Proposition: The transported response is not a first VI solution-jet coefficient

For $t\in (\epsilon ,T)$,

Its other components vanish on that interval. In particular, $h_{\lambda }$ fails (eq:VI-vacuum). For every nonzero phase $\zeta$, the real coefficient $\mathrm{Re}(\zeta h_{\lambda })$ also fails that equation on the slab.

Consequently, placing this response as VI's first metric coefficient does not produce an element of ${\mathcal{J}}_T$, whatever higher metric and scalar coefficients are chosen.

Proof. For a spatially homogeneous tensor $h$ with $h_{0i}=0$, write $H_s={\sum }_k{h}_{kk}$. Substitution in (eq:Einstein) gives

For example, ${\overline{h}}_{-}^{00}=(h_{00}+H_s)/2$; the $h_{00}^{\prime \prime }$ terms cancel from the spatial equation. This verifies the formula without imposing any gauge condition.

From (eq:htransport) and (eq:V), $h_{\lambda }=({\mathcal{P}}_{+}{V}_{\lambda }{)}^{♭}$. On the plateau,

so

It follows that

This proves (eq:Einstein-witness).

The matrix $n_i{n}_j-{\delta }_{ij}$ is nonzero. For every $\zeta \ne 0$, $\mathrm{Re}(\zeta e^{2i\lambda t})$ is not identically zero on an open time interval. Thus the corresponding real Einstein coefficient is nonzero. The first metric equation in ${\mathcal{J}}_T$ is independent of higher coefficients, which proves the final assertion.