Manuscript VI-B@1 · VI-B-v1.tex · section · 33798723

Explicit curvature definition of the quadratic metric term

Explicit curvature definition of the quadratic metric term · lines 2193–2245

VI-B@1#eq:G2explicit

This appendix makes the symbol $G^{(2)} [h, h]$ a fully specified differential expression.

For a symmetric $h$ , let

H_{μν σ} = \partial_{μ} h_{ν σ} + \partial_{ν} h_{μ σ} - \partial_{σ} h_{μν},

Γ_{1}^{ρ}_{μν} = \frac{1}{2} η^{ρ σ} H_{μν σ}, Γ_{2}^{ρ}_{μν} = - \frac{1}{2} h^{ρ σ} H_{μν σ} .

Define

R_{1, μν} = \partial_{ρ} Γ_{1}^{ρ}_{μν} - \partial_{ν} Γ_{1}^{ρ}_{μ ρ},

R_{2, μν} = \partial_{ρ} Γ_{2}^{ρ}_{μν} - \partial_{ν} Γ_{2}^{ρ}_{μ ρ} + Γ_{1}^{ρ}_{ρ λ} Γ_{1}^{λ}_{μν} - Γ_{1}^{ρ}_{ν λ} Γ_{1}^{λ}_{μ ρ} .

Then

G^{(2)} [h, h]_{μν} = R_{2, μν} - \frac{1}{2} η_{μν} (η^{ab} R_{2, ab} - h^{ab} R_{1, ab}) - \frac{1}{2} h_{μν} η^{ab} R_{1, ab} .

These are the coefficients obtained by inserting $η + κh$ into the coordinate Christoffel and Ricci formulas. Terms linear in $j$ give exactly $G^{(1)} (j)$ . Thus no hidden second-order gravitational source normalization is contained in the notation.

Displayed formulas in this passage

unlabeled · line 2219
$R_{2, μν} = \partial_{ρ} Γ_{2}^{ρ}_{μν} - \partial_{ν} Γ_{2}^{ρ}_{μ ρ} + Γ_{1}^{ρ}_{ρ λ} Γ_{1}^{λ}_{μν} - Γ_{1}^{ρ}_{ν λ} Γ_{1}^{λ}_{μ ρ} .$
VI-B@1#eq:G2explicit · line 2229
$G^{(2)} [h, h]_{μν} = R_{2, μν} - \frac{1}{2} η_{μν} (η^{ab} R_{2, ab} - h^{ab} R_{1, ab}) - \frac{1}{2} h_{μν} η^{ab} R_{1, ab} .$
unlabeled · line 2200
$H_{μν σ} = \partial_{μ} h_{ν σ} + \partial_{ν} h_{μ σ} - \partial_{σ} h_{μν},$
unlabeled · line 2206
$Γ_{1}^{ρ}_{μν} = \frac{1}{2} η^{ρ σ} H_{μν σ}, Γ_{2}^{ρ}_{μν} = - \frac{1}{2} h^{ρ σ} H_{μν σ} .$
unlabeled · line 2214
$R_{1, μν} = \partial_{ρ} Γ_{1}^{ρ}_{μν} - \partial_{ν} Γ_{1}^{ρ}_{μ ρ},$