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

The second-order Einstein–scalar coefficient space

The second-order Einstein–scalar coefficient space · lines 424–453

VI-B@1#eq:L, VI-B@1#eq:G1

All indices in coefficient formulas are raised with $η$ . For a symmetric tensor $a$ , put

a ˉ_{μν} = a_{μν} - \frac{1}{2} η_{μν} tr_{η} a, D (a)^{ν} = \partial_{μ} a ˉ^{μν} .

For a vector $ξ$ ,

(K ξ)_{μν} = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ}, X_{ξ} = ξ^{μ} \partial_{μ} .

Set

L_{a} u G^{(1)} (a)_{μν} = - a^{μν} \partial_{μ} \partial_{ν} u - D (a)^{ν} \partial_{ν} u, = - \frac{1}{2} □ a ˉ_{μν} + \partial_{(μ} D (a)_{ν)} - \frac{1}{2} η_{μν} \partial_{ρ} D (a)^{ρ} .

Displayed formulas in this passage

VI-B@1#eq:L · line 442
$L_{a} u G^{(1)} (a)_{μν} = - a^{μν} \partial_{μ} \partial_{ν} u - D (a)^{ν} \partial_{ν} u, = - \frac{1}{2} □ a ˉ_{μν} + \partial_{(μ} D (a)_{ν)} - \frac{1}{2} η_{μν} \partial_{ρ} D (a)^{ρ} .$
VI-B@1#eq:G1 · line 442
$L_{a} u G^{(1)} (a)_{μν} = - a^{μν} \partial_{μ} \partial_{ν} u - D (a)^{ν} \partial_{ν} u, = - \frac{1}{2} □ a ˉ_{μν} + \partial_{(μ} D (a)_{ν)} - \frac{1}{2} η_{μν} \partial_{ρ} D (a)^{ρ} .$
unlabeled · line 429
$a ˉ_{μν} = a_{μν} - \frac{1}{2} η_{μν} tr_{η} a, D (a)^{ν} = \partial_{μ} a ˉ^{μν} .$
unlabeled · line 436
$(K ξ)_{μν} = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ}, X_{ξ} = ξ^{μ} \partial_{μ} .$