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Local evolution of the adopted electrovacuum model

1 · Prerequisites

2 · Summary

Scope

This page develops local and maximal Cauchy evolution in the selected smooth source-free Einstein-Maxwell action branch. It derives the field-strength equations from the adopted compact-variation action, then applies the conditional CP02 local and global mathematical suppliers. Two local mathematical inputs make the normal collar and fixed bundle-sector arguments explicit. Initial constraints, SI signs, periodic Sobolev regularity, and the compatible normalized circle-bundle sector remain explicit. These model deductions neither prove that the adopted action describes nature nor assert geodesic completeness.

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.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck pendingjudge pass (gpt-6-luna)Open item page →

Local normal collar charts at a smooth spacelike hypersurface

Statement

Let $(M, g, N)$ be a smooth time-oriented Lorentzian four-manifold, let $Σ$ be a smooth three-manifold, and let $i : Σ \to M$ be a smooth spacelike embedding. Suppose a smooth future unit normal field $n$ along $i (Σ)$ is supplied. For every $p \in Σ$ there are an open neighborhood $U_{p} \subseteq Σ$ , a number $ϵ_{p} > 0$ , and an open $V_{p} \subseteq M$ such that the restricted map $Φ_{p} : (- ϵ_{p}, ϵ_{p}) \times U_{p} \to V_{p}$ given by

$Φ_{p} (t, x) = γ_{i (x), n_{i (x)}} (t)$

is a diffeomorphism, where $γ_{q, v}$ is the maximal affinely parameterized metric geodesic with initial data $(q, v)$ . At its center,

$d (Φ_{p})_{(0, p)} (a \partial_{t} + X) = a n_{i (p)} + d i_{p} (X) .$

The neighborhoods and time widths may depend on $p$ ; no globally injective normal exponential map or uniform width is asserted.

Facts & Assumptions

Given: The smooth Lorentzian manifold, the spacelike embedding, and the supplied smooth future unit normal.

[F1]

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model fixes the smooth spacetime structure and time orientation. Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons then makes $i (Σ)$ an embedded three-dimensional submanifold with positive definite induced metric; the supplied future unit normal is timelike and orthogonal to its tangent space.

[F2]

The geodesic initial-value problem and the exponential map of the metric-selected connection: geodesics depend smoothly on time, base point and initial tangent on an open set of admissible data.

[L1]

The smooth inverse function theorem on manifolds: a smooth map whose differential is an isomorphism at a point restricts to a diffeomorphism between neighborhoods of that point and its image.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): the tangent-bundle smooth-structure convention used by [F2] is carried under ${A C}_{ω}$ . No stronger choice principle is used in this local construction.

Proof

Given: The geometric data in the Statement and the choice convention [A1].

1.1givenA1F2algebra

The map $(t, x) \mapsto (t, i (x), n_{i (x)})$ is smooth and sends $(0, p)$ to admissible geodesic initial data. By [F2] that datum has an open neighborhood on which the geodesic evaluation map is smooth. Their composition is therefore the normal exponential map $Φ_{p}$ on a neighborhood of $(0, p)$ .

2.1step 1.1F1L1algebraconstruct∎

Differentiating the geodesic at zero gives the displayed differential: the time direction maps to $n_{i (p)}$ , and the slice direction maps to $d i_{p} (X)$ . If $a n_{i (p)} + d i_{p} (X) = 0$ , its inner product with $n_{i (p)}$ is $- a = 0$ because $n$ is orthogonal to $i (Σ)$ ; then $d i_{p} (X) = 0$ and $X = 0$ because an embedding is an immersion. Thus the differential is an isomorphism between four-dimensional tangent spaces. Apply [L1], then shrink its source neighborhood to a product $(- ϵ_{p}, ϵ_{p}) \times U_{p}$ ; the restriction is still a diffeomorphism onto its open image.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck pendingjudge pass (gpt-6-luna)Open item page →

Closed two-forms agreeing on a Cauchy slice differ by an exact form

Statement

Assume ${A C}_{ω}$ and Dependent Choice. Let $(M, g, N)$ be a smooth time-oriented globally hyperbolic Lorentzian four-manifold, and let $i : Σ ↪ M$ be a smooth spacelike Cauchy embedding. If $F, \overset{ˉ}{F} \in Ω^{2} (M)$ are smooth closed two-forms with $i^{*} F = i^{*} \overset{ˉ}{F}$ , then $F - \overset{ˉ}{F}$ is exact: there is a smooth $α \in Ω^{1} (M)$ such that $d α = F - \overset{ˉ}{F}$ .

