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For $\alpha >0$ and $t\ne 0$, the conformal factor $t^2/\alpha$ makes spatial translation a Killing vector field. Using proper time, its conserved momentum is $P=(t^2/\alpha )dx/d\tau$. With $v=dx/dt$ and $d{s}^2=-c^{2}d{\tau }^2$ this becomesThus $v/\sqrt{1-v^2/c^2}$ alone is not constant. Set $b=\sqrt{\alpha }P/c$. Every unparametrized timelike geodesic in a connected nonzero-$t$ patch has the formThe nonvertical paths have $v=cb/\sqrt{t^2+b^2}$ and proper-time parametrizationThe chart excludes $t=0$, where its metric is degenerate; a vertical family must not be lost by dividing by the conserved momentum.

For the plane-fronted gravitational wave $d{s}^2=2dudv+d{\mathbf{x}}^2+{\mathbf{x}}^{T}M(u)\mathbf{x}d{u}^2$, synchronized neighboring geodesics parametrized by $u$ have transverse separation $\mathbf{\xi }$ satisfying the displayed equation. Along the central timelike geodesic $(u,v,\mathbf{x})=(\tau ,-\tau /2,0)$ the coordinate basis is parallel transport, so this is also the transverse Jacobi field equation in a transported frame. Vacuum requires $\mathrm{tr}M=0$, producing equal and opposite instantaneous transverse tidal acceleration eigenvalues. The ordinary coordinate equation ${\ddot{\xi }}^v=0$ holds on that central reference, but generally fails on off-axis reference curves.