Imported finite varied-action formula

For a compactly supported diffeomorphism $F$ in the interior, X-A's Route II gives

where $F_{\mathrm{dens}}^{\ast }r-r={\partial }_{\mu }{Z_F}^{\mu }[r]$. For the flow of $\kappa \xi +{\kappa }^2\zeta$,

The proof of (eq:RouteII) is not repeated.

The unswitched Einstein–Hilbert density is natural. The same divergence lemma therefore shows ${\delta }_F{S}_{\mathrm{EH}}=0$ for compact interior $F$. Consequently (eq:RouteII) is also the finite variation of $S_{\mathrm{inv}}$. There is no analogous assertion for $S_{\mathrm{gf}}$.