Here a closed form has zero exterior derivative, and exactness means being the exterior derivative of a globally defined smooth form, as in De rham cohomology.

Facts & Assumptions

Given: The globally hyperbolic Lorentzian manifold, Cauchy embedding, two closed forms, and the two stated choice principles.

[F1]

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model and Smooth Cauchy temporal splitting: a global temporal function with Cauchy levels and an orthogonal complete flow: the spacetime is smooth and time-oriented; under Countable Choice and Dependent Choice it has a smooth temporal function $T$ and a complete future-directed timelike flow $Φ_{t}$ with $T (Φ_{t} (p)) = T (p) + t$ . The published temporal-splitting carrier has verification precheck pending; this proof uses its exact stated flow interface conditionally and does not certify that supplier.

[F2]

Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons: a Cauchy embedding meets every inextendible causal curve at exactly one parameter value, and the induced metric on its spacelike image is positive definite.

[F3]

Lorentzian exterior calculus and Hodge star: the exterior derivative has the coordinate formula $d ω = (k + 1) \partial_{[a_{0}} ω_{a_{1} \dots a_{k}]}$ on $k$ -forms, and $d^{2} = 0$ .

[F4]

De rham cohomology: a smooth form is exact precisely when it is $d α$ for a globally defined smooth form of one lower degree.

[L1]

The smooth inverse function theorem on manifolds: a smooth map with invertible differential at a point is a diffeomorphism on sufficiently small neighborhoods of that point.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) and The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain are the only choice principles assumed.

Proof

Given: The data in the Statement.

1.1givenA1F1F2L1algebraconstruct

Let $X$ and $Φ_{t}$ be the complete timelike field and its flow from [F1], and define $Ψ : R \times Σ \to M$ by $Ψ (t, x) = Φ_{t} (i (x))$ . Since $T (Φ_{t} (p)) = T (p) + t$ , each orbit has no endpoint at either infinite end and is an inextendible timelike curve. Thus [F2] makes its intersection with $i (Σ)$ unique. Hence $Ψ$ is surjective and injective: if two images agree, their points lie on the same orbit, their slice intersections agree, and the flow times agree because the temporal function increases at unit rate along $X$ . The flow is smooth, so $Ψ$ is smooth. If $d Ψ_{(t, x)} (a \partial_{t} + V) = 0$ , applying $d Φ_{- t}$ gives $a X_{i (x)} + d i_{x} (V) = 0$ . The first vector is timelike when $a \neq 0$ , while the second is tangent to a spacelike hypersurface; therefore $a = 0$ and $V = 0$ . Thus $d Ψ$ is an isomorphism, and [L1] shows that $Ψ$ is a local diffeomorphism. Its bijectivity makes it a global diffeomorphism, with $Ψ (0, x) = i (x)$ .

2.1step 1.1F3algebraconstruct

Pull back $G : = F - \overset{ˉ}{F}$ to the product and write its unique time/spatial decomposition as $\tilde{G} : = Ψ^{*} G = d t \land η_{t} + β_{t}$ , where $η_{t}$ is a smooth family of spatial one-forms and $β_{t}$ a smooth family of spatial two-forms on $Σ$ . Pullback commutes with $d$ : in local coordinates the first-derivative terms in $d (Ψ^{*} G)$ are the pullback of $d G$ , while the second derivatives of $Ψ$ cancel in pairs because mixed partials commute and $G$ is antisymmetric. Since $d G = 0$ by hypothesis and [F3], the product-coordinate formula for $d \tilde{G}$ gives $\partial_{t} β_{t} = d_{Σ} η_{t}$ . The slice equality implies $β_{0} = 0$ . Define the spatial one-form ${\tilde{α}}_{t} : = \int_{0}^{t} η_{s} d s$ and give $\tilde{α}$ zero $d t$ component. The integral of a smooth parameter family is smooth, and direct differentiation gives $d \tilde{α} = d t \land η_{t} + d_{Σ} {\tilde{α}}_{t} = d t \land η_{t} + \int_{0}^{t} d_{Σ} η_{s} d s = d t \land η_{t} + β_{t} = \tilde{G}$ .

3.1step 1.1step 2.1F4algebra∎

Set $α : = (Ψ^{- 1})^{*} \tilde{α}$ . The same coordinate pullback identity gives $d α = G = F - \overset{ˉ}{F}$ . By [F4], this is exact on the whole spacetime, not merely on coordinate neighborhoods.

