Manuscript XX-B@1 · XX-B.tex · proposition · 56d708a5

Proposition: Complete two-jet Euler expansion

The complete second-order classical field equations · lines 673–709 · proof included or linked

XX-B@1#prop:Euler2

Equations (eq:J)–(eq:scalar2), together with (eq:Gamma1)–(eq:scalar1), give every coefficient of the coupled Euler equations through $s^{2}$ . The second coefficient separates uniquely into the linear operator applied to $(k, ψ)$ and a homogeneous quadratic source in $(h, φ)$ .

Proof. The exact connection-difference formula is

Γ [g_{s}]^{ρ}_{μν} - Γ [g ˉ]^{ρ}_{μν} = \frac{1}{2} g_{s}^{ρ σ} (\nabla_{μ} (g_{s} - g ˉ)_{ν σ} + \nabla_{ν} (g_{s} - g ˉ)_{μ σ} - \nabla_{σ} (g_{s} - g ˉ)_{μν}) .

Its Taylor expansion gives (eq:C). The exact curvature-difference formula is its covariant derivative plus the two connection products in (eq:R2). Contracting with the inverse metric gives the scalar curvature coefficients.

Expanding the stress tensor gives (eq:U) and (eq:T2). Finally,

\nabla^{g_{s}}_{μ} \partial_{ν} ϕ_{s} = \nabla_{μ} \partial_{ν} ϕ_{s} - C_{s}^{ρ}_{μν} \partial_{ρ} ϕ_{s}

gives (eq:A) and (eq:scalar2). Every occurrence of $k, ψ$ is linear; setting them to zero leaves the homogeneous quadratic source.

Displayed formulas in this passage

unlabeled · line 685
$Γ [g_{s}]^{ρ}_{μν} - Γ [g ˉ]^{ρ}_{μν} = \frac{1}{2} g_{s}^{ρ σ} (\nabla_{μ} (g_{s} - g ˉ)_{ν σ} + \nabla_{ν} (g_{s} - g ˉ)_{μ σ} - \nabla_{σ} (g_{s} - g ˉ)_{μν}) .$
unlabeled · line 701
$\nabla^{g_{s}}_{μ} \partial_{ν} ϕ_{s} = \nabla_{μ} \partial_{ν} ϕ_{s} - C_{s}^{ρ}_{μν} \partial_{ρ} ϕ_{s}$

Proposition: Complete two-jet Euler expansion

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