Why do all the curvature invariants