabla_{n} H_{n}$ and $E (n, e_{i}) = \frac{1}{2} \nabla_{n} H_{i}$ there. The endpoint Einstein constraints from step 4.1 give $\nabla_{n} H = 0$ . The same local wave-energy uniqueness argument of [F1], valid for this positive symmetric wave block with bounded coefficient derivatives, yields $H = 0$ near the endpoint slice. Therefore $V$ is a full solution.

6.1step 5.1step 2.1F2algebra∎

On the overlap before $T$ , compare $V$ and the old solution $U$ using the difference-energy estimate [F2] on $[t, t^{'}]$ with $t < t^{'} < T$ , in the reverse time direction. The compact state sets and $H^{s}$ norms of both solutions have common bounds, so its constant is uniform as $t^{'} ↑ T$ . Their strong $H^{s - 1}$ traces both tend to $U_{T}$ , giving $∥ U (t) - V (t) ∥_{H^{s - 1}} \leq C e^{C (t^{'} - t)} ∥ U (t^{'}) - V (t^{'}) ∥_{H^{s - 1}} \to 0$ . Thus the solutions agree on the overlap, and $V$ extends the original reduced solution through $T$ , and the full development when the additional hypotheses hold. The same estimate identifies any two continuations. Time reversal gives the left-endpoint case. On each common interval the same [F2] estimate gives the stated $H^{s - 1}$ continuous dependence, with constants depending on the coefficient time slab, compact state set, norm bound and $s$ .

TheoremStatement: AI-adaptedProof: AI-adaptedprecheck pendingOpen item page →

Smooth local geometric existence and uniqueness for source-free Einstein–Maxwell data

Statement

Assume Countable Choice and Dependent Choice. Let $Σ$ be a smooth oriented three-manifold without boundary, with smooth Riemannian metric $h$ , smooth symmetric covariant tensor $K$ , and smooth spatial vectors $e, b$ . Fix $κ > 0$ , $μ_{0} > 0$ , $Λ \in R$ , use signature $(-, +, +, +)$ and $K (X, Y) = g (\nabla_{X} n, Y)$ , and suppose the four Einstein–Maxwell constraints of Gauss-Codazzi and Maxwell initial constraints hold. In SI convention use length coordinate $x^{0} = c t$ , dimensionless $g, h$ , $K$ in $m^{- 1}$ , and $e = E / c$ , $b = B$ in tesla. Reconstruct the complete initial two-form on $R N \oplus T Σ$ by $F (N, e_{i}) = - e_{i}, F (e_{i}, e_{j}) = ϵ_{i j k} b_{k},$ where $N$ is the future unit normal symbol and $(e_{i})$ is a positively oriented $h$ -orthonormal frame. Then the data have a smooth time-oriented local source-free Einstein–Maxwell development on an open variable-width neighborhood of the entire initial surface. After restriction to a smaller collar, the interior of its intrinsic domain of dependence is a globally hyperbolic neighborhood with $Σ$ Cauchy. Any two developments of this same model and complete data are related near the entire initial surface by a data-fixing, time-orientation-preserving isometry carrying $F$ to $F$ . There is no uniform positive time width for noncompact $Σ$ and no global-in-time conclusion.

Facts & Assumptions

Given: Countable Choice, Dependent Choice, the smooth oriented data and constants in the Statement, and the specified tetrad, field and extrinsic-curvature conventions.

[L1]

Draft model Electrovacuum tensor Cauchy system fixes $G + Λ g = κ T$ , $T_{a b} = μ_{0}^{- 1} (F_{a c} F_{b}^{c} - \frac{1}{4} g_{a b} F_{c d} F^{c d})$ , $d F = d ⋆_{g} F = 0$ , and the curvature and orientation/sign conventions. Its item file is listed in batch 6; its status is draft and its precheck is pending; the application remains conditional.

