Manuscript I-B-VP-1@1 · I-B-VP-1-v1.tex · proposition · 6d700be9

Proposition: A computed contraction defect

Dilation residues and compatibility of presentations › The angular current is a smooth contraction, not singular lowering · lines 1173–1210 · proof included or linked

I-B-VP-1@1#prop:contraction, I-B-VP-1@1#eq:contractiondefect, I-B-VP-1@1#eq:generalangular

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For the derivative-compatible current,

α^{2} R_{i} (J_{L})^{i} = j U_{L} - \frac{j σ _{α}}{2} δ_{W} .

For the general current (eq:allJ),

J_{φ}^{ext} = j U_{L} - (\frac{j σ _{α}}{2} + 2 α^{2} w) δ_{W} .

In contrast, the direct componentwise current obeys

α^{2} R_{i} (J_{L}^{c})^{i} = j U_{L} .

Thus a contact can preserve this latter contraction identity, but not simultaneously the unmodified derivative identity (eq:Jhat) when $j \neq = 0$ .

Proof. Since $R_{i} ϵ^{i}_{k} = x_{k}$ ,

α^{2} R_{i} (J_{L})^{i} = - \frac{j}{2} E U_{L} .

Equation (eq:EulerU) gives (eq:contractiondefect). Also $x^{i} \partial_{i} δ_{W} = - 2 δ_{W}$ , proving (eq:generalangular). Equation (eq:currentrepair) then gives the direct finite-part contraction. Choosing $w = - j σ_{α} / (4 α^{2})$ removes the contraction defect, but changes (eq:Jhat) by that nonzero contact.

Displayed formulas in this passage

I-B-VP-1@1#eq:contractiondefect · line 1176
$α^{2} R_{i} (J_{L})^{i} = j U_{L} - \frac{j σ _{α}}{2} δ_{W} .$
I-B-VP-1@1#eq:generalangular · line 1182
$J_{φ}^{ext} = j U_{L} - (\frac{j σ _{α}}{2} + 2 α^{2} w) δ_{W} .$
unlabeled · line 1188
$α^{2} R_{i} (J_{L}^{c})^{i} = j U_{L} .$
unlabeled · line 1198
$α^{2} R_{i} (J_{L})^{i} = - \frac{j}{2} E U_{L} .$

Proposition: A computed contraction defect

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