Physics theoremStatement: Literature-sourcedProof: AI-adaptedprecheck pendingOpen item page →

Local evolution of the adopted electrovacuum model

Physical scope

Conditional local evolution in the selected smooth, source-free, zero-matter-action Einstein-Maxwell branch. It covers the stated periodic high-Sobolev class and arbitrary smooth constrained data locally, subject to magnetic compatibility with one supplied normalized circle-bundle sector; electric initial data remain independent smooth data. There is no uniform noncompact time width, global-time, maximal-development, or empirical-validity claim.

Statement

Assume Countable Choice and Dependent Choice. Let $Σ$ be a nonempty smooth oriented three-manifold without boundary, with smooth Riemannian metric $h$ , smooth symmetric covariant tensor $K$ , and smooth tangent vector fields $e, b$ . Fix $κ > 0$ , $μ_{0} > 0$ , and $Λ \in R$ , and use the signature, SI conventions, and electromagnetic stress tensor of Einstein–Maxwell primitive quantities and SI conventions and Electrovacuum tensor Cauchy system. Assume $(Σ, h, K, e, b)$ satisfies all four source-free constraints of Constrained electrovacuum initial data.

In the periodic case $Σ = T_{2 π L_{0}}^{3}$ with fixed $L_{0} > 0$ in metres, for an integer $s \geq 5$ , also consider smooth data with $h \in H^{s + 1}$ , $K, e, b \in H^{s}$ , and the uniform lower eigenvalue bound $h \geq c_{0} I$ for some $c_{0} > 0$ specified by Periodic high-Sobolev state spaces. Here the length-frame metric and curvature arrays and the orthonormal electric/magnetic arrays are composed with $[θ] \mapsto [L_{0} θ]$ for Sobolev norms on $T_{2 π}^{3}$ ; geometric spatial derivatives are $\partial_{X^{i}} = L_{0}^{- 1} \partial_{θ^{i}}$ . On its common local interval the field-strength solution has $g \in L^{\infty} H^{s + 1} \cap C^{0} H^{s}$ , $\partial_{t} g \in L^{\infty} H^{s} \cap C^{0} H^{s - 1}$ , and $F \in L^{\infty} H^{s} \cap C^{0} H^{s - 1}$ , with the time coordinate measured in length units as in Einstein–Maxwell primitive quantities and SI conventions.

For the magnetic data define the smooth two-form $β_{b} (X, Y) : = {vol}_{h} (b, X, Y)$ . Assume the following explicit initial bundle compatibility: there are a countable open cover $(U_{i})$ of $Σ$ , smooth transition maps $g_{i j} : U_{i} \cap U_{j} \to U (1)$ , smooth local real one-forms $a_{i}$ , and one constant $κ_{0} > 0$ . Require $g_{i i} = 1$ , $g_{i j} g_{j k} = g_{i k}$ on triple overlaps, and $a_{j} = a_{i} + λ (g_{i j})$ on every overlap, where $λ (g) = - i g^{- 1} d g$ ; also require $κ_{0} d a_{i} = β_{b}$ on $U_{i}$ . Here $κ_{0}$ has units weber and the normalized local forms have the units in Local potentials and smooth circle-bundle connection data. This is the fixed normalized circle-bundle sector of Local potentials and smooth circle-bundle connection data; it imposes no global real-valued potential and no condition on the independent electric data beyond the stated constraints.

In the action model of Interior stationary Einstein–Maxwell–matter action, choose the zero-matter branch: the matter configuration space is a singleton and $S_{m} \equiv 0$ , so its current and Hilbert tensor vanish. Then there is a smooth time-oriented local development of the source-free system on a neighborhood of the entire initial surface, after restriction to a possibly variable-width globally hyperbolic Cauchy neighborhood. Its electromagnetic two-form is the curvature of a smooth connection in the supplied normalized circle-bundle sector, and its initial electric and magnetic fields are exactly $e, b$ . Any two such developments are related near the initial surface by a data-fixing, time-orientation-preserving isometry carrying the metric and two-form to one another. This is a conditional model deduction; it asserts neither physical validity, a uniform positive time width for noncompact $Σ$ , global-time existence, nor a maximal-development result.

Facts & Assumptions

Given: The smooth constrained data and constants above, the normalized initial transition and potential data, the zero-matter action specialization, and ${A C}_{ω} + D C$ .

[F1]

Interior stationary Einstein–Maxwell–matter action: the adopted total action is varied independently under compactly supported metric, fixed-sector connection, and matter variations. Its current and stress are defined by those variations. The action carrier is draft and its current Step-1 authoring record is ready; this is not an independent proof or physical acceptance.

