Manuscript Em01-F@3 · Em01-F-v3.tex · section · 225e59af

Unintegrated density representative

Explicit cubic density and source audit › Unintegrated density representative · lines 1104–1159

Em01-F@3#eq:ell3

For a covariant symmetric field $h$ , let

S (h)_{σ μν} = \partial_{μ} h_{σ ν} + \partial_{ν} h_{σ μ} - \partial_{σ} h_{μν},

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

For $n = 1, 2, 3$ , set

Γ_{n}^{ρ}_{μν} R_{n, μν} r_{n} = \frac{1}{2} u_{n - 1}^{ρ σ} S (h)_{σ 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 α} Γ_{b}^{α}_{μν} - Γ_{a}^{ρ}_{ν α} Γ_{b}^{α}_{μ ρ}), = i = 0 \sum^{n - 1} u_{i}^{μν} R_{n - i, μν} .

With

b_{1} [h] = \frac{1}{2} tr_{η} h, b_{2} [h] = \frac{1}{8} (tr_{η} h)^{2} - \frac{1}{4} h_{ρ σ} h^{ρ σ},

the declared cubic density is

ℓ_{3} [h] = - 2 (r_{3} + b_{1} [h] r_{2} + b_{2} [h] r_{1}) .

It is the coefficient of $λ$ in the unintegrated Einstein–Hilbert density of (eq:action). The source is calculated from this density with unit weight. No density-weighted integration-by-parts equivalence is assumed.

At zero scalar field, write the lowered metric Euler equation as

- □_{η} P γ + λ J_{2} [γ] + O (λ^{2}) = 0.

The source is homogeneous quadratic. The label $J_{3}^{app}$ may equivalently denote $J_{2}$ , emphasizing that it comes from the cubic action coefficient, not from a cubic field-degree source.

Displayed formulas in this passage

unlabeled · line 1120
$Γ_{n}^{ρ}_{μν} R_{n, μν} r_{n} = \frac{1}{2} u_{n - 1}^{ρ σ} S (h)_{σ μν}, = \partial_{ρ} Γ_{n}^{ρ}_{μν} - \partial_{ν} Γ_{n}^{ρ}_{μ ρ} + a + b = n a, b \geq 1 \sum (Γ_{a}^{ρ}_{ρ α} Γ_{b}^{α}_{μν} - Γ_{a}^{ρ}_{ν α} Γ_{b}^{α}_{μ ρ}), = i = 0 \sum^{n - 1} u_{i}^{μν} R_{n - i, μν} .$
Em01-F@3#eq:ell3 · line 1141
$ℓ_{3} [h] = - 2 (r_{3} + b_{1} [h] r_{2} + b_{2} [h] r_{1}) .$
unlabeled · line 1107
$S (h)_{σ μν} = \partial_{μ} h_{σ ν} + \partial_{ν} h_{σ μ} - \partial_{σ} h_{μν},$
unlabeled · line 1114
$u_{0}^{μν} = η^{μν}, u_{1}^{μν} = - h^{μν}, u_{2}^{μν} = h^{μ ρ} h_{ρ}^{ν} .$
unlabeled · line 1136
$b_{1} [h] = \frac{1}{2} tr_{η} h, b_{2} [h] = \frac{1}{8} (tr_{η} h)^{2} - \frac{1}{4} h_{ρ σ} h^{ρ σ},$
unlabeled · line 1151
$- □_{η} P γ + λ J_{2} [γ] + O (λ^{2}) = 0.$