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

Exact metric derivative and its expansion

Metric Euler derivative through coupling order two › Exact metric derivative and its expansion · lines 637–667

XI-A@1#eq:Hdef, XI-A@1#eq:J1

Define the Einstein tensor-density coefficients by

- q G^{μν} (q) = κ H_{1}^{μν} + κ^{2} H_{2}^{μν} + κ^{3} H_{3}^{μν} + O (κ^{4}) .

Here $H_{1} = G_{lin} (γ)$ . Appendix app:coefficients gives finite explicit formulas for all three coefficients, without an external curvature convention.

Let

- q T_{q}^{μν} = T_{0}^{μν} + κ J_{1}^{μν} + O (κ^{2}) .

Direct expansion gives

J_{1}^{μν} [γ, ϕ] = b_{1} T_{0}^{μν} - γ^{μα} p_{α} p^{ν} - γ^{ν α} p_{α} p^{μ} + γ^{μν} ℓ_{0} + \frac{1}{2} η^{μν} γ^{α β} p_{α} p_{β} .

Displayed formulas in this passage

XI-A@1#eq:Hdef · line 640
$- q G^{μν} (q) = κ H_{1}^{μν} + κ^{2} H_{2}^{μν} + κ^{3} H_{3}^{μν} + O (κ^{4}) .$
XI-A@1#eq:J1 · line 657
$J_{1}^{μν} [γ, ϕ] = b_{1} T_{0}^{μν} - γ^{μα} p_{α} p^{ν} - γ^{ν α} p_{α} p^{μ} + γ^{μν} ℓ_{0} + \frac{1}{2} η^{μν} γ^{α β} p_{α} p_{β} .$
unlabeled · line 652
$- q T_{q}^{μν} = T_{0}^{μν} + κ J_{1}^{μν} + O (κ^{2}) .$

Exact metric derivative and its expansion

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