Manuscript Em01-G@5 · Em01-G-v5.tex · section · 3fbfe951

Frozen component arrays, source, and crossed kernels

Frozen component arrays, source, and crossed kernels · lines 2059–2174

Em01-G@5#eq:density-array, Em01-G@5#eq:source

For $I = (i, j)$ , $0 \leq i \leq j \leq 3$ , define

(e_{I})_{μν} = \frac{δ _{μ i} δ _{ν j} + δ _{μ j} δ _{ν i}}{1 + δ _{ij}}, (e^{I})^{μν} = \frac{1}{2} (δ_{i}^{μ} δ_{j}^{ν} + δ_{j}^{μ} δ_{i}^{ν}) .

Then $e^{I} (e_{J}) = δ^{I}_{J}$ . The dual basis is algebraic, not the raised basis.

Set

P H = H - \frac{1}{2} η tr_{η} H, P^{2} = 1,

A^{μν, I} = \frac{1}{2} η^{μν} tr_{η} e_{I} - (e_{I})^{μν}, b_{I} = \frac{1}{2} tr_{η} e_{I}, P^{L}_{M} = (e^{L})^{μν} (P e_{M})_{μν} .

A lower component label means evaluation on its basis tensor.

The density mixed derivative is

D_{dens}^{μν} (G, H) = [\frac{1}{4} (tr_{η} G) (tr_{η} H) - \frac{1}{2} G_{ρ σ} H^{ρ σ}] η^{μν} - \frac{1}{2} [(tr_{η} G) H^{μν} + (tr_{η} H) G^{μν}] + G^{μ ρ} H_{ρ}^{ν} + H^{μ ρ} G_{ρ}^{ν}, D^{μν, I, J} = D_{dens}^{μν} (e_{J}, e_{I}) .

It is the mixed derivative, not a half-polarization. The stipulated free tensor Hessian and inverse are $- □_{η} P$ and $- P E_{0}^{R}$ .

For symmetric fields $H, G$ , define

S (H)_{σ μν} = \partial_{μ} H_{σ ν} + \partial_{ν} H_{σ μ} - \partial_{σ} H_{μν}, C (H)_{μν}^{ρ} = \frac{1}{2} η^{ρ σ} S (H)_{σ μν},

D (H, G)_{μν}^{ρ} = - \frac{1}{2} [H^{ρ σ} S (G)_{σ μν} + G^{ρ σ} S (H)_{σ μν}],

R_{1} (H)_{μν} = \partial_{ρ} C (H)_{μν}^{ρ} - \partial_{ν} C (H)_{μ ρ}^{ρ}, r_{1} (H) = η^{μν} R_{1} (H)_{μν}, G_{1} (H) = R_{1} (H) - \frac{1}{2} η r_{1} (H),

U (H, G)_{μν} = \partial_{ρ} D (H, G)_{μν}^{ρ} - \partial_{ν} D (H, G)_{μ ρ}^{ρ} + C (H)_{ρ α}^{ρ} C (G)_{μν}^{α} + C (G)_{ρ α}^{ρ} C (H)_{μν}^{α} - C (H)_{ν α}^{ρ} C (G)_{μ ρ}^{α} - C (G)_{ν α}^{ρ} C (H)_{μ ρ}^{α},

s (H, G) = η^{μν} U (H, G)_{μν} - H^{μν} R_{1} (G)_{μν} - G^{μν} R_{1} (H)_{μν},

E (H, G)_{μν} = U (H, G)_{μν} - \frac{1}{2} η_{μν} s (H, G) - \frac{1}{2} [H_{μν} r_{1} (G) + G_{μν} r_{1} (H)] .

The frozen source is

Q (H, G)_{μν} = 2 [E (H, G)_{μν} + \frac{1}{2} (tr_{η} H) G_{1} (G)_{μν} + \frac{1}{2} (tr_{η} G) G_{1} (H)_{μν} - H_{μ}^{α} G_{1} (G)_{α ν} - H_{ν}^{α} G_{1} (G)_{μα} - G_{μ}^{α} G_{1} (H)_{α ν} - G_{ν}^{α} G_{1} (H)_{μα}] .

Its component coefficients are

q_{α β}^{L, I, J} = [p^{α} q^{β}] (e^{L})^{μν} Q (e_{I}, e_{J}; p, q)_{μν} .

