Manuscript XVI-H@1 · XVI-H.tex · section · af895076

Displayed switch-correction coefficients

Degree audit and real polarization › Displayed switch-correction coefficients · lines 556–598

XVI-H@1#eq:source-coefficients, XVI-H@1#eq:CM

To avoid confusing expansion coefficients with response fields or completion coefficients, write the quoted source coefficients temporarily as $u_{j}$ and $b_{j}$ :

u_{0}^{μν} b_{0} = η^{μν}, = 1, u_{1}^{μν} b_{1} = - γ^{μν}, = \frac{1}{2} tr_{η} γ, u_{2}^{μν} b_{2} = γ^{μ}_{ℓ} γ^{ℓ ν}, = \frac{1}{8} (tr_{η} γ)^{2} - \frac{1}{4} γ_{μν} γ^{μν} .

The remaining quoted coefficients are

C_{r}^{μν α β} Γ_{n}^{ρ}_{α β} M_{2}^{μν ρ} = i + j + k = r \sum b_{i} (u_{j}^{μν} u_{k}^{α β} - u_{j}^{μ (α} u_{k}^{β) ν}), r = 0, 1, 2, = \frac{1}{2} u_{n - 1}^{ρ ℓ} (\partial_{α} γ_{β ℓ} + \partial_{β} γ_{α ℓ} - \partial_{ℓ} γ_{α β}), n = 1, 2, = C_{0}^{μν α β} Γ_{2}^{ρ}_{α β} + C_{1}^{μν α β} Γ_{1}^{ρ}_{α β} .

These are the formulas reproduced in XVI-H-v3. Their homogeneity, rather than a new source derivation, is audited here.

Displayed formulas in this passage

XVI-H@1#eq:source-coefficients · line 561
$u_{0}^{μν} b_{0} = η^{μν}, = 1, u_{1}^{μν} b_{1} = - γ^{μν}, = \frac{1}{2} tr_{η} γ, u_{2}^{μν} b_{2} = γ^{μ}_{ℓ} γ^{ℓ ν}, = \frac{1}{8} (tr_{η} γ)^{2} - \frac{1}{4} γ_{μν} γ^{μν} .$
XVI-H@1#eq:CM · line 574
$C_{r}^{μν α β} Γ_{n}^{ρ}_{α β} M_{2}^{μν ρ} = i + j + k = r \sum b_{i} (u_{j}^{μν} u_{k}^{α β} - u_{j}^{μ (α} u_{k}^{β) ν}), r = 0, 1, 2, = \frac{1}{2} u_{n - 1}^{ρ ℓ} (\partial_{α} γ_{β ℓ} + \partial_{β} γ_{α ℓ} - \partial_{ℓ} γ_{α β}), n = 1, 2, = C_{0}^{μν α β} Γ_{2}^{ρ}_{α β} + C_{1}^{μν α β} Γ_{1}^{ρ}_{α β} .$

Displayed switch-correction coefficients

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