[F2]

Einstein–Maxwell equations from compact action stationarity: under the compact-variation hypotheses, stationarity is equivalent to the Einstein equation, the sourced Maxwell equation $\nabla_{a} F^{a b} = - μ_{0} j^{b}$ , and the selected matter Euler–Lagrange condition. Its draft carrier and Step-1 record are current; the equivalence is used only at that inspected interface.

[F3]

Einstein–Maxwell primitive quantities and SI conventions: the spacetime signature is $(-, +, +, +)$ ; $F$ is a smooth real two-form with SI frame components in tesla, and the zero-matter Einstein coupling is $κ = 8 π G_{N} / c^{4}$ .

[F4]

Electrovacuum tensor Cauchy system: the source-free model is $G_{a b} + Λ g_{a b} = κ T_{a b}^{E M}$ , $d F = 0 = d (⋆_{g} F)$ , with $T_{a b}^{E M} = μ_{0}^{- 1} (F_{a c} F_{b}^{c} - \frac{1}{4} g_{a b} F_{c d} F^{c d})$ . The full initial two-form is reconstructed by $F (n, e_{i}) = - e_{i}$ and $F (e_{i}, e_{j}) = ϵ_{i j k} b_{k}$ .

[F5]

Constrained electrovacuum initial data: the four constraints are the electric and magnetic divergences, the scalar Hamiltonian constraint, and the covector momentum constraint with the stated $K$ and $e \times_{h} b$ signs.

[F15]

Periodic high-Sobolev state spaces: $H^{s}$ is the real componentwise Fourier Sobolev space on $T_{2 π}^{3}$ ; the regularity orders and positive metric lower bound in the Statement match its definition.

[F6]

Local high-Sobolev and smooth electrovacuum: smooth constrained data on an arbitrary smooth initial three-manifold have a smooth full development locally near each initial point; the periodic $H^{s}$ , $s \geq 5$ , statement and its regularity bounds are separate clauses. This current draft supplier has verification.precheck: pending and a current Step-1-ready record; the local argument is conditional on its exact inspected interface, not independent acceptance.

[F7]

Smooth local geometric existence and uniqueness for source-free Einstein–Maxwell data: complete smooth data for this same field-strength system have local smooth developments, a smaller globally hyperbolic Cauchy neighborhood, and data-fixing geometric uniqueness near the entire initial surface. Its actual draft argument is conditional on its inspected CP02 suppliers; its current verification.precheck is pending and its Step-1 record is ready. This consumer uses that conditional interface.

[F8]

Local potentials and smooth circle-bundle connection data: compatible local forms and circle transition functions with $a_{j} = a_{i} + λ (g_{i j})$ define one normalized bundle sector; the curvature is represented locally by $κ_{0} d a_{i}$ .

[F9]

Circle gluing, curvature and connection variations: such data on a four-manifold glue to a smooth principal $U (1)$ bundle with a connection and global normalized curvature; adding one global base one-form to its normalized local connection forms preserves the transition functions and bundle sector.

[F10]

Local normal collar charts at a smooth spacelike hypersurface: the normal exponential map is a diffeomorphism on a neighborhood of each point of a smooth spacelike initial slice, with differential $(a, X) \mapsto a n + d i (X)$ . Its proof uses the actual smooth geodesic and inverse-function suppliers. This is a current local draft result with a complete conditional argument. Its frontmatter precheck field remains pending for engine recertification.

[F11]

Closed two-forms agreeing on a Cauchy slice differ by an exact form: on a globally hyperbolic development, two closed two-forms with the same pullback to the Cauchy surface differ by a global exact form. Its use is conditional on the current temporal-splitting supplier. The local draft proof is complete and uses that exact interface conditionally; its frontmatter precheck field remains pending for engine recertification.

[F12]

Smooth Cauchy temporal splitting: a global temporal function with Cauchy levels and an orthogonal complete flow: under ${A C}_{ω}$ and DC, a globally hyperbolic smooth spacetime has a complete timelike flow and a product splitting by Cauchy slices. Its current published carrier has verification.precheck pending; this proof records and conditionally uses its exact interface, not an independent acceptance.

[F13]

The smooth inverse function theorem on manifolds: a smooth map with invertible differential is locally a diffeomorphism; this is the local step used in [F10] and [F11].

[F14]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) and The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain: the assumptions are respectively ${A C}_{ω}$ and DC. No full Axiom of Choice is used.

[A1]

Choose the zero-matter specialization of [F1]. The zero functional has zero metric and connection variations, so $T^{m} = 0$ and $j = 0$ ; its singleton matter sector has no additional field equation.

