Manuscript X-A-conv@1 · X-A-conv.tex · section · c76fffa3

The fixed divergence relating the gravitational densities

Independent action variation and the plateau Noether identity › The fixed divergence relating the gravitational densities · lines 1690–1726

X-A-conv@1#eq:Bdensity, X-A-conv@1#eq:Hilbertsplit

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Define the formal vector density

B^{μ} (κ, h) = - \frac{2}{κ ^{2}} A (q)^{α β} Γ^{μ}_{α β} + \frac{2}{κ ^{2}} A (q)^{μα} Γ^{β}_{α β} .

Direct integration by parts in the coordinate formula for $R$ gives

- \frac{2}{κ ^{2}} d (q) R [q] = ℓ_{grav} + \partial_{μ} B^{μ} .

To verify the identity, expand the derivative terms in $d (q) q^{α β} R_{α β}$ , move the two derivatives of $Γ$ into the displayed divergence, and use metric compatibility to differentiate $A (q)^{α β}$ . The remaining terms are exactly

A (q)^{α β} (Γ^{ρ}_{α σ} Γ^{σ}_{β ρ} - Γ^{ρ}_{α β} Γ^{σ}_{ρ σ}),

with the coefficient $- 2/ κ^{2}$ . This proves (eq:Hilbertsplit) without altering the switched representative.

The formal series $B^{μ}$ may begin at $κ^{- 1}$ . The density $ℓ$ in (eq:ellseries) begins at $κ^{0}$ .

Displayed formulas in this passage

X-A-conv@1#eq:Bdensity · line 1693
$B^{μ} (κ, h) = - \frac{2}{κ ^{2}} A (q)^{α β} Γ^{μ}_{α β} + \frac{2}{κ ^{2}} A (q)^{μα} Γ^{β}_{α β} .$
X-A-conv@1#eq:Hilbertsplit · line 1702
$- \frac{2}{κ ^{2}} d (q) R [q] = ℓ_{grav} + \partial_{μ} B^{μ} .$
unlabeled · line 1713
$A (q)^{α β} (Γ^{ρ}_{α σ} Γ^{σ}_{β ρ} - Γ^{ρ}_{α β} Γ^{σ}_{ρ σ}),$