[L2]

Draft constraint result Gauss-Codazzi and Maxwell initial constraints states, with this $K$ and field convention, $D_{i} e^{i} = D_{i} b^{i} = 0, R (h) + ({tr}_{h} K)^{2} - ∣ K ∣_{h}^{2} = 2 Λ + \frac{κ}{μ_{0}} (∣ e ∣_{h}^{2} + ∣ b ∣_{h}^{2}), D^{j} K_{j i} - D_{i} {tr}_{h} K = - \frac{κ}{μ_{0}} (e \times_{h} b)_{i} .$ Its item file is present but listed in batch 6 and has verification precheck pending, so the interface remains conditional.

[L3]

Draft stress identity Maxwell stress algebra and covariant conservation gives $\nabla^{a} T_{a b} = - F_{b c} j^{c}$ for smooth fields satisfying $d F = 0$ and $\nabla_{a} F^{a b} = - μ_{0} j^{b}$ . The exact source-free Hodge conversion is $\nabla_{a} F^{a b} = \frac{1}{3!} ε^{b a c d} (d ⋆_{g} F)_{a c d} = \frac{1}{\sqrt{∣ \det g ∣}} \partial_{a} (\sqrt{∣ \det g ∣} F^{a b}), ε_{a b c d} : = ({v o l}_{g})_{a b c d},$ with indices raised by $g^{- 1}$ . It follows by expanding $d ⋆_{g} F = 3 \nabla_{[a} (⋆_{g} F)_{c d]}$ , using metric-compatibility $\nabla_{a} ε_{b c d e} = 0$ , $(⋆_{g} F)_{c d} = \frac{1}{2} ε_{c d e f} F^{e f}$ and $ε^{b a c d} ε_{c d e f} = - 2 (δ_{e}^{b} δ_{f}^{a} - δ_{f}^{b} δ_{e}^{a})$ ; the coordinate divergence uses $Γ^{a}_{a c} = \partial_{c} \log \sqrt{∣ \det g ∣}$ and the vanishing contraction $Γ^{b}_{a c} F^{a c}$ . Hence the source-free equations in [L1] meet both hypotheses with $j = 0$ before the unreduced Einstein equation is used. The file exists but is listed in batch 6 and its precheck is pending; this input remains conditional.

[L4]

Draft harmonic initialization Harmonic initial data relative to a fixed smooth background specifies all initial metric first derivatives, including auxiliary reduction variables, so $H = 0$ and the initial reduced state is compatible with the chosen smooth background. Its file exists but is listed in batch 6 and its precheck is pending; this use remains conditional.

[F1]

Positive symmetric Maxwell field-strength reduction gives the symmetric Maxwell principal block and positive time matrix; Principal part of the reduced Einstein operator and its first-order symmetric hyperbolic reduction gives the metric wave block; Coupled reduced Einstein–Maxwell local evolution combines them, verifies smooth coefficients and no derivative coupling through the algebraic stress, and applies the conditional local solver.

[F2]

Local high-Sobolev and smooth electrovacuum includes the local smooth existence clause on arbitrary smooth initial slices. It proves the derivative-variable compatibility, local Maxwell constraint propagation and Einstein harmonic recovery, conditional on the CP01 carriers named above.

[F3]

Maxwell Gauss constraints propagate by form transport proves local recovery of $d F = d ⋆_{g} F = 0$ from the normal-slot equations $ι_{n} d F = ι_{n} d ⋆_{g} F = 0$ and zero initial Gauss divergences for $C^{1}$ metric, field and transverse future unit normal; the smooth local developments here satisfy these hypotheses. Einstein harmonic gauge propagation with Maxwell on shell proves local $H = 0$ and recovery of the unreduced Einstein equation from the normal constraints and conserved stress.

[F4]

Local geometric development for a fixed covariant Einstein-matter model with complete constrained data applies to a fixed covariant matter model with a symmetric-hyperbolic reduction, complete smooth matter initialization, propagated auxiliary compatibility, on-shell stress conservation, and the Einstein constraints. Its actual published carrier has precheck pending; its statement and full local proof were inspected.

[F5]

Causal and Cauchy-surface inputs for smooth electrovacuum developments collects the exact local causal-neighborhood, globally hyperbolic Cauchy-neighborhood, and choice interfaces used in the local-development conclusion.

Proof

1.1givenL1L2algebra

On the abstract initial bundle $R N \oplus T Σ$ , the displayed formulas assign the three normal-spatial components and the three spatial-spatial components of an antisymmetric two-form. These are all six independent components of a two-form in four dimensions, so they determine a unique complete initial $F$ without choosing a potential or assuming global exactness. They are smooth because $h, e, b$ are smooth and the reconstruction is tensorial under oriented orthonormal-frame changes.

1.2givenL4F1F2F3F4algebra

In each relatively compact initial coordinate ball choose product coordinates $(x^{0}, x^{i})$ with $x^{0} = 0$ on the slice and $\partial_{0} = n$ there. Choose a smooth positive spatial background $γ$ and put $Q^{i} : = h^{j k} (Γ (h)^{i}_{j k} - Γ (γ)^{i}_{j k})$ . Relative to $\overset{ˉ}{g} = - (d x^{0})^{2} + γ_{i j} (x) d x^{i} d x^{j}$ , prescribe $g_{i j} = h_{i j}$ , $g_{00} = - 1$ , $g_{0 i} = 0$ , $\partial_{0} g_{i j} = 2 K_{i j}$ , $\partial_{0} g_{00} = - 2 {tr}_{h} K$ , and $\partial_{0} g_{0 i} = h_{i j} Q^{j}$ . The same Christoffel computation as the periodic construction [L4], now carried out on this local chart, gives $H^{0} = - {tr}_{h} K + {tr}_{h} K = 0$ and $H^{i} = - Q^{i} + h^{j k} (Γ (h)^{i}_{j k} - Γ (γ)^{i}_{j k}) = 0$ . This local jet is smooth and compatible with the first-order reduction. The metric block has time matrix $diag (I, - g^{00} I, g^{i j} I)$ by [F1]'s reduced Einstein principal-part supplier, and the Maxwell block has time matrix $α^{- 1} I_{6}$ by [F1]. On the open state neighborhood where $g^{00} < 0$ , $(g^{i j}) > 0$ and $α > 0$ , their block sum is positive definite; every spatial matrix is symmetric. Since $T (g, F)$ is algebraic in the state, it contributes only to lower-order terms and creates no off-diagonal derivative blocks. The prescribed local high-Sobolev construction [F2] applies after the compact chart localization in its proof and yields a reduced solution. Its derivative relations propagate; [F3] restores both Maxwell equations and the full Einstein equation on a smaller local domain. Every smooth physical solution in the same harmonic chart extracts to this same coefficient system because the component equations are precisely the harmonic-reduced Einstein equation and field-strength Maxwell evolution. This verifies the local symmetric-hyperbolic, recovery and compatibility hypotheses of [F4].

2.1step 1.1L1F4algebra

The equations $d F = 0$ and $d ⋆_{g} F = 0$ are covariant: pullback commutes with exterior differentiation, and for an orientation-preserving diffeomorphism $ϕ$ , $ϕ^{*} (⋆_{g} F) = ⋆_{ϕ^{*} g} (ϕ^{*} F)$ . The algebraic stress $T (g, F)$ is likewise natural under pullback. Thus the matter field, equations, stress and initial reconstruction define one fixed covariant model of the kind required by [F4]; no coordinate-dependent potential gauge is introduced.

2.2step 1.1step 1.2L1L2L3F3F4

Step 1.2, using [F3], has already recovered $d F = d ⋆_{g} F = 0$ on the local development; [L1] fixes the corresponding source-free model and stress tensor. The exact Hodge conversion in [L3] gives $\nabla_{a} F^{a b} = 0$ , so the stress identity applies with $j = 0$ and yields $\nabla^{a} T_{a b} = 0$ , independently of the unreduced Einstein equation. The constraints in the Statement are exactly [L2]'s four normal-projection constraints with the same sign, orientation, $κ, μ_{0}, Λ$ and $K$ conventions. This verifies the conservation and Einstein-constraint hypotheses of [F4].

3.1step 1.1step 2.1step 2.2step 1.2L2L3L4F4F5∎

We have now checked every specific input of [F4]: a fixed covariant model by step 2.1, complete matter data and smooth initialization by step 1.1 and the local harmonic jet in step 1.2, a smooth symmetric-hyperbolic reduction and reconstruction by step 1.2, on-shell stress conservation by step 2.2, and the model's Einstein constraints by [L2]. The choice assumptions are those in the Statement. Applying [F4] gives the smooth local development, variable-width collar, intrinsic globally hyperbolic Cauchy neighborhood and data-fixing isometry matching $F$ . The causal qualifications in [F5] confirm that no compactness or uniform collar width is used.