[A2]

On a countable cover of $Σ$ , the supplied $g_{i j}$ , $a_{i}$ , and $κ_{0}$ satisfy the cocycle, transition, and curvature equations stated in the Statement. Only the magnetic two-form $β_{b}$ is curvature data; $e$ is independent normal electric data.

[A3]

The choices in [F14] are adopted for the local existence, uniqueness, and temporal-splitting suppliers. They are model assumptions for this deduction, not empirical premises.

Proof

Given: The data, assumptions, and supplier interfaces above.

1.1givenA2F3F4F5algebra

The four constraints in [F5] are exactly the initial constraints for [F4] with the same $K$ , orientation, stress normalization, and definitions of $e, b$ . The Cauchy data prescribe the magnetic two-form $β_{b} (X, Y) = {vol}_{h} (b, X, Y)$ and the electric covector determined by $e$ . In any resulting future-normal orthonormal frame, [F4] records their components as $F (n, e_{i}) = - e_{i}$ and $F (e_{i}, e_{j}) = ϵ_{i j k} b_{k}$ . These are the six independent components of the initial field strength and require no global potential.

2.1step 1.1A3F5F6F7F10F14F15

Apply [F6] to these smooth constrained data and choice assumptions [A3]. Its arbitrary-slice clause gives a source-free solution locally near each initial point. In the periodic case, [F15] fixes the stated real Sobolev data convention and [F6] gives the displayed Sobolev regularity bounds. The stronger geometric patching and uniqueness supplier [F7] uses the same complete data and exact field-strength conventions; it yields a single smooth development on a variable-width neighborhood of the entire initial surface and a smaller globally hyperbolic Cauchy neighborhood. No positive uniform width is inferred. The resulting spacelike Cauchy embedding has a future unit normal by the definition of spacelike hypersurface; [F10] then supplies normal geodesic charts at each point, again with no uniform width. These charts justify the local observer-frame reconstruction after the spacetime exists; they are not used to assume local existence.

3.1A2A3step 2.1F4F8F9F10F11F12F13

Let $i : Σ ↪ M$ be the Cauchy embedding of the globally hyperbolic neighborhood supplied in step 2.1, and let $Φ_{t}$ be the complete flow from [F12]. The product-map construction in proof step 1.1 of [F11] defines $Ψ : R \times Σ \to M$ by $Ψ (t, x) = Φ_{t} (i (x))$ and proves that it is a diffeomorphism with $Ψ (0, x) = i (x)$ . Pull the initial transition functions and local forms in [A2] to $R \times Σ$ by the projection ${pr}_{Σ}$ , construct the reference connection there by [F8]–[F9], and transport it to $M$ by $Ψ$ . Its curvature $\overset{ˉ}{F}$ satisfies $i^{*} \overset{ˉ}{F} = β_{b} = i^{*} F$ . Both $F$ and $\overset{ˉ}{F}$ are closed by [F4] and [F9]. Apply [F11] on this globally hyperbolic Cauchy neighborhood to obtain a global one-form $α$ with $F - \overset{ˉ}{F} = d α$ . Adding $α / κ_{0}$ to the normalized local connection representatives preserves their transition functions by [F9] and changes the physical curvature by $κ_{0} d (α / κ_{0}) = d α$ , hence to exactly $F$ . Thus the field-strength solution lies in the supplied fixed circle-bundle sector; the temporal splitting and inverse-function inputs are exactly those used in [F10]–[F11].

3.2step 1.1step 2.1A1F2F7

For any other solution of the same model with the same complete data, [F2] gives the same source-free field-strength system. [F7] then supplies a data-fixing, time-orientation-preserving isometry on a neighborhood of the initial surface carrying $g$ and $F$ to the other development. The argument asserts uniqueness of these field-strength fields, not uniqueness of a local connection representative.

4.1A1F1F2F4step 2.1step 3.1

By [A1], the selected matter variations vanish and $j = T^{m} = 0$ . The solution from step 2.1 satisfies the field equations in [F4], and step 3.1 realizes its $F$ as the curvature of a connection in the fixed sector. The converse direction of [F2] therefore makes this configuration stationary for the adopted action under every allowed compact variation. This is a conditional deduction from the action branch, not a proof that the model describes nature.

5.1step 2.1step 3.1step 4.1step 3.2∎

Steps 2.1, 3.1, 4.1 and 3.2 prove the stated local existence, bundle sector, action stationarity and geometric uniqueness. No empirical premise, global-time claim, or maximal-development assertion has entered.

