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

Proposition: Identified density derivative

Density and scalar vertex · lines 974–1046 · proof included or linked

Em01-F@3#prop:density, Em01-F@3#eq:Ddensity, Em01-F@3#eq:Didentified

The expansions are

a_{λ} (γ) = η^{- 1} + λ a_{1} [γ] + λ^{2} a_{2} [γ] + O (λ^{3}), b_{λ} (γ) = 1 + λ b_{1} [γ] + O (λ^{2}),

with

a_{1}^{μν} [h] = \frac{1}{2} η^{μν} tr_{η} h - h^{μν}, b_{1} [h] = \frac{1}{2} tr_{η} h,

a_{2}^{μν} [h] = (\frac{1}{8} (tr_{η} h)^{2} - \frac{1}{4} h_{ρ σ} h^{ρ σ}) η^{μν} - \frac{1}{2} (tr_{η} h) h^{μν} + h^{μ ρ} h_{ρ}^{ν} .

The mixed derivative is

D_{dens}^{μν} (G, h) = (\frac{1}{4} (tr_{η} G) (tr_{η} h) - \frac{1}{2} G_{ρ σ} h^{ρ σ}) η^{μν} - \frac{1}{2} ((tr_{η} G) h^{μν} + (tr_{η} h) G^{μν}) + G^{μ ρ} h_{ρ}^{ν} + h^{μ ρ} G_{ρ}^{ν} .

For the defined insertion,

D^{α μ, I, J} = D_{dens}^{α μ} (e_{J}, e_{I}) .

The full mixed metric-field variation carries $λ^{2} D_{dens} (G, h)$ .

Proof. Expand the inverse matrix and square-root determinant:

(I + λ H)^{- 1} = I - λ H + λ^{2} H^{2} + O (λ^{3}),

det (I + λ H) = 1 + \frac{λ}{2} tr H + λ^{2} [\frac{1}{8} (tr H)^{2} - \frac{1}{4} tr (H^{2})] + O (λ^{3}), H = η^{- 1} h .

Multiplication gives $a_{1}, a_{2}, b_{1}$ . Differentiation of $a_{2} [G + t h]$ at $t = 0$ gives (eq:Ddensity). The insertion uses this derivative, not its half-polarization.

Displayed formulas in this passage

Em01-F@3#eq:Ddensity · line 1001
$D_{dens}^{μν} (G, h) = (\frac{1}{4} (tr_{η} G) (tr_{η} h) - \frac{1}{2} G_{ρ σ} h^{ρ σ}) η^{μν} - \frac{1}{2} ((tr_{η} G) h^{μν} + (tr_{η} h) G^{μν}) + G^{μ ρ} h_{ρ}^{ν} + h^{μ ρ} G_{ρ}^{ν} .$
Em01-F@3#eq:Didentified · line 1016
$D^{α μ, I, J} = D_{dens}^{α μ} (e_{J}, e_{I}) .$
unlabeled · line 977
$a_{λ} (γ) = η^{- 1} + λ a_{1} [γ] + λ^{2} a_{2} [γ] + O (λ^{3}), b_{λ} (γ) = 1 + λ b_{1} [γ] + O (λ^{2}),$
unlabeled · line 986
$a_{1}^{μν} [h] = \frac{1}{2} η^{μν} tr_{η} h - h^{μν}, b_{1} [h] = \frac{1}{2} tr_{η} h,$
unlabeled · line 992
$a_{2}^{μν} [h] = (\frac{1}{8} (tr_{η} h)^{2} - \frac{1}{4} h_{ρ σ} h^{ρ σ}) η^{μν} - \frac{1}{2} (tr_{η} h) h^{μν} + h^{μ ρ} h_{ρ}^{ν} .$
unlabeled · line 1028
$(I + λ H)^{- 1} = I - λ H + λ^{2} H^{2} + O (λ^{3}),$
unlabeled · line 1033
$det (I + λ H) = 1 + \frac{λ}{2} tr H + λ^{2} [\frac{1}{8} (tr H)^{2} - \frac{1}{4} tr (H^{2})] + O (λ^{3}), H = η^{- 1} h .$

Proposition: Identified density derivative

Displayed formulas in this passage

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