Transverse tidal equation of a Brinkmann plane wave ($\ddot{\mathbf{\xi }}=M(u)\mathbf{\xi }$)

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.