Manuscript XII-C@1 · XII-C.tex · lemma · dead1004

Lemma: Weighted curvature variation

C1: the density representative and actual cubic source › The curvature variation needed for the expansion · lines 622–663 · proof included or linked

XII-C@1#lem:weighted-curvature, XII-C@1#eq:weighted-full-variation

For a fixed smooth scalar $σ$ , compact covariant metric variation gives

\frac{δ}{δ γ _{μν}} (- \frac{2}{κ ^{2}} \int σ - q R [q] d^{4} x) = \frac{2}{κ} - q [σ G^{μν} + q^{μν} □_{q} σ - \nabla^{μ} \nabla^{ν} σ] .

This identity is used for coefficient extraction, not to select the full-density model.

Proof. For $h_{μν} = δ q_{μν}$ , write $h = q^{μν} h_{μν}$ and $h^{μν} = q^{μα} q^{ν β} h_{α β}$ . The Christoffel variation gives

δ R = - R^{μν} h_{μν} + \nabla_{μ} \nabla_{ν} h^{μν} - □_{q} h .

Together with $δ - q = \frac{1}{2} - q h$ , two compact integrations by parts give

δ \int σ - q R = \int - q [- σ G^{μν} + \nabla^{μ} \nabla^{ν} σ - q^{μν} □_{q} σ] h_{μν} d^{4} x .

Multiply by $- 2/ κ^{2}$ and use $h_{μν} = κ δ γ_{μν}$ .

Displayed formulas in this passage

XII-C@1#eq:weighted-full-variation · line 626
$\frac{δ}{δ γ _{μν}} (- \frac{2}{κ ^{2}} \int σ - q R [q] d^{4} x) = \frac{2}{κ} - q [σ G^{μν} + q^{μν} □_{q} σ - \nabla^{μ} \nabla^{ν} σ] .$
unlabeled · line 647
$δ R = - R^{μν} h_{μν} + \nabla_{μ} \nabla_{ν} h^{μν} - □_{q} h .$
unlabeled · line 653
$δ \int σ - q R = \int - q [- σ G^{μν} + \nabla^{μ} \nabla^{ν} σ - q^{μν} □_{q} σ] h_{μν} d^{4} x .$