Lemma: First rod variations

On (eq:scalar-metric),

Thus original and modified rods are not identical functional prescriptions.

Proof. On an original clock slice,

Expanding ${\Delta }_{\gamma }f=(\det \gamma {)}^{-1/2}{\partial }_i((\det \gamma {)}^{1/2}{\gamma }^{ij}{\partial }_{j}f)$ gives

The first harmonic equation is $\Delta {Y_1}^i=-{\partial }_{i}D$, which proves the first formula.

A modified slice through background time $\tau$ has embedding

because $v={\dot{T}}_0$. Its induced spatial metric at first order is

The contribution involving ${\partial }_{i}t{\partial }_{j}t$ starts at order $e^2$; the lapse perturbation $A$ therefore does not alter this first-order calculation. The modified harmonic equation is $\Delta {Z_1}^i=0$ with decaying boundary data, so ${Z_1}^i=0$. For any $D$ such that ${\partial }_i{G}_{\Delta }D$ is nonzero, the prescriptions differ.