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Positive symmetric Maxwell field-strength reduction — Examples

1 · Prerequisites

2 · Summary

This companion checks a nontrivial constrained torus datum and records the exact scope of maximality. Nonzero parallel Maxwell fields satisfy torus constraints with isotropic initial curvature takes flat $T^{3}$ , $Λ = 0$ , nonzero parallel constant electric and magnetic data, and $K = H h$ with $6 H^{2} = κ μ_{0}^{- 1} (∣ e ∣^{2} + ∣ b ∣^{2})$ . It verifies every Einstein and Maxwell constraint and then invokes the conditional local development theorem. For each nonzero component $b_{i}$ , the oriented coordinate two-torus orthogonal to that axis has magnetic period $(2 π L_{0})^{2} b_{i}$ , where $L_{0} > 0$ is the fixed length scale of the example; hence a nonzero component rules out a global potential on a neighborhood of the full initial torus. The example makes no later-time isotropy claim.

Maximality of the Einstein–Maxwell GHD does not assert geodesic completeness states the logical boundary of the smooth maximal globally hyperbolic development result: its universal embedding property, using the affine geodesic equation and maximal-interval convention, does not assert geodesic completeness. No concrete incomplete Einstein–Maxwell maximal development is claimed here. Neither item reports an observation or an experiment.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck pendingOpen item page →

Nonzero parallel Maxwell fields satisfy torus constraints with isotropic initial curvature

Example

Fix $L_{0} > 0$ and dimensionless periodic coordinates $θ^{i} \in R / (2 π Z)$ , with length coordinates $X^{i} = L_{0} θ^{i}$ as in Periodic high-Sobolev state spaces. Let $h = \sum_{i} d X^{i} \otimes d X^{i} = L_{0}^{2} \sum_{i} d θ^{i} \otimes d θ^{i}$ on the flat periodic torus $T_{2 π}^{3}$ (thus $h_{i j} = δ_{i j}$ in the length-coordinate frame), take $Λ = 0$ , and choose two nonzero vectors with constant components $e, b \in R^{3}$ in its oriented orthonormal frame $\partial_{X^{i}}$ parallel to one another. In the SI convention $e = E / c$ and $b = B$ , set $H : = \sqrt{\frac{κ}{6 μ_{0}} (∣ e ∣^{2} + ∣ b ∣^{2})}, K_{i j} : = H h_{i j},$ where $x^{0} = c t$ is a length coordinate. Then $(h, K, e, b)$ satisfies the source-free Einstein–Maxwell Hamiltonian, momentum and Gauss constraints with $K (X, Y) = g (\nabla_{X} n, Y)$ . Consequently, conditional on the local theorem's stated CP01 suppliers, these data have a nontrivial smooth source-free local Einstein–Maxwell development.

Facts & Assumptions

Given: The flat torus, $Λ = 0$ , constant parallel $e, b$ , and the fixed positive constants $κ, μ_{0}$ .

[L1]

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

[F1]

Local high-Sobolev and smooth electrovacuum gives a smooth full local solution from smooth constrained data, conditional on the model, stress and hyperbolic suppliers identified in that item's Facts & Assumptions. The inspected current GR Sobolev suppliers used there are published with precheck pending.

Verification

1.1givenL1algebra

The flat torus has $R (h) = 0$ and zero Christoffel symbols. Constant $e, b$ therefore have $D_{i} e^{i} = D_{i} b^{i} = 0$ ; their collinearity gives $e \times_{h} b = 0$ . Since $K = H h$ with constant $H$ , $D K = 0$ , ${tr}_{h} K = 3 H$ , and $∣ K ∣_{h}^{2} = 3 H^{2}$ . Hence the Hamiltonian left side is $0 + 9 H^{2} - 3 H^{2} = 6 H^{2}$ , which by the definition of $H$ equals $κ μ_{0}^{- 1} (∣ e ∣^{2} + ∣ b ∣^{2})$ , exactly the right side of [L1] for $Λ = 0$ . The momentum left side and right side both vanish. Thus all four constraints hold. The scalar under the square root is positive since both fields are nonzero, and $H$ is finite and real.

2.1step 1.1F1algebra

The data are smooth and $h$ is uniformly positive on the compact torus. For any integer $s \geq 5$ they belong to the Sobolev classes in [F1]; their verified constraints meet its hypotheses. Applying [F1] yields a smooth local full source-free solution. The reconstructed initial two-form has nonzero electric and magnetic components because $e, b \neq 0$ , so the solution is nontrivial. The argument is a model deduction, not an empirical claim.

