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Gauge-compensated charged Killing energy — Examples

1 · Prerequisites

2 · Summary

This example places a stationary exact electromagnetic field on a spacetime with a periodic spatial direction. A nonzero period of its Killing contraction prevents a global scalar charged equilibrium ansatz, even though the field has a global potential and is stationary.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-generatedVerification: AI-generatedOpen item page →

Electric circulation obstructs a global charged equilibrium scalar

Example

Let $L > 0$ and let $O : = R_{x^{0}} \times S_{L}^{1} \times R_{x^{2}, x^{3}}^{2}$ , where $S_{L}^{1} : = R / L Z$ . Let $ϑ$ be the smooth global closed one-form on $S_{L}^{1}$ induced by $d \tilde{θ}$ on its universal cover, normalized by $\oint_{S_{L}^{1}} ϑ = L$ . Give $O$ the flat product metric $g = - (d x^{0})^{2} + ϑ \otimes ϑ + (d x^{2})^{2} + (d x^{3})^{2}$ and time orientation $ξ : = \partial_{x^{0}}$ . Then $ξ$ is a future timelike Killing field with $N = 1$ . For a nonzero constant $E_{0}$ in tesla, define the global potential and field $A : = E_{0} x^{0} ϑ, F : = d A = E_{0} d x^{0} \land ϑ .$ For a charged massive species with $q E_{0} \neq 0$ , take $T = T_{K} > 0$ constant and $β : = ξ / (k_{B} T_{K})$ . Although $F$ is exact and stationary, no smooth single-valued scalar $ζ : O \to R$ makes $f (x, P) = \exp (ζ (x) + g (β, P))$ stationary under the charged Liouville transport of Charged particle and Liouville data on a stationary Lorentzian domain.

Facts & Assumptions

Given: The product Lorentzian manifold, its smooth global one-form $ϑ$ , metric, Killing field, potential $A$ , field $F$ , and constants $L > 0$ , $E_{0} \neq 0$ , $q \neq 0$ , and $T_{K} > 0$ in the Example.

[F1]

Charged particle and Liouville data on a stationary Lorentzian domain: for any smooth scalar $ζ$ and smooth vector field $β$ , $f = \exp (ζ + g (β, P))$ is the charged exponential ansatz on the future massive shell.

[F2]

Global electrochemical redshift from charged Liouville stationarity: without assuming a global potential compensator, $f$ is stationary on the entire future massive shell if and only if $β$ is Killing and $d ζ = - c q ι_{β} F$ .

[F3]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): every sequence of nonempty sets has a choice function. This is inherited from the actual draft smooth tangent-bundle data; no stronger choice principle is used in this example.

[A1]

$O = R \times S_{L}^{1} \times R^{2}$ is connected, $ξ = \partial_{x^{0}}$ is future timelike Killing, and its lapse is $N = 1$ .

[A2]

$A = E_{0} x^{0} ϑ$ is global, $F = d A = E_{0} d x^{0} \land ϑ$ , $ι_{ξ} F = E_{0} ϑ$ , and $q E_{0} \neq 0$ .

Verification

Given: [A1], [A2], [F1], and the theorem's local stationarity criterion [F2].

1.1givenA1A2algebra

The product metric is smooth, has signature $(-, +, +, +)$ , and is invariant under translations of $x^{0}$ ; hence $ξ = \partial_{x^{0}}$ is a future timelike Killing field and $N = \sqrt{- g (ξ, ξ)} = 1$ . The form $ϑ$ is globally defined, closed, and independent of $x^{0}$ , so $A = E_{0} x^{0} ϑ$ is a global one-form and differentiation gives $F = d A = E_{0} d x^{0} \land ϑ$ . Since $L_{ξ} d x^{0} = d (ξ (x^{0})) = d 1 = 0$ and $L_{ξ} ϑ = 0$ , the product rule gives $L_{ξ} F = 0$ ; direct contraction gives $ι_{ξ} F = E_{0} ϑ$ .

1.2givenA2algebra

The product rule gives $L_{ξ} A = E_{0} ξ (x^{0}) ϑ + E_{0} x^{0} L_{ξ} ϑ = E_{0} ϑ$ , whose period around the circle is $E_{0} L \neq 0$ . Every exact one-form $d Λ$ for a global smooth real $Λ$ has zero integral around this closed circle, since its line integral is the endpoint difference of $Λ$ . Thus no global $Λ_{ξ}$ satisfies $L_{ξ} A = d Λ_{ξ}$ . Under a gauge change $A \mapsto A + d χ$ , the coordinate formula gives $(L_{ξ} d χ)_{a} = ξ^{b} \partial_{b} \partial_{a} χ + (\partial_{a} ξ^{b}) \partial_{b} χ = \partial_{a} (ξ^{b} \partial_{b} χ)$ ; this is exact and has zero circle period. Hence the nonzero period cannot be removed by a single-valued gauge.

2.1givenA2F1F2F3algebra∎

Here $β = ξ / (k_{B} T_{K})$ is Killing. If a global smooth $ζ$ made $f$ from [F1] stationary, [F2] would give $d ζ = - c q ι_{β} F = - \frac{c q E_{0}}{k_{B} T_{K}} ϑ .$ Integrating around the positive circle gives $0 = \oint d ζ = - c q E_{0} L / (k_{B} T_{K})$ , which contradicts $q E_{0} \neq 0$ and $T_{K} > 0$ . Hence no single-valued global $ζ$ exists.

Remarks

This example has a global potential and exact field strength, but the contraction $ι_{ξ} F$ is closed with nonzero period and is not exact. The obstruction is global topology, not a singular gauge. If $q = 0$ or $E_{0} = 0$ , the displayed period vanishes and this example gives no obstruction.