Manuscript X-E-ST@1 · X-E-ST.tex · section · 44614221

A definite second-order metric equation

E1: action, Palatini variation, and switched Noether identity › A definite second-order metric equation · lines 633–702

X-E-ST@1#eq:Glin, X-E-ST@1#eq:metric-expansion

The linearized Einstein tensor is

G_{lin, μν} [l] = \frac{1}{2} (\partial_{ρ} \partial_{μ} l^{ρ}_{ν} + \partial_{ρ} \partial_{ν} l^{ρ}_{μ} - □ l_{μν} - \partial_{μ} \partial_{ν} tr_{η} l - η_{μν} (\partial_{α} \partial_{β} l^{α β} - □ tr_{η} l)) .

It satisfies $\partial_{μ} G_{lin}^{μν} [l] = 0$ .

To specify the nonlinear coefficient without an unspecified vertex, define

G_{2}^{μν} [h, h] = \frac{1}{2} \frac{d ^{2}}{d s ^{2}}_{s = 0} (J_{η + s h} G^{μν} [η + s h]) .

This is an explicit differential polynomial. It can equivalently be calculated from

Γ_{1}^{ρ}_{μν} = \frac{1}{2} η^{ρ λ} (\partial_{μ} h_{λ ν} + \partial_{ν} h_{λ μ} - \partial_{λ} h_{μν}), Γ_{2}^{ρ}_{μν} = - h^{ρ}_{λ} Γ_{1}^{λ}_{μν},

and

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

followed by the trace, inverse-metric, and density expansions (eq:densities).

The invariant metric equation through its second geometric order is

J_{q} G^{μν} [q] - κ^{2} g J_{q} T^{μν} [q, ϕ] = κ G_{lin}^{μν} [h] + κ^{2} (G_{lin}^{μν} [k] + G_{2}^{μν} [h, h] - g T_{0}^{μν} [ϕ_{0}]) + O (κ^{3}) .

Equivalently,

E_{h}^{μν} = \frac{1}{2} G_{lin}^{μν} [h] + \frac{κ}{2} (G_{lin}^{μν} [k] + G_{2}^{μν} [h, h] - g T_{0}^{μν} [ϕ_{0}]) + O (κ^{2}) .

Displayed formulas in this passage

X-E-ST@1#eq:Glin · line 636
$G_{lin, μν} [l] = \frac{1}{2} (\partial_{ρ} \partial_{μ} l^{ρ}_{ν} + \partial_{ρ} \partial_{ν} l^{ρ}_{μ} - □ l_{μν} - \partial_{μ} \partial_{ν} tr_{η} l - η_{μν} (\partial_{α} \partial_{β} l^{α β} - □ tr_{η} l)) .$
X-E-ST@1#eq:metric-expansion · line 682
$J_{q} G^{μν} [q] - κ^{2} g J_{q} T^{μν} [q, ϕ] = κ G_{lin}^{μν} [h] + κ^{2} (G_{lin}^{μν} [k] + G_{2}^{μν} [h, h] - g T_{0}^{μν} [ϕ_{0}]) + O (κ^{3}) .$
unlabeled · line 653
$G_{2}^{μν} [h, h] = \frac{1}{2} \frac{d ^{2}}{d s ^{2}}_{s = 0} (J_{η + s h} G^{μν} [η + s h]) .$
unlabeled · line 660
$Γ_{1}^{ρ}_{μν} = \frac{1}{2} η^{ρ λ} (\partial_{μ} h_{λ ν} + \partial_{ν} h_{λ μ} - \partial_{λ} h_{μν}), Γ_{2}^{ρ}_{μν} = - h^{ρ}_{λ} Γ_{1}^{λ}_{μν},$
unlabeled · line 670
$R_{2, μν} = \partial_{ρ} Γ_{2}^{ρ}_{μν} - \partial_{ν} Γ_{2}^{ρ}_{ρ μ} + Γ_{1}^{ρ}_{ρ λ} Γ_{1}^{λ}_{μν} - Γ_{1}^{ρ}_{ν λ} Γ_{1}^{λ}_{ρ μ},$
unlabeled · line 694
$E_{h}^{μν} = \frac{1}{2} G_{lin}^{μν} [h] + \frac{κ}{2} (G_{lin}^{μν} [k] + G_{2}^{μν} [h, h] - g T_{0}^{μν} [ϕ_{0}]) + O (κ^{2}) .$