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Positive-parameter Harrison charge limits in the original end chart

Statement

Under the hypotheses of the two end lemmas, with the same orientation and $0 < q < 1$ , the original-coordinate end spheres satisfy $\lim_{R \to \infty} (4 π)^{- 1} \int_{S_{R}} F_{q} = 0$ and $\lim_{R \to \infty} (4 π)^{- 1} \int_{S_{R}} ⋆_{g_{q}} F_{q} = \sqrt{1 - q^{2}}$ . These are flux integrals of forms on the original spheres; no asymptotically orthonormal change of the transformed time coordinate is required.

Facts & Assumptions

Given: the hypotheses in the Statement; work in ZFC.

Goal: establish the Statement.

[F1]

The oriented seed flux and fixed-positive-parameter spatial error have been established. The mass-one Killing form has electric seed flux minus four pi Harrison spatial two-form differs from the seed by an integrable tail

[F2]

The electromagnetic definition proves the exact identity star of the real form equals the imaginary part of the complex form, using the orientation-preserving Harrison bundle map. Harrison electromagnetic form

[F3]

Integration is by pullback of differential forms, with the geometric charge normalization. Geometric electrovacuum equations and charge convention Integration on an oriented embedded submanifold

Proof

Given: the hypotheses in the Statement; work in ZFC.

Goal: establish the Statement.

[F1] The oriented seed flux and fixed-positive-parameter spatial error have been established. The mass-one Killing form has electric seed flux minus four pi Harrison spatial two-form differs from the seed by an integrable tail

[F2] The electromagnetic definition proves the exact identity star of the real form equals the imaginary part of the complex form, using the orientation-preserving Harrison bundle map. Harrison electromagnetic form

[F3] Integration is by pullback of differential forms, with the geometric charge normalization. Geometric electrovacuum equations and charge convention Integration on an oriented embedded submanifold

Proof technique: direct.

1.1F1F2F3

Restrict the error identity of the preceding lemma to $S_{R}$ and integrate. The two-form coefficient estimate gives $\int_{S_{R}} E_{q} = O_{q} (R^{- 1})$ by the sphere rescaling argument. By the exact Hodge identity, $\int_{S_{R}} (F_{q} + i ⋆_{g_{q}} F_{q}) = \int_{S_{R}} H_{q} = - a_{q} \int_{S_{R}} (F + i ⋆_{g} F) + O_{q} (R^{- 1})$ . This uses the actual metric $g_{q}$ on the left, not the original star.

2.1F1F3step 1.1∎

Taking real and imaginary parts and using the seed limits gives magnetic limit zero and electric limit $- a_{q} (- 4 π) = 4 π a_{q}$ . Dividing by the prescribed $4 π$ proves the two charge limits. Pullback integrals need no metric-dependent sphere area factor in this calculation, since that information is already in the Hodge two-form. This proves the Statement.

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