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Causal and Cauchy-surface inputs for smooth electrovacuum developments

Statement

Assume the Axioms of Countable Choice and Dependent Choice. Let $(M, g, N)$ be a time-oriented smooth globally hyperbolic Lorentzian four-manifold with the causal-curve and global-hyperbolicity conventions of Chronology, causality, strong causality, global hyperbolicity and time functions and Causal curves, chronological and causal reachability, and inextendibility. The following interfaces are available for the local and global geometry of smooth electrovacuum developments:

Every point has an arbitrarily small strictly convex normal neighborhood. For a future causal curve $σ$ in such a neighborhood from $p$ , its length satisfies $L_{g} (σ) \leq \sqrt{- g_{p} (v, v)}$ , where $v = \exp_{p}^{- 1} (σ (1))$ . Equality is characterized by the reparametrized radial geodesic when $v$ is future timelike and the curve does not return to $p$ after its start. The future causal and timelike sets are the exponential images of their tangent cones.
Each connected component admits a complete auxiliary Riemannian metric. The causal relation is closed, and the causal hull $J^{+} (K_{1}) \cap J^{-} (K_{2})$ of two compact sets is compact.
Let $h$ be the complete auxiliary metric $h'$ of The auxiliary Riemannian metric h=g+2n-flat tensor n-flat is complete after a proper-exhaustion conformal factor, and its closed bounded sets are compact clause (b), and let $α_{k} : [0, b_{k}] \to M$ , $0 < b_{k} < \infty$ , be future causal curves parametrized by $h$ -arclength, with $α_{k} (0) \to x$ and $α_{k} (b_{k}) \to z$ in $M$ . After passage to a subsequence with $b_{k} \to b \in [0, \infty]$ , the bounded-length, zero-length and unbounded-length conclusions of Two-endpoint limit curve theorem: bounded lengths give joining limits, unbounded lengths give two inextendible tails with the closure relation hold: for $0 < b < \infty$ the common-domain reparametrizations converge uniformly to a joining causal curve, for $b = 0$ the curves converge uniformly on their varying domains to $x = z$ , and for $b = \infty$ the forward and backward tails converge on compact parameter intervals to the respective inextendible curves with the supplier's sequential closure relation. Strong causality excludes the unbounded case when all images lie in one compact set.
Lorentzian time separation is finite on causal pairs in a globally hyperbolic spacetime, is attained by a causal geodesic, is upper semicontinuous, and is continuous at timelike-related points, with the degenerate coincident-endpoint case as specified in Time separation in a globally hyperbolic manifold: finiteness, existence of maximizing geodesics and the radial formula on causally convex normal regions.
Every smooth embedded spacelike hypersurface has a variable-width temporal neighborhood whose intrinsic domain-of-dependence interior is globally hyperbolic with that hypersurface Cauchy. Every globally hyperbolic spacetime also admits a smooth Cauchy temporal splitting as in Smooth Cauchy temporal splitting: a global temporal function with Cauchy levels and an orthogonal complete flow.

These are mathematical causal inputs. They add no field equations, empirical assumptions, completeness conclusion for causal geodesics, or uniform time-width assertion for noncompact hypersurfaces.

Facts & Assumptions

Given: Countable Choice and Dependent Choice, a smooth time-oriented Lorentzian metric, and the causal definitions in the Statement.

[F1]

Existence of strictly convex normal neighborhoods: small coordinate balls are normal neighborhoods of each of their points provides strict convex normal neighborhoods and unique contained joining geodesics; Local causality in a convex normal ball: causal reachability is the exponential cone, the Lorentzian length is bounded by the displacement radius, and null endpoints force the null geodesic gives the causal/timelike exponential-cone characterizations and local length maximality.

