Clock, lapse, and relational pullback

Let $q$ have signature $(-,+,+,+)$, and let $T$ be a smooth scalar with

The clock orientation is chosen so that $T$ increases to the future. Its future unit normal and raw lapse are

Thus $n_T(T)=\sqrt{S}$ and $N{n}_T(T)=1$.

The lapse is neither the scalar value $T$ nor its normal derivative. It is a functional of the metric and the first derivatives of $T$.

Let ${\Delta }_{q,T}$ be the Laplace–Beltrami operator of the Riemannian metric induced on a $T$-level set. The original rods satisfy

These are slice-elliptic equations, not spacetime wave equations. The original and modified choices in BFHPR (arXiv equations (28)–(35)) must be distinguished. The supplied Em02-E record explicitly selects the original rods CheungEm02E.

For a fixed background clock $T_0(t)$ with ${\dot{T}}_0>0$, let

The relational lapse is

The fixed clock calibration in (eq:relmap) is part of this definition.