The symbol is homogeneous of total degree two:

Q (H, G)^{L} = α, β, I, J \sum q_{α β}^{L, I, J} (\partial^{α} G_{J}) (\partial^{β} H_{I}) .

Multiindex derivatives are coordinate derivatives.

Displayed formulas in this passage

Em01-G@5#eq:density-array · line 2089
$D_{dens}^{μν} (G, H) = [\frac{1}{4} (tr_{η} G) (tr_{η} H) - \frac{1}{2} G_{ρ σ} H^{ρ σ}] η^{μν} - \frac{1}{2} [(tr_{η} G) H^{μν} + (tr_{η} H) G^{μν}] + G^{μ ρ} H_{ρ}^{ν} + H^{μ ρ} G_{ρ}^{ν}, D^{μν, I, J} = D_{dens}^{μν} (e_{J}, e_{I}) .$
unlabeled · line 2128
$U (H, G)_{μν} = \partial_{ρ} D (H, G)_{μν}^{ρ} - \partial_{ν} D (H, G)_{μ ρ}^{ρ} + C (H)_{ρ α}^{ρ} C (G)_{μν}^{α} + C (G)_{ρ α}^{ρ} C (H)_{μν}^{α} - C (H)_{ν α}^{ρ} C (G)_{μ ρ}^{α} - C (G)_{ν α}^{ρ} C (H)_{μ ρ}^{α},$
Em01-G@5#eq:source · line 2148
$Q (H, G)_{μν} = 2 [E (H, G)_{μν} + \frac{1}{2} (tr_{η} H) G_{1} (G)_{μν} + \frac{1}{2} (tr_{η} G) G_{1} (H)_{μν} - H_{μ}^{α} G_{1} (G)_{α ν} - H_{ν}^{α} G_{1} (G)_{μα} - G_{μ}^{α} G_{1} (H)_{α ν} - G_{ν}^{α} G_{1} (H)_{μα}] .$
unlabeled · line 2063
$(e_{I})_{μν} = \frac{δ _{μ i} δ _{ν j} + δ _{μ j} δ _{ν i}}{1 + δ _{ij}}, (e^{I})^{μν} = \frac{1}{2} (δ_{i}^{μ} δ_{j}^{ν} + δ_{j}^{μ} δ_{i}^{ν}) .$
unlabeled · line 2075
$P H = H - \frac{1}{2} η tr_{η} H, P^{2} = 1,$
unlabeled · line 2078
$A^{μν, I} = \frac{1}{2} η^{μν} tr_{η} e_{I} - (e_{I})^{μν}, b_{I} = \frac{1}{2} tr_{η} e_{I}, P^{L}_{M} = (e^{L})^{μν} (P e_{M})_{μν} .$
unlabeled · line 2106
$S (H)_{σ μν} = \partial_{μ} H_{σ ν} + \partial_{ν} H_{σ μ} - \partial_{σ} H_{μν}, C (H)_{μν}^{ρ} = \frac{1}{2} η^{ρ σ} S (H)_{σ μν},$
unlabeled · line 2114
$D (H, G)_{μν}^{ρ} = - \frac{1}{2} [H^{ρ σ} S (G)_{σ μν} + G^{ρ σ} S (H)_{σ μν}],$
unlabeled · line 2119
$R_{1} (H)_{μν} = \partial_{ρ} C (H)_{μν}^{ρ} - \partial_{ν} C (H)_{μ ρ}^{ρ}, r_{1} (H) = η^{μν} R_{1} (H)_{μν}, G_{1} (H) = R_{1} (H) - \frac{1}{2} η r_{1} (H),$
unlabeled · line 2137
$s (H, G) = η^{μν} U (H, G)_{μν} - H^{μν} R_{1} (G)_{μν} - G^{μν} R_{1} (H)_{μν},$
unlabeled · line 2142
$E (H, G)_{μν} = U (H, G)_{μν} - \frac{1}{2} η_{μν} s (H, G) - \frac{1}{2} [H_{μν} r_{1} (G) + G_{μν} r_{1} (H)] .$
unlabeled · line 2162
$q_{α β}^{L, I, J} = [p^{α} q^{β}] (e^{L})^{μν} Q (e_{I}, e_{J}; p, q)_{μν} .$
unlabeled · line 2168
$Q (H, G)^{L} = α, β, I, J \sum q_{α β}^{L, I, J} (\partial^{α} G_{J}) (\partial^{β} H_{I}) .$

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section: Frozen component arrays, source, and crossed kernels