[F2]

The auxiliary Riemannian metric h=g+2n-flat tensor n-flat is complete after a proper-exhaustion conformal factor, and its closed bounded sets are compact constructs a complete Riemannian metric on each connected component. Under strong causality and compact diamonds, In a globally hyperbolic manifold the causal relation is closed and truncated causal cones over compact sets are compact proves closedness of the causal relation and compactness of causal hulls of compact sets.

[F3]

Two-endpoint limit curve theorem: bounded lengths give joining limits, unbounded lengths give two inextendible tails with the closure relation supplies the bounded and unbounded limit-curve conclusions for curves parametrized by auxiliary Riemannian arclength, including its explicit endpoint and strong-causality hypotheses.

[F4]

Time separation in a globally hyperbolic manifold: finiteness, existence of maximizing geodesics and the radial formula on causally convex normal regions defines time separation by a supremum of Lorentzian lengths and proves finiteness, attainment, causal-geodesic maximizers, upper semicontinuity, and continuity at timelike-related points; its zero-length coincident endpoint is treated as a degenerate maximizer.

[F5]

Local Cauchy neighborhoods in a conically bounded product chart gives a variable-width neighborhood of any smooth embedded spacelike hypersurface whose intrinsic domain-of-dependence interior has that hypersurface Cauchy; The existence of a Cauchy set implies strong causality and compact causal diamonds makes the neighborhood globally hyperbolic. Smooth Cauchy temporal splitting: a global temporal function with Cauchy levels and an orthogonal complete flow supplies a global smooth Cauchy temporal function, complete flow and product splitting on a globally hyperbolic spacetime.

Proof

1.1givenF1

Apply [F1] at each point of $M$ , with an arbitrary prescribed open neighborhood to obtain the local normal neighborhood. The first supplier gives the unique geodesic segment and strict convexity; the second applies to that same set and gives exactly the exponential-cone, null-rigidity and length conclusions in clause 1.

1.2givenF2algebra

Apply [F2]. The time orientation gives a smooth timelike field needed for its auxiliary Riemannian construction, and the assumed global hyperbolicity supplies strong causality and compact diamonds. The resulting complete auxiliary metric is precisely the metric required by the causal-structure lemma; its conclusions give clause 2.

1.3givenF1F2F3F4

The assumed manifold is globally hyperbolic, so [F4] applies to every fixed endpoint $q$ . Its definition of $τ_{q}$ as a supremum and its compact-diamond argument give finiteness and attainment; the maximizing-curve argument gives a causal geodesic; its local normal-neighborhood formula gives continuity on timelike-related points, while the endpoint-sequence argument gives upper semicontinuity on compact sets. The $r = q$ case has length zero by definition and is not silently treated as a nondegenerate curve. This proves clause 4 with exactly its endpoint qualifications.

2.1givenstep 1.2F3

For the sequence in clause 3, both endpoint limits and the exact auxiliary metric required by [F3] are stipulated. Together with Countable Choice, Dependent Choice and the smooth time-oriented metric, these are all its hypotheses. Apply its positive bounded-length, zero-length and unbounded-length alternatives after taking a subsequence with $b_{k} \to b$ ; the zero-length limit is constant and is not called a causal curve with nonzero velocity. Global hyperbolicity supplies strong causality for its compact-imprisonment conclusion. This proves clause 3 with precisely the stated endpoint conditions.

3.1givenF5∎

For any smooth embedded spacelike hypersurface, apply [F5]'s local-neighborhood clause. Its width is a positive function on the hypersurface, and its domain of dependence is intrinsic to that neighborhood; the Cauchy and global-hyperbolicity conclusions are therefore precisely those in clause 5, without compactness or uniform-width assumptions. For any globally hyperbolic spacetime, the temporal-splitting supplier applies under the same choice assumptions and gives its product decomposition.

