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Electrovacuum tensor Cauchy system — Examples

1 · Prerequisites

2 · Summary

These examples check three separate interfaces of the tensor formulation. Zero field on Minkowski spacetime is an exact coupled solution; a nonzero Maxwell test field on a fixed flat metric generally fails the Einstein constraint; and a closed magnetic field on a torus can have a nonzero period, allowing local potentials but obstructing a single-valued global potential. The magnetic-period example is explicitly a Maxwell-sector example on a prescribed background, not a coupled flat solution.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck pendingjudge pass (gpt-6-luna)Open item page →

Flat zero-field data and Minkowski development

Example

Take $M = R^{1, 3}$ with coordinates $(x^{0}, x^{1}, x^{2}, x^{3})$ , Minkowski metric $g = - (d x^{0})^{2} + (d x^{1})^{2} + (d x^{2})^{2} + (d x^{3})^{2}$ , the standard orientation and future direction, and the hypersurface $Σ = {x^{0} = 0}$ . Set $F = 0$ and $Λ = 0$ , with the fixed positive parameters $κ, μ_{0}$ of Electrovacuum tensor Cauchy system. The induced data are $h = δ$ , $K = 0$ , and $e = b = 0$ . This is a smooth exact solution and its data satisfy all four electrovacuum constraints.

Facts & Assumptions

Given: The source-free tensor system and initial-data conventions of Electrovacuum tensor Cauchy system and Constrained electrovacuum initial data.

[F1]

Electrovacuum tensor Cauchy system: the equations are $G + Λ g = κ μ_{0}^{- 1} Q (g, F)$ , $d F = 0$ , and $d (⋆_{g} F) = 0$ .

[F2]

Constrained electrovacuum initial data: the induced data are tested by the two Maxwell divergences, the Hamiltonian constraint, and the momentum constraint with $K = + \frac{1}{2} L_{n} h$ .

Verification

Given: $g = η$ , $F = 0$ , $Λ = 0$ , and $κ, μ_{0} > 0$ .

1.1F1algebra

In the displayed global coordinates the metric coefficients are constant, so every Christoffel symbol and every curvature component is zero; hence $G_{a b} = 0$ .

2.1F1step 1.1algebra

Since $F = 0$ , one has $d F = 0$ , $d (⋆_{g} F) = 0$ , and $Q_{a b} = T_{a b}^{E M} = 0$ . Thus $G + Λ g = κ T^{E M}$ holds identically, proving that $(g, F)$ is an exact electrovacuum solution.

3.1F2step 1.1algebra∎

The slice has future unit normal $n = \partial_{0}$ , induced metric $h = δ$ , second fundamental form $K = 0$ , and observer fields $e = b = 0$ . Its scalar curvature and all four constraint expressions vanish, so the induced data satisfy Constrained electrovacuum initial data.

CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck pendingjudge pass (gpt-6-luna)Open item page →

Nonzero flat test field is not a coupled flat solution

Statement refuted

For $Λ = 0$ , every smooth source-free Maxwell field on a prescribed Minkowski metric is an electrovacuum solution with that same flat metric.

Facts & Assumptions

Given: The tensor system and constraint conventions of Electrovacuum tensor Cauchy system and Constrained electrovacuum initial data, with $κ > 0$ , $μ_{0} > 0$ , $Λ = 0$ , and $B_{0} \neq 0$ .

[F1]

Electrovacuum tensor Cauchy system: a coupled solution must satisfy both Maxwell equations and $G + Λ g = κ T^{E M}$ .

[F2]

Maxwell stress algebra and covariant conservation: for $e = 0$ and $b = (0, 0, B_{0})$ , $T^{E M} (n, n) = B_{0}^{2} / (2 μ_{0})$ .

[F3]

Gauss-Codazzi and Maxwell initial constraints: any smooth coupled solution induces data satisfying the Hamiltonian constraint.

Counterexample

Given: The system and constraint conventions above; $κ, μ_{0} > 0$ , $Λ = 0$ , and a fixed $B_{0} \neq 0$ (tesla in the SI interpretation).

1.1F1algebra

Take $M = R^{4}$ with length coordinates $(x^{0}, x^{1}, x^{2}, x^{3})$ , metric $g = - (d x^{0})^{2} + (d x^{1})^{2} + (d x^{2})^{2} + (d x^{3})^{2}$ , orientation $d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ , and future field $N = \partial_{x^{0}}$ ; in the SI interpretation $x^{0} = c t$ . Set $F = B_{0} d x^{1} \land d x^{2}$ . The metric coefficients and those of $F$ and $⋆_{g} F$ are constant, so the curvature vanishes and $d F = d (⋆_{g} F) = 0$ . Hence this is a smooth source-free Maxwell field on the prescribed Minkowski background.

2.1F1F2F3algebra∎

On the slice $x^{0} = 0$ , the data are $h = δ$ , $K = 0$ , $e = 0$ , and $b = (0, 0, B_{0})$ . The Hamiltonian constraint from [F3] would require $0 = (κ / μ_{0}) B_{0}^{2}$ , because $R (h) = 0$ and $Λ = 0$ . This is impossible since $κ, μ_{0} > 0$ and $B_{0} \neq 0$ . Hence the given flat Maxwell test field is not a coupled electrovacuum solution with that metric.

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Magnetic period obstructs a global potential

Example

Let $M = R_{x^{0}, x^{3}}^{2} \times T_{L}^{2}$ , where $T_{L}^{2} = (R / L Z)^{2}$ carries its standard global translation-invariant one-forms $d x^{1}, d x^{2}$ , and give $M$ the flat product Lorentz metric

$g = - (d x^{0})^{2} + (d x^{1})^{2} + (d x^{2})^{2} + (d x^{3})^{2} .$

Take future direction $\partial_{x^{0}}$ and orient $M$ by $d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ . In the SI interpretation use the length-coordinate convention $x^{0} = c t$ of Electrovacuum tensor Cauchy system, with $c > 0$ and $x^{0}, x^{3}$ and local lifts of $x^{1}, x^{2}$ measured in metres; $L > 0$ is measured in metres. For a fixed $B_{0} \neq 0$ tesla, define the smooth two-form $F = B_{0} d x^{1} \land d x^{2}$ . Then $F$ and its Hodge dual have constant coefficients, so $d F = d (⋆_{g} F) = 0$ on the flat background. On the oriented spatial torus $S : = {x^{0} = x_{*}^{0}, x^{3} = x_{*}^{3}} ≅ T_{L}^{2}$ , oriented by $d x^{1} \land d x^{2}$ , its period is $\int_{S} F = B_{0} L^{2} \neq 0$ . The field has local potentials but no single-valued global one-form potential on $M$ . This example concerns the Maxwell field-strength sector on a prescribed background; it is not asserted to solve the coupled Einstein equation with that flat metric.

Facts & Assumptions

Given: The exterior-calculus and source-free Maxwell conventions of Electrovacuum tensor Cauchy system and the local potential conclusion of Local potential homotopy and Lorenz gauge reachability.

[F1]

Electrovacuum tensor Cauchy system: the source-free Maxwell equations are $d F = 0$ and $d (⋆_{g} F) = 0$ .

[F2]

Local potential homotopy and Lorenz gauge reachability: every smooth closed two-form has a potential on a star-shaped coordinate neighborhood.

Verification

Given: $L > 0$ , $B_{0} \neq 0$ , flat product coordinates, and $F = B_{0} d x^{1} \land d x^{2}$ .

1.1F1algebra

The global invariant one-forms $d x^{1}, d x^{2}$ on $T_{L}^{2}$ are closed and $B_{0}$ is constant, so $d F = 0$ . The flat Hodge dual has constant coefficients and is closed as well. On $S$ at fixed $x^{0}, x^{3}$ , direct integration over a fundamental square gives $\int_{S} F = \int_{[0, L]^{2}} B_{0} d x^{1} d x^{2} = B_{0} L^{2} \neq 0$ .

2.1step 1.1F2algebra∎

On each star-shaped coordinate neighborhood, [F2] supplies a local potential. If a global smooth one-form $A$ on $M$ satisfied $F = d A$ , its components $A_{1}, A_{2}$ are periodic in the torus variables. At fixed $x_{*}^{0}, x_{*}^{3}$ , direct integration over a fundamental square and periodicity give $\int_{S} F = \int_{[0, L]^{2}} (\partial_{1} A_{2} - \partial_{2} A_{1}) d x^{1} d x^{2} = \int_{0}^{L} (A_{2} (L, x^{2}) - A_{2} (0, x^{2})) d x^{2} - \int_{0}^{L} (A_{1} (x^{1}, L) - A_{1} (x^{1}, 0)) d x^{1} = 0$ . This contradicts the nonzero period in step 1.1, so no global potential exists although local potentials do.