3.1givenstep 2.1algebra∎

(A nonzero magnetic period.) The initial spatial two-form is $ω_{b} = \frac{1}{2} ϵ_{i j k} b_{k} d X^{i} \land d X^{j} = \frac{1}{2} L_{0}^{2} ϵ_{i j k} b_{k} d θ^{i} \land d θ^{j}$ . Write the development from step 2.1 as $M = (- T, T) \times T_{2 π}^{3}$ with initial embedding $ι_{Σ} (θ) = (0, θ)$ . For each $i$ , choose the cyclic pair $(j, k)$ with $ϵ_{i j k} = 1$ and orient the coordinate two-torus $T_{j k}^{2} = {θ^{i} = 0}$ by $d θ^{j} \land d θ^{k}$ . Its magnetic period, the integral of the restricted two-form, is $\int_{T_{j k}^{2}} ω_{b} = (2 π L_{0})^{2} b_{i}$ . Since $b \neq 0$ and $L_{0} > 0$ , some such period is nonzero. If $F = d A$ for a globally defined smooth one-form on a neighborhood of the full initial torus, then $ω_{b} = d (ι_{Σ}^{*} A)$ . Write $ι_{Σ}^{*} A = \sum_{ℓ} A_{ℓ} (θ) d θ^{ℓ}$ with smooth periodic coefficients. Integrating $\partial_{j} A_{k} - \partial_{k} A_{j}$ over the coordinate two-torus gives zero by periodicity in each variable, contradicting its nonzero period. Thus no such global potential exists.

Remarks

If instead $K = 0$ while $h = δ$ , $Λ = 0$ , and either field is nonzero, the Hamiltonian constraint fails: its left side is zero and its right side is strictly positive. This obstruction distinguishes coupled initial data from a test Maxwell field placed on a fixed Minkowski background. No later isotropy or long-time behavior is claimed for the development above.

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

Maximality of the Einstein–Maxwell GHD does not assert geodesic completeness

Statement

For the maximal smooth globally hyperbolic development in Source-free smooth maximal globally hyperbolic Einstein–Maxwell development, maximality means that every development of the same complete source-free data admits a data-fixing development embedding into it, in the sense of Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality. By The geodesic initial-value problem and the exponential map of the metric-selected connection, an affinely parametrized geodesic for its metric-selected connection is a smooth curve $γ : I \to M$ on an open interval $I \subseteq R$ satisfying $D_{λ} \dot{γ} = 0$ ; in coordinates $y^{μ}$ this is ${\ddot{y}}^{μ} + Γ^{μ}_{α β} (y) {\dot{y}}^{α} {\dot{y}}^{β} = 0$ . That theorem supplies a unique maximal solution $γ_{p, v} : I_{p, v} \to M$ through each initial datum $(p, v)$ , with $I_{p, v}$ an open interval; here inextendible means that the curve is such a maximal solution, so no geodesic on a strictly larger interval with the same affine parameter restricts to it. Geodesic completeness means that every such maximal geodesic has parameter domain all of $R$ . The maximality conclusion by itself asserts no geodesic-completeness property. This remark does not claim that a particular Einstein–Maxwell maximal development is geodesically incomplete.

Facts & Assumptions

Given: The maximal development and the definitions of development embedding and maximality.

[F1]

Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality defines maximality solely by the existence of a data-fixing development embedding from every development of the same data. Its definition imposes no condition on affine parameters of geodesics.

[F2]

Source-free smooth maximal globally hyperbolic Einstein–Maxwell development supplies the maximal smooth globally hyperbolic Einstein–Maxwell development within that definition. The original source Existence and uniqueness of the smooth maximal globally hyperbolic development of constrained initial data in the selected matter sectors is Sbierski's smooth global gluing theorem after its local model obligations are discharged; Sbierski's statement has no geodesic-completeness conclusion.

[F3]

The geodesic initial-value problem and the exponential map of the metric-selected connection defines geodesics by $D_{λ} \dot{γ} = 0$ and supplies the unique maximal solution on an open interval through each initial position-velocity datum. Its proof gives the coordinate equation displayed in the Statement and fixes maximality as absence of a same-parameter geodesic extension to a strictly larger interval.

Proof

1.1givenF1

By [F1], the maximality predicate quantifies over extensions that are themselves developments of the same initial data and asks for data-fixing embeddings. Its clauses do not quantify over geodesics, affine parameters, or endpoint behavior.

2.1step 1.1F1F2F3algebra∎

Geodesic completeness, as defined in the Statement, is a separate assertion about the parameter domain of every inextendible affinely parametrized geodesic. Neither [F1]'s maximality definition nor the conclusion [F2] supplies such an assertion. Therefore maximality alone yields no completeness conclusion; because no explicit incomplete Einstein–Maxwell development is constructed here, none is claimed.

Positive symmetric Maxwell field-strength reduction — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

Nonzero parallel Maxwell fields satisfy torus constraints with isotropic initial curvature

Example

Facts & Assumptions

Verification

Remarks

Maximality of the Einstein–Maxwell GHD does not assert geodesic completeness

Statement

Facts & Assumptions

Proof

Sources