Manuscript VI@1 · VI.tex · section · b5b60afb

The metric source factor

Sign, coefficient, and switching audit › The metric source factor · lines 2512–2540

Variation of the gravitational and matter actions with respect to $γ^{μν}$ gives

δ S_{grav} = \frac{2}{κ ^{2}} \int - γ G_{μν} δ γ^{μν}, δ S_{m} = - \frac{1}{2} \int - γ T_{μν} δ γ^{μν} .

Therefore

G_{μν} = \frac{κ ^{2}}{4} T_{μν} .

The second-order coefficient equation is consequently

G^{(1)} (j) + G^{(2)} [h, h] = \frac{1}{4} T (η, ϕ_{0}) .

A convention with right-hand side $T$ uses a different normalization.

The normalization displayed in Figure fig:VI-coefficient-domain is used throughout.

Displayed formulas in this passage

unlabeled · line 2516
$δ S_{grav} = \frac{2}{κ ^{2}} \int - γ G_{μν} δ γ^{μν}, δ S_{m} = - \frac{1}{2} \int - γ T_{μν} δ γ^{μν} .$
unlabeled · line 2528
$G_{μν} = \frac{κ ^{2}}{4} T_{μν} .$
unlabeled · line 2532
$G^{(1)} (j) + G^{(2)} [h, h] = \frac{1}{4} T (η, ϕ_{0}) .$