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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.

Maximal smooth Cauchy development of adopted electrovacuum

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources