The full geometric input

Suppose a smooth Lorentzian metric $q_{\epsilon }$ and a scalar clock ${\mathcal{T}}_{\epsilon }$ are supplied on a region where $\mathrm{d}{\mathcal{T}}_{\epsilon }$ is timelike. Define the positive raw lapse

Supply original slice-harmonic rods and suppose the resulting coordinate map

is locally invertible, with inverse ${\alpha }_{\epsilon }$. Then the relational lapse is

In these relational coordinates the full metric has ADM form

Indeed, writing the metric as $A\mathrm{d}{\tau }^2+2B_i\mathrm{d}\tau \mathrm{d}{y}^i+{\gamma }_{ij}\mathrm{d}{y}^i\mathrm{d}{y}^j$, set ${\beta }^i={\gamma }^{ij}{B}_j$ and $n^2=-A+{\gamma }^{ij}{B}_i{B}_j$; completing the square gives (eq:adm), and inversion gives $q^{\tau \tau }=-n^{-2}$.

Thus a lapse alone does not specify the metric. Along a curve of constant rods, proper-time rate is

when the curve is timelike. Along a normal curve $\mathrm{d}{y}^i=-{{\beta }_{\epsilon }}^i\mathrm{d}\tau$, it is $n_{\epsilon }$. Constant rods and normal observers must not be silently identified.

If $q_{\epsilon },{\phi }_{\epsilon }$ are required to satisfy Einstein–scalar equations, those equations are additional geometric input. Merely supplying $n_{\epsilon }$ does not impose them.