Physics theoremStatement: Literature-sourcedProof: AI-adaptedprecheck pendingOpen item page →

Electromagnetic conservation in source-free curved spacetime

Physical scope

A local model deduction for the smooth zero-matter-action Einstein-Maxwell branch on a Lorentzian four-manifold. It asserts covariant conservation of electromagnetic stress on stationary source-free configurations only; it does not assert existence, global energy conservation, or empirical validity.

Statement

Adopt the zero-matter specialization of Interior stationary Einstein–Maxwell–matter action: the matter functional is identically zero on a singleton matter configuration space, so its current and Hilbert tensor vanish. Let $(M, g, N, o)$ be a smooth, time-oriented, oriented Lorentzian four-manifold of signature $(-, +, +, +)$ , and let $F$ be the smooth physical curvature two-form of a connection in a fixed normalized circle-bundle sector, with the local $F = κ_{0} d a_{i}$ convention of Local potentials and smooth circle-bundle connection data. If the adopted compact-variation action is stationary, then its electromagnetic stress tensor

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

obeys

$\nabla^{a} T_{a b}^{E M} = 0.$ This is a local conditional deduction for the stated classical model. It does not assert a global conserved energy, existence of a stationary configuration, or empirical validity.

Facts & Assumptions

Given: The smooth geometry, fixed-sector connection, adopted action, and zero-matter specialization in the Statement.

[F1]

Interior stationary Einstein–Maxwell–matter action: stationarity is adopted for independent compactly supported metric, fixed-sector connection, and matter variations. The selected current and stress are the matter first-variation coefficients.

[F2]

Einstein–Maxwell equations from compact action stationarity: for that action and compact variations, stationarity is equivalent to its Einstein, Maxwell, and matter Euler–Lagrange equations. The connection equation is $\nabla_{a} F^{a b} = - μ_{0} j^{b}$ , and $d F = 0$ because $F$ is a connection curvature.

[F3]

Maxwell stress algebra and covariant conservation: for a smooth field satisfying $d F = 0$ and $\nabla_{a} F^{a b} = - μ_{0} J^{b}$ , $\nabla^{a} T_{a b}^{E M} = - F_{b c} J^{c}$ ; its proof derives the identity by contracting the homogeneous Maxwell equation and applying the product rule. This supplier is a current draft with Step-1 readiness and verification.precheck: pending; its exact identity is used conditionally.

[F4]

Electrovacuum tensor Cauchy system and Einstein–Maxwell primitive quantities and SI conventions: $T^{E M}$ has the displayed SI normalization on the adopted Lorentzian signature.

[F5]

Local potentials and smooth circle-bundle connection data and Circle gluing, curvature and connection variations: the normalized local representatives glue in one fixed circle-bundle sector, and their physical curvature is $F = κ_{0} d a_{i}$ .

[A1]

The matter functional is $S_{m} \equiv 0$ on a singleton configuration space. Hence every matter variation vanishes and the two coefficients $T^{m}, j$ in [F1] are zero. This choice introduces no empirical premise.

Proof

Given: The data and assumptions in the Statement.

1.1A1F1algebra

Since $S_{m}$ is the zero functional, its first variation is zero for every admissible compactly supported metric, connection, or matter variation. Consequently the matter stress and current defined by [F1] satisfy $T_{a b}^{m} = 0$ and $j^{a} = 0$ .

2.1step 1.1F2F3F4F5algebra

Apply [F2] to the stationary configuration. Its connection equation and the value $j = 0$ from step 1.1 give $\nabla_{a} F^{a b} = 0$ ; the homogeneous equation $d F = 0$ also holds because $F$ is the physical curvature of a smooth connection with the normalization in [F5]. These are exactly the source-free Maxwell hypotheses of [F3].

3.1step 2.1F3algebra∎

Apply [F3] with $J = 0$ . Its pointwise identity reduces to $\nabla^{a} T_{a b}^{E M} = 0$ , as claimed. This is covariant local conservation, not a statement about a global energy charge.

Physics theoremStatement: Literature-sourcedProof: AI-adaptedprecheck pendingOpen item page →

Maximal smooth Cauchy development of adopted electrovacuum

Physical scope

Conditional universal smooth globally hyperbolic development for the zero-matter-action Einstein-Maxwell field-strength model, with initial data in one supplied normalized circle-bundle sector. Maximality is within smooth globally hyperbolic developments; no geodesic completeness, rough or non-globally-hyperbolic extension maximality, global-time estimate, or empirical-validity claim is made.

