Manuscript XI-A@1 · XI-A.tex · section · b8211ecc

Finite formulas for the gravitational Euler coefficients

Finite formulas for the gravitational Euler coefficients · lines 1319–1380

XI-A@1#eq:Gamma-n, XI-A@1#eq:R-n, XI-A@1#eq:r-n, XI-A@1#eq:C-n, XI-A@1#eq:H-n

This appendix makes (eq:Hdef) an explicit coefficient prescription. All contractions use $η$ unless a coefficient $u_{i}$ is written.

Set

u_{0}^{μν} = η^{μν}, u_{1}^{μν} = - γ^{μν}, u_{2}^{μν} = γ^{μ}_{ρ} γ^{ρ ν} .

For $n = 1, 2, 3$ define

Γ_{n}^{ρ}_{μν} = \frac{1}{2} u_{n - 1}^{ρ σ} (\partial_{μ} γ_{ν σ} + \partial_{ν} γ_{μ σ} - \partial_{σ} γ_{μν}) .

These are precisely the first three coefficients of the Christoffel symbols of $q = η + κγ$ .

Define, for $1 \leq n \leq 3$ ,

R_{n, μν} = r_{n} C_{n, 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 + j = n i \geq 0, j \geq 1 \sum u_{i}^{μν} R_{j, μν}, r_{0} = 0, = R_{n, μν} - \frac{1}{2} η_{μν} r_{n} - \frac{1}{2} γ_{μν} r_{n - 1} .

Only $u_{0}, u_{1}, u_{2}$ occur in these sums. Let $b_{0} = 1$ and use $b_{1}, b_{2}$ from (eq:b). Then

H_{n}^{μν} = i + j + ℓ + s = n i, j, ℓ \geq 0, s \geq 1 \sum b_{i} u_{j}^{μα} u_{ℓ}^{ν β} C_{s, α β}, 1 \leq n \leq 3.

Again, no index above two is required for $b_{i}, u_{j}, u_{ℓ}$ . Equations (eq:Gamma-n)–(eq:H-n) are finite local differential formulas, not placeholders for unknown vertices.

Displayed formulas in this passage

XI-A@1#eq:Gamma-n · line 1333
$Γ_{n}^{ρ}_{μν} = \frac{1}{2} u_{n - 1}^{ρ σ} (\partial_{μ} γ_{ν σ} + \partial_{ν} γ_{μ σ} - \partial_{σ} γ_{μν}) .$
XI-A@1#eq:R-n · line 1345
$R_{n, μν} = r_{n} C_{n, μν} \partial_{ρ} Γ_{n}^{ρ}_{μν} - \partial_{ν} Γ_{n}^{ρ}_{μ ρ} + a + b = n a, b \geq 1 \sum (Γ_{a}^{ρ}_{ρ λ} Γ_{b}^{λ}_{μν} - Γ_{a}^{ρ}_{ν λ} Γ_{b}^{λ}_{μ ρ}), = i + j = n i \geq 0, j \geq 1 \sum u_{i}^{μν} R_{j, μν}, r_{0} = 0, = R_{n, μν} - \frac{1}{2} η_{μν} r_{n} - \frac{1}{2} γ_{μν} r_{n - 1} .$
XI-A@1#eq:r-n · line 1345
$R_{n, μν} = r_{n} C_{n, μν} \partial_{ρ} Γ_{n}^{ρ}_{μν} - \partial_{ν} Γ_{n}^{ρ}_{μ ρ} + a + b = n a, b \geq 1 \sum (Γ_{a}^{ρ}_{ρ λ} Γ_{b}^{λ}_{μν} - Γ_{a}^{ρ}_{ν λ} Γ_{b}^{λ}_{μ ρ}), = i + j = n i \geq 0, j \geq 1 \sum u_{i}^{μν} R_{j, μν}, r_{0} = 0, = R_{n, μν} - \frac{1}{2} η_{μν} r_{n} - \frac{1}{2} γ_{μν} r_{n - 1} .$
XI-A@1#eq:C-n · line 1345
$R_{n, μν} = r_{n} C_{n, μν} \partial_{ρ} Γ_{n}^{ρ}_{μν} - \partial_{ν} Γ_{n}^{ρ}_{μ ρ} + a + b = n a, b \geq 1 \sum (Γ_{a}^{ρ}_{ρ λ} Γ_{b}^{λ}_{μν} - Γ_{a}^{ρ}_{ν λ} Γ_{b}^{λ}_{μ ρ}), = i + j = n i \geq 0, j \geq 1 \sum u_{i}^{μν} R_{j, μν}, r_{0} = 0, = R_{n, μν} - \frac{1}{2} η_{μν} r_{n} - \frac{1}{2} γ_{μν} r_{n - 1} .$
XI-A@1#eq:H-n · line 1368
$H_{n}^{μν} = i + j + ℓ + s = n i, j, ℓ \geq 0, s \geq 1 \sum b_{i} u_{j}^{μα} u_{ℓ}^{ν β} C_{s, α β}, 1 \leq n \leq 3.$
unlabeled · line 1327
$u_{0}^{μν} = η^{μν}, u_{1}^{μν} = - γ^{μν}, u_{2}^{μν} = γ^{μ}_{ρ} γ^{ρ ν} .$

Finite formulas for the gravitational Euler coefficients

Displayed formulas in this passage

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