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Source-free smooth maximal globally hyperbolic Einstein–Maxwell development

Statement

Assume Countable Choice and Dependent Choice. Let $Σ$ be a nonempty connected smooth oriented three-manifold without boundary with smooth Riemannian metric $h$ , smooth symmetric tensor $K$ , and smooth spatial vectors $e, b$ . Fix the source-free Einstein–Maxwell model, its constants $κ > 0$ , $μ_{0} > 0$ , $Λ \in R$ , signature $(-, +, +, +)$ , orientation and $K (X, Y) = g (\nabla_{X} n, Y)$ convention as in Electrovacuum tensor Cauchy system. Assume the four constraints of Gauss-Codazzi and Maxwell initial constraints and reconstruct the complete initial two-form by $F (N, e_{i}) = - e_{i}$ , $F (e_{i}, e_{j}) = ϵ_{i j k} b_{k}$ . Then there is a smooth connected time-oriented globally hyperbolic development $(M_{\max}, g_{\max}, F_{\max}, ι)$ into which every other smooth source-free Einstein–Maxwell development of these complete data embeds by a time-orientation-preserving isometric embedding fixing $ι$ and carrying $F$ to $F_{\max}$ . The maximal development is unique up to a unique such data-fixing isometry. This maximality is within the smooth globally hyperbolic development class for this fixed two-form model; it asserts no geodesic completeness, no maximality among rough or non-globally-hyperbolic extensions, and no global-in-time regularity estimate. Here an affinely parametrized metric geodesic solves ${\ddot{x}}^{a} + Γ^{a}_{b c} {\dot{x}}^{b} {\dot{x}}^{c} = 0$ on an interval; maximality of that solution forbids extension to a strictly larger interval with the same affine parameter. Geodesic completeness means every maximal affine geodesic has parameter domain all of $R$ .

Facts & Assumptions

Given: Countable Choice and Dependent Choice, connected smooth constrained source-free data, the fixed covariant model and its complete two-form data.

[L1]

Draft model Electrovacuum tensor Cauchy system fixes the equations, stress, constants and pullback conventions. The initial $F$ data in the Statement specify all six independent components on $R N \oplus T Σ$ , with no global potential requirement. Its item file is listed in batch 6; its status is draft and its precheck is pending; this input remains conditional.

[L2]

Draft constraint result Gauss-Codazzi and Maxwell initial constraints assumes an existing smooth source-free solution near a smooth spacelike embedded hypersurface and proves that its induced data satisfy the four Einstein and Maxwell constraints in the selected $K, e, b$ convention. It proves necessity, not existence from constrained data. Its item file is listed in batch 6; its status is draft and its precheck is pending; this input remains conditional.

[L3]

Draft identity Maxwell stress algebra and covariant conservation gives $\nabla^{a} T_{a b} = - F_{b c} j^{c}$ for smooth fields satisfying $d F = 0$ and $\nabla_{a} F^{a b} = - μ_{0} j^{b}$ . The exact Hodge conversion is $\nabla_{a} F^{a b} = \frac{1}{3!} ε^{b a c d} (d ⋆_{g} F)_{a c d} = \frac{1}{\sqrt{∣ \det g ∣}} \partial_{a} (\sqrt{∣ \det g ∣} F^{a b}), ε_{a b c d} : = ({v o l}_{g})_{a b c d},$ with indices raised by $g^{- 1}$ . It follows from metric-compatibility $\nabla_{a} ε_{b c d e} = 0$ , $(⋆_{g} F)_{c d} = \frac{1}{2} ε_{c d e f} F^{e f}$ , the antisymmetrized covariant derivative, and $ε^{b a c d} ε_{c d e f} = - 2 (δ_{e}^{b} δ_{f}^{a} - δ_{f}^{b} δ_{e}^{a})$ ; the coordinate divergence follows from the trace Christoffel identity and antisymmetry of $F$ . Thus $d ⋆_{g} F = 0$ supplies $\nabla_{a} F^{a b} = 0$ , and the model's $d F = 0$ supplies the other hypothesis, so the stress identity applies with $j = 0$ before Einstein recovery. Its item file is listed in batch 6; its status is draft and its precheck is pending; this input remains conditional.

[F1]

Smooth local geometric existence and uniqueness for source-free Einstein–Maxwell data proves smooth local existence and geometric uniqueness for this exact covariant model and complete data on arbitrary smooth initial slices. Its proof verifies the local symmetric-hyperbolic reduction, full model recovery, stress conservation and constraints, conditional on the actual CP01 draft statements [L1]–[L3], whose files are present but whose prechecks are pending although their IDs are listed in the batch-6 manifest.