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 tangent vector fields $e, b$ . Fix $κ > 0$ , $μ_{0} > 0$ , $Λ \in R$ , and the SI Lorentzian conventions of Einstein–Maxwell primitive quantities and SI conventions. Assume the four constraints of Constrained electrovacuum initial data. Define $β_{b} (X, Y) = {vol}_{h} (b, X, Y)$ , and supply a countable cover $(U_{i})$ , transition maps $g_{i j} : U_{i} \cap U_{j} \to U (1)$ , local real one-forms $a_{i}$ , and $κ_{0} > 0$ satisfying $g_{i i} = 1$ , $g_{i j} g_{j k} = g_{i k}$ , $a_{j} = a_{i} + λ (g_{i j})$ , and $κ_{0} d a_{i} = β_{b}$ , where $λ (g) = - i g^{- 1} d g$ . These are the normalized local data of Local potentials and smooth circle-bundle connection data restricted to the initial slice. Adopt the zero-matter action branch $S_{m} \equiv 0$ of Interior stationary Einstein–Maxwell–matter action.

Then there is a smooth connected time-oriented globally hyperbolic development $(M_{\max}, g_{\max}, F_{\max}, i_{\max})$ of the source-free Electrovacuum tensor Cauchy system with the complete data $(h, K, e, b)$ . The two-form $F_{\max} = κ_{0} f_{\max}$ is the physical curvature of a smooth connection in the bundle sector obtained by transporting the supplied initial sector over a Cauchy product. Here $f_{\max} \in Ω^{2} (M_{\max})$ is its normalized curvature, locally $f_{\max} = d a_{i}$ ; the supplied constant $κ_{0} > 0$ has units Wb, so multiplication gives the SI physical curvature. Every other smooth globally hyperbolic development with the same complete data for which the adopted action is stationary embeds into this one by a time-orientation-preserving isometric embedding that fixes $Σ$ and carries $F$ to $F_{\max}$ . Any two maximal developments are related by exactly one data-fixing isometry of $(g, F)$ . Maximality is within the smooth globally hyperbolic solution class; it asserts no geodesic completeness, no maximality among rough or non-globally-hyperbolic extensions, no global-in-time regularity estimate, and no empirical validity. 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: The fixed source-free model and SI constants, smooth connected constrained data, compatible initial circle-connection data, the zero-matter action branch, and the choice assumptions stated above.

[F1]

Interior stationary Einstein–Maxwell–matter action: the adopted action uses independent compact metric, fixed-sector connection and matter variations.

[F2]

Einstein–Maxwell equations from compact action stationarity: stationary configurations of that action satisfy the Einstein and sourced Maxwell equations, and conversely those equations plus the selected matter equations imply stationarity.

[F3]

Electrovacuum tensor Cauchy system: the source-free field-strength system is $G + Λ g = κ T^{E M}$ and $d F = d (⋆_{g} F) = 0$ , with the declared SI stress tensor and pullback conventions.

[F4]

Constrained electrovacuum initial data: the complete initial fields and their four constraints have the $K, e, b$ signs and normalization stated in the Statement.

[F5]

Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality: a development includes the complete smooth field data and a Cauchy embedding; a data-fixing development embedding is a time-orientation-preserving isometric embedding onto an open subset, and maximality means every other development embeds into it. The direction of this embedding is from the candidate development into the maximal one.

[F6]

Source-free smooth maximal globally hyperbolic Einstein–Maxwell development: under ${A C}_{ω}$ and DC, smooth connected constrained source-free field-strength data have a smooth globally hyperbolic maximal development, universal for other smooth globally hyperbolic field-strength developments, unique up to a unique data-fixing isometry carrying $F$ . Its draft item and Step-1 readiness are current; its verification.precheck is pending. Its proof is conditional on the exact CP02 local existence/uniqueness interface and the inspected published GR maximal theorem, whose verification precheck is also pending.

[F7]

Local potentials and smooth circle-bundle connection data and Circle gluing, curvature and connection variations: the supplied local transition and normalized one-form data glue to a smooth fixed circle bundle; its physical curvature is $κ_{0}$ times the normalized curvature. Adding one global normalized base one-form changes the connection in that same bundle without changing its transition functions.

[F8]

Closed two-forms agreeing on a Cauchy slice differ by an exact form: on a globally hyperbolic development, a closed two-form agreeing on the Cauchy-slice restriction with the curvature of a reference connection differs from it by the exterior derivative of one global one-form. The lemma uses the current draft temporal-splitting exactness argument and the published temporal-splitting carrier, whose verification precheck is pending. This batch's local proof uses the exact interface conditionally; its frontmatter precheck remains pending for engine recertification.

