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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.

Local evolution of the adopted electrovacuum model

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources