Appendices A and B: boundaries and variations

A. Boundary integrations and density inversion

With the convention (17) the affine derivative variation is

$$\delta R_{\mu\nu}=\partial_\alpha\delta A^\alpha{}_{\nu\mu} -\partial_\nu\delta A^\alpha{}_{\alpha\mu} +\delta\left(A^\alpha{}_{\alpha\rho}A^\rho{}_{\nu\mu} -A^\alpha{}_{\nu\rho}A^\rho{}_{\alpha\mu}\right). \tag{65}$$

Integration against the symmetric density yields

$$\Theta^\rho=B^{\mu\nu}\delta A^\rho{}_{\nu\mu} -B^{\mu\rho}\delta A^\alpha{}_{\alpha\mu}. \tag{66}$$

The bulk derivative coefficient is $J(B)$ in (33). The constant-density divergence in the source action is

$$\lambda^{-1}\eta^{\mu\nu}D_{\mu\nu}(c)= \partial_\rho\!\left[\lambda^{-1} \left(\eta^{\mu\nu}c^\rho{}_{\nu\mu} -\eta^{\mu\rho}c^\alpha{}_{\alpha\mu}\right)\right]. \tag{67}$$

Without lower-index symmetry, $c^\alpha{}_{\alpha\mu}$ cannot be exchanged for a different trace. Equations (66) and (67) give the matched boundary integrations in stress and recovery. A different boundary functional changes the variational problem and must be stated explicitly.

In dimension $n\gt2$, $B=vg^{-1}$ implies $|\det B|=v^{n-2}$, giving

$$v=|\det B|^{1/(n-2)},\qquad g^{-1}=B/v . \tag{68}$$

The determinant identity uses $|\det g|=v^2$. Its derivative gives $\delta v=(n-2)^{-1}g_{\mu\nu}\delta B^{\mu\nu}$ and reduces to (48) for $n=4$. In dimension two the determinant does not recover the conformal scale. The connected nondegenerate component fixes signature, not the absolute determinant.

B. Variation domains and the fixed-transport obstruction

Work locally on a smooth patch, with finite regularity high enough for all displayed derivatives or in the smooth category. The differentiability-at-constants theorem uses the $C^1$ Banach topology. The reconstruction bound uses controlled $C^2$ paths and bounded second derivatives. Joint differentiability in base point and path perturbation is part of the realization. Ordinary finite jets cannot replace the domain hypotheses.

$A$ and regular transport are inverse by uniqueness for the Volterra equation. Their differentials are inverse by (11) and its short-path derivative. The density map is a smooth local diffeomorphism on the nondegenerate branch by determinant inversion. Together these cover compactly supported affine, density and matter variations. The projective quotient removes exactly the directions $P\omega$ and $P^TE_c=0$ identically. A moving volume section contributes no additional connection Euler term. Other distortions are eliminated only on shell.

The fixed-transport error can be derived before using any field equation. In the auxiliary metric-density variation,

$$\delta C=-GJ(\delta\psi)+P\zeta,\qquad \left.\delta c_\psi\right|_T=-\lambda^{-1}\delta C . \tag{69}$$

Contract the latter with $E_c^{(2)}$ and integrate the adjoint:

$$\left.\delta S_2\right|_{T,B=\psi+\lambda h} =\int\delta\psi^{\mu\nu}\left[\tau_{\mu\nu}+ \lambda^{-1}D_{\mu\nu}(GE_c^{(2)})\right]. \tag{70}$$

The projective term vanishes since $P^TE_c^{(2)}=0$. If $B$ is frozen too, relative $h$ varies by $-\delta\psi/\lambda$ and gives the additional term $-E_h^{(2)}/\lambda$. Neither fixed-source operation gives the horizontal stress off shell, even though the defects can vanish on special solutions.