[F9]

Smooth Cauchy temporal splitting: a global temporal function with Cauchy levels and an orthogonal complete flow: the temporal flow is complete and a Cauchy slice can be used as the zero slice of a smooth product diffeomorphism; the latter alignment is proved in [F8]. Its published proof is present, but its verification precheck is pending.

[F10]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) and The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain: the exact set-theoretic assumptions are ${A C}_{ω}$ and DC; no full AC is used.

[A1]

Select the zero-matter action specialization. Then the matter variations vanish and the current and matter stress are zero.

[A2]

The initial transition functions and local potentials satisfy the normalized cocycle and curvature equations from Local potentials and smooth circle-bundle connection data, with $κ_{0} d a_{i} = β_{b}$ for the magnetic two-form $β_{b}$ .

Proof

Given: The data and interfaces in the Statement and Facts & Assumptions.

1.1givenA1A2F1F2F3F4algebra

The four constraints and complete initial field-strength data are exactly those of [F3]–[F4] with the fixed signature, SI constants, orientation, and $K$ sign. The normalized local potential data [A2] place the magnetic two-form in the supplied circle-bundle sector, while $e$ is independent normal electric data. The zero-matter action choice [A1] specializes [F1] and its field-equation interface [F2] to the source-free system [F3].

2.1step 1.1F3F4F6F10

The constrained data in step 1.1 meet the exact smooth connected data hypotheses of [F6], with precisely its Countable Choice and Dependent Choice assumptions [F10]. Applying [F6] gives a smooth globally hyperbolic maximal field-strength development $M_{\max}$ , universal for all smooth globally hyperbolic field-strength developments of these data and unique up to the stated data-fixing isometry. This constructs the field-strength development; its fixed-bundle connection realization is proved next.

3.1step 2.1A2F3F7F8F9

Let $i_{\max} : Σ ↪ M_{\max}$ be the Cauchy embedding and let $Φ_{t}$ be the complete flow from [F9]. As in the product-map construction in proof step 1.1 of [F8], $Ψ_{\max} (t, x) = Φ_{t} (i_{\max} (x))$ is a diffeomorphism $R \times Σ \to M_{\max}$ with $Ψ_{\max} (0, x) = i_{\max} (x)$ . Pull the initial local connection data [A2] to that product by ${pr}_{Σ}$ , construct its reference connection by [F7], and transport it to $M_{\max}$ by $Ψ_{\max}$ . Its curvature $\overset{ˉ}{F}$ restricts to $β_{b}$ on the initial slice. Since $F_{\max}$ is closed by [F3] and agrees with $\overset{ˉ}{F}$ on that slice, [F8] gives a global one-form $α$ with $F_{\max} - \overset{ˉ}{F} = d α$ . Adding $α / κ_{0}$ to the normalized local connection forms retains their transition maps and changes the physical curvature by $κ_{0} d (α / κ_{0}) = d α$ , giving $F_{\max}$ . Thus the maximal field-strength development is realized in the supplied initial bundle sector.

4.1step 2.1step 3.1A1F2F3F5F6F7F8

Let another smooth globally hyperbolic development have the same complete data and be stationary for the adopted action. By [F2] and the zero-matter choice [A1], it solves the source-free system [F3], so the universal embedding from [F6] carries its metric and field strength into $M_{\max}$ . On its Cauchy product, pull back the same initial transition data; as in step 3.1 they give a reference connection restricting to $β_{b}$ . The other solution's field is closed and has that same slice restriction, so [F8] supplies one global exact adjustment and [F7] realizes it in the transported initial bundle sector. By the development definition [F5], this proves maximality within the stated smooth globally hyperbolic stationary class.

5.1step 2.1F5F6∎

The uniqueness clause of [F6] gives exactly one data-fixing isometry between any two maximal metric/field-strength developments. Connection representatives need not be unique and are not additional fields in the compared development. The definition [F5] imposes no geodesic-completeness condition, so no such conclusion follows.

5 · Examples, experiments and counterexamples

None yet.

Local evolution of the adopted electrovacuum model

1 · Prerequisites

2 · Summary

Scope

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Local normal collar charts at a smooth spacelike hypersurface

Statement

Facts & Assumptions

Proof

Closed two-forms agreeing on a Cauchy slice differ by an exact form

Statement

Facts & Assumptions

Proof

Local evolution of the adopted electrovacuum model

Statement

Facts & Assumptions

Proof

Electromagnetic conservation in source-free curved spacetime

Statement

Facts & Assumptions

Proof

Maximal smooth Cauchy development of adopted electrovacuum

Statement

Facts & Assumptions

Proof

5 · Examples, experiments and counterexamples

Sources