[F2]

Causal and Cauchy-surface inputs for smooth electrovacuum developments records the causal, Cauchy-surface and choice interfaces used in the globally hyperbolic development construction, with their exact hypotheses.

[F3]

Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality defines a development, a data-fixing development embedding, and maximality by universal embedding into a smooth globally hyperbolic development. It makes no completeness requirement.

[F4]

Existence and uniqueness of the smooth maximal globally hyperbolic development of constrained initial data in the selected matter sectors proves the set-sized gluing, quotient, universality and uniqueness result for any fixed admissible covariant model once constrained local existence and geometric uniqueness hold at every spacelike restart. The actual inspected carrier is published with verification precheck pending. Its proof is sourced to Sbierski, whose original theorem is vacuum-only; therefore the current GR theorem's additional-model clause applies here only after [F1] discharges the EMG local obligations.

Proof

1.1givenL1L2F1algebra

The initial bundle formulas specify the three normal-spatial and three spatial-spatial components of $F$ and hence all independent matter data, as in [F1]'s complete-data reconstruction. The four equations displayed in [L2] are assumed for the initial data in the Statement, with the same stress normalization, orientation and sign convention. They are precisely the constraint hypotheses of the local existence theorem [F1]. Thus the data are complete constrained data for the one fixed EMG model; no assumption that $F = d A$ is introduced. The necessity theorem [L2] will supply the constraints on spacelike restart surfaces of existing solutions.

2.1step 1.1L2L3F1F4

By [F1], every smooth constrained initial data set for this fixed model has a smooth local development and data-fixing geometric uniqueness. The same statement applies to every smooth spacelike restart: the full Einstein and Maxwell equations hold there; their normal projections give [L2]'s constraints; the normal electric and spatial magnetic components determine the complete restarted two-form; and the Hodge conversion in [L3] lets the stress identity apply with $j = 0$ after Maxwell propagation. Therefore the local existence and uniqueness obligations are available at every restart used in [F4].

3.1step 2.1F1F2F4algebra

In particular, [F1] provides at least one local development. Its local Cauchy-neighborhood clause, also covered by [F2], restricts it to an open globally hyperbolic neighborhood with $Σ$ Cauchy, so the nonempty-development hypothesis of [F4] holds. The model and complete data remain fixed under all comparisons, and the Countable Choice and Dependent Choice assumptions match [F4].

4.1step 1.1step 2.1step 3.1F2F3F4∎

Apply [F4] to this fixed model, data, local development and restart theorem. Its universality clause gives a data-fixing development embedding of every smooth globally hyperbolic development into one maximal development; its uniqueness clause gives exactly one data-fixing isometry between any two maximal developments. By [F3], this is precisely maximality in the stated smooth globally hyperbolic class. The definition imposes no condition on affine-parameter endpoints of geodesics, so this conclusion contains no geodesic-completeness assertion.

5 · Examples, experiments and counterexamples

None yet.

Positive symmetric Maxwell field-strength reduction

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Maxwell stress divergence for classical fields

Statement

Facts & Assumptions

Proof

Positive symmetric Maxwell field-strength reduction

Statement

Facts & Assumptions

Proof

Maxwell Gauss constraints propagate by form transport

Statement

Facts & Assumptions

Proof

Coupled reduced Einstein–Maxwell local evolution

Statement

Facts & Assumptions

Proof

Einstein harmonic gauge propagation with Maxwell on shell

Statement

Facts & Assumptions

Proof

Local high-Sobolev and smooth electrovacuum

Statement

Facts & Assumptions

Proof

Local Sobolev stability and continuation for the reduced electrovacuum system

Statement

Facts & Assumptions

Proof

Smooth local geometric existence and uniqueness for source-free Einstein–Maxwell data

Statement

Facts & Assumptions

Proof

Causal and Cauchy-surface inputs for smooth electrovacuum developments

Statement

Facts & Assumptions

Proof

Source-free smooth maximal globally hyperbolic Einstein–Maxwell development

Statement

Facts & Assumptions

Proof

5 · Examples, experiments and counterexamples

Sources