Manuscript I-B-VP-0@1 · I-B-VP-0-v1.tex · section · 8a9d0682

Components, scaling, and charge conjugation

Properties and information carried by the current › Components, scaling, and charge conjugation · lines 803–841

I-B-VP-0@1#eq:components

The positive angular orthonormal coframe is $e^{φ^} = α r d φ$ . Thus

J_{φ} J_{φ^} J^{φ} = \frac{β ( 1 - β ) ( 2 β - 1 )}{12 π ^{2} α ^{2} r ^{2}}, = \frac{β ( 1 - β ) ( 2 β - 1 )}{12 π ^{2} α ^{3} r ^{3}}, = \frac{β ( 1 - β ) ( 2 β - 1 )}{12 π ^{2} α ^{4} r ^{4}} .

The $r^{- 2}$ statement concerns the covariant component in a dimensionless angular coordinate. The physical orthonormal current has the expected $r^{- 3}$ scaling.

Under

δ_{λ} (t, r, φ, z) = (λ t, λ r, φ, λ z),

the one-form satisfies

δ_{λ}^{*} J = λ^{- 2} J .

Directly,

\nabla_{μ} J^{μ} = \frac{1}{α r} \partial_{φ} (α r J^{φ}) = 0,

in agreement with Proposition prop:well-defined.

Displayed formulas in this passage

I-B-VP-0@1#eq:components · line 807
$J_{φ} J_{φ^} J^{φ} = \frac{β ( 1 - β ) ( 2 β - 1 )}{12 π ^{2} α ^{2} r ^{2}}, = \frac{β ( 1 - β ) ( 2 β - 1 )}{12 π ^{2} α ^{3} r ^{3}}, = \frac{β ( 1 - β ) ( 2 β - 1 )}{12 π ^{2} α ^{4} r ^{4}} .$
unlabeled · line 826
$δ_{λ} (t, r, φ, z) = (λ t, λ r, φ, λ z),$
unlabeled · line 831
$δ_{λ}^{*} J = λ^{- 2} J .$
unlabeled · line 835
$\nabla_{μ} J^{μ} = \frac{1}{α r} \partial_{φ} (α r J^{φ}) = 0,$

Components, scaling, and charge conjugation

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