Theorem: Original and modified rods change the quadratic lapse

Let $\psi \in {C_c}^{\infty }({\mathbb{R}}^3;\mathbb{R})$ be nonconstant and let $b$ be a smooth time bump supported strictly inside the comparison interval. In (eq:scalar-metric), choose

Let $n_{2,Y}$ and $n_{2,Z}$ be the second coefficients of the relational lapse using original and modified rods, respectively, with the same clock $T_e=T_0$. Then

The difference is detected by a nonnegative compact spacetime test. Hence the quadratic relational lapse depends on the rod choice.

Proof. Since $G_{\Delta }\Delta \psi =\psi$ for compact $\psi$, Lemma lem:rod-first gives

There is no clock perturbation, so $u_1=u_2=0$. For (eq:scalar-metric), the raw lapse is

and therefore

The local raw second coefficient is the same for the two rod choices. Proposition prop:lapse-composition consequently gives

Choose a nonnegative test supported where $b\ne 0$ and $\nabla \psi \ne 0$, positive on a nonempty open subset. Its pairing with the displayed difference is strictly negative.