X iv : g r - qc / 9 40 40 37 v 1 1 9 A pr 1 99 4 Why do all the curvature invariants of a gravitational wave vanish ?

We prove the theorem valid for (Pseudo)-Riemannian manifolds Vn: ”Let x ∈ Vn be a fixed point of a homothetic motion which is not an isometry then all curvature invariants vanish at x.” and get the Corollary: ”All curvature invariants of the plane wave metric ds = 2 du dv + a(u) dw + b(u) dz identically vanish.” Analysing the proof we see: The fact that for definite signature flatness can be characterized by the vanishing of a curvature invariant, essentially rests on the compactness of the rotation group SO(n). For Lorentz signature, however, one has the non-compact Lorentz group SO(3, 1) instead of it. A further and independent proof of the corollary uses the fact, that the Geroch limit does not lead to a Hausdorff topology, so a sequence of gravitational waves can converge to the flat space-time, even if each element of the sequence is the same pp-wave. AMS number: 53 B 30 Lorentz metrics, indefinite metrics PACS number: 0430 Gravitational waves: theory