Proposition: Equivariance and classical invariance

Through the retained order,

and therefore

Proof. The scalar clock transforms by pullback. A diffeomorphism maps its level hypersurfaces and their induced metrics into the corresponding level hypersurfaces and metrics of the transformed configuration.

The pullback of each harmonic spatial coordinate is harmonic for the pulled-back slice metric. Because $\chi$ is the identity at spatial infinity, it has the same asymptotic Cartesian boundary data. The coefficientwise uniqueness proved in Proposition prop:rods identifies it with the coordinate constructed for ${\chi }^{\ast }\Gamma$.

Invert the resulting scalar-coordinate identity to obtain the transformation law for $\alpha$. The lapse is a scalar built from the metric and scalar gradient, so $N_{\varphi ,{\chi }^{\ast }\Gamma }=N_{\varphi ,\Gamma }\circ \chi$. The two compositions cancel. Integration against the fixed label-space density $f$ preserves the identity.