= Transverse tidal equation of a Brinkmann plane wave
{title2=$\ddot{\boldsymbol\xi}=M(u)\boldsymbol\xi$}
For the <plane-fronted gravitational wave> $ds^2=2du\,dv+d\mathbf x^2+\mathbf x^TM(u)\mathbf x\,du^2$, synchronized neighboring <geodesics> parametrized by $u$ have transverse separation $\boldsymbol\xi$ 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 $\operatorname{tr}M=0$, 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.
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