Why Do All the Curvature Invariants of a Gravitational Wave Vanish ?

We prove the theorem valid for (Pseudo)-Riemannian manifolds V n : "Let x 2 V n be a xed 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 = 2 du dv + a 2 (u) dw 2 + b 2 (u) dz 2 identically vanish." Analysing the proof we see: The fact that for deenite signature atness can be characterized by the vanishing of a curvature invariant, essentially rests on the compactness of the rotation group S O(n). For Lorentz signature, however, one has the non-compact Lorentz group S O(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 Hausdorr topology, so a sequence of gravitational waves can converge to the at space-time, even if each element of the sequence is the same pp-wave.