Manuscript XVI-J@1 · XVI-J.tex · section · 6c1f306d

The unintegrated cubic density

Finite source and density definitions › The unintegrated cubic density · lines 618–655

XVI-J@1#eq:Gamma, XVI-J@1#eq:Ricci, XVI-J@1#eq:ell

The following finite formulas fix the density without integration by parts:

Γ_{n}^{ρ}_{μν} R_{n, μν} r_{n} = \frac{1}{2} u_{n - 1}^{ρ a} (\partial_{μ} h_{ν a} + \partial_{ν} h_{μ a} - \partial_{a} h_{μν}), = \partial_{ρ} Γ_{n}^{ρ}_{μν} - \partial_{ν} Γ_{n}^{ρ}_{μ ρ} + a + b = n a, b \geq 1 \sum (Γ_{a}^{ρ}_{ρ λ} Γ_{b}^{λ}_{μν} - Γ_{a}^{ρ}_{ν λ} Γ_{b}^{λ}_{μ ρ}), = i = 0 \sum^{n - 1} u_{i}^{μν} R_{n - i, μν}, ℓ_{3} = - 2 (r_{3} + b_{1} r_{2} + b_{2} r_{1}) . n = 1, 2, 3,

The $u_{i}, b_{i}$ in these equations are evaluated on $h$ .

Every monomial is cubic in field jets and has total derivative order two. This is the supplied XII-C unintegrated representative XIIC; changing it by a total divergence would change the constant-slot sector unless a separate equivalence proof were supplied.

Displayed formulas in this passage

XVI-J@1#eq:Gamma · line 622
$Γ_{n}^{ρ}_{μν} R_{n, μν} r_{n} = \frac{1}{2} u_{n - 1}^{ρ a} (\partial_{μ} h_{ν a} + \partial_{ν} h_{μ a} - \partial_{a} h_{μν}), = \partial_{ρ} Γ_{n}^{ρ}_{μν} - \partial_{ν} Γ_{n}^{ρ}_{μ ρ} + a + b = n a, b \geq 1 \sum (Γ_{a}^{ρ}_{ρ λ} Γ_{b}^{λ}_{μν} - Γ_{a}^{ρ}_{ν λ} Γ_{b}^{λ}_{μ ρ}), = i = 0 \sum^{n - 1} u_{i}^{μν} R_{n - i, μν}, ℓ_{3} = - 2 (r_{3} + b_{1} r_{2} + b_{2} r_{1}) . n = 1, 2, 3,$
XVI-J@1#eq:Ricci · line 622
$Γ_{n}^{ρ}_{μν} R_{n, μν} r_{n} = \frac{1}{2} u_{n - 1}^{ρ a} (\partial_{μ} h_{ν a} + \partial_{ν} h_{μ a} - \partial_{a} h_{μν}), = \partial_{ρ} Γ_{n}^{ρ}_{μν} - \partial_{ν} Γ_{n}^{ρ}_{μ ρ} + a + b = n a, b \geq 1 \sum (Γ_{a}^{ρ}_{ρ λ} Γ_{b}^{λ}_{μν} - Γ_{a}^{ρ}_{ν λ} Γ_{b}^{λ}_{μ ρ}), = i = 0 \sum^{n - 1} u_{i}^{μν} R_{n - i, μν}, ℓ_{3} = - 2 (r_{3} + b_{1} r_{2} + b_{2} r_{1}) . n = 1, 2, 3,$
XVI-J@1#eq:ell · line 622
$Γ_{n}^{ρ}_{μν} R_{n, μν} r_{n} = \frac{1}{2} u_{n - 1}^{ρ a} (\partial_{μ} h_{ν a} + \partial_{ν} h_{μ a} - \partial_{a} h_{μν}), = \partial_{ρ} Γ_{n}^{ρ}_{μν} - \partial_{ν} Γ_{n}^{ρ}_{μ ρ} + a + b = n a, b \geq 1 \sum (Γ_{a}^{ρ}_{ρ λ} Γ_{b}^{λ}_{μν} - Γ_{a}^{ρ}_{ν λ} Γ_{b}^{λ}_{μ ρ}), = i = 0 \sum^{n - 1} u_{i}^{μν} R_{n - i, μν}, ℓ_{3} = - 2 (r_{3} + b_{1} r_{2} + b_{2} r_{1}) . n = 1, 2, 3,$

The unintegrated cubic density

Displayed formulas in this passage

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