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A global potential compensator makes the contracted field exact

Statement

Let $O$ be a smooth time-oriented Lorentzian domain, let $ξ$ be a smooth future-timelike Killing field on $O$ , and let $A$ be a global smooth one-form with $F = d A$ . Suppose a global smooth scalar $Λ_{ξ}$ satisfies $L_{ξ} A = d Λ_{ξ}$ . Define $Ψ_{ξ} : = ξ^{a} A_{a} - Λ_{ξ}$ and $ω : = ι_{ξ} F$ , with $(ι_{ξ} F)_{a} = ξ^{b} F_{b a}$ . Then

$ω = - d Ψ_{ξ},$

so $ω$ is exact. This identity uses the global potential symmetry and assumes no charged trajectory or conservation law.

Statement references

Charged particle and Liouville data on a stationary Lorentzian domain

Closed and exact differential forms

Facts & Assumptions

Given: A smooth time-oriented Lorentzian domain $O$ , a smooth future-timelike Killing field $ξ$ , a global smooth one-form $A$ with $F = d A$ , and a global smooth scalar $Λ_{ξ}$ satisfying $L_{ξ} A = d Λ_{ξ}$ .

[F1]

Charged particle and Liouville data on a stationary Lorentzian domain: $F = d A$ , $(ι_{ξ} F)_{a} = ξ^{b} F_{b a}$ , $(L_{ξ} A)_{a} = ξ^{b} \partial_{b} A_{a} + A_{b} \partial_{a} ξ^{b}$ , and $Ψ_{ξ} = ξ^{a} A_{a} - Λ_{ξ}$ for the supplied global potential data.

[F2]

Closed and exact differential forms: a one-form is exact when it is the exterior derivative of a globally defined smooth scalar.

[A1]

The global compensator equation $L_{ξ} A = d Λ_{ξ}$ holds on all of $O$ .

Proof

1.1givenA1F1algebra

In any smooth chart, the definitions give $(ι_{ξ} F)_{a} = ξ^{b} (\partial_{b} A_{a} - \partial_{a} A_{b})$ and $\partial_{a} Ψ_{ξ} = A_{b} \partial_{a} ξ^{b} + ξ^{b} \partial_{a} A_{b} - \partial_{a} Λ_{ξ}$ . By [A1] and [F1], $\partial_{a} Λ_{ξ} = ξ^{b} \partial_{b} A_{a} + A_{b} \partial_{a} ξ^{b}$ . Substitution yields $(ι_{ξ} F)_{a} = - \partial_{a} Ψ_{ξ}$ . Since this calculation holds in every chart, it proves the stated one-form identity.

2.1givenF2step 1.1algebra∎

The scalar $Ψ_{ξ} = ξ^{a} A_{a} - Λ_{ξ}$ is global and smooth because it is the contraction of globally defined smooth fields minus a global smooth scalar. Thus step 1.1 gives $ω = - d Ψ_{ξ} = d (- Ψ_{ξ})$ globally, which is exact by [F2].

A global potential compensator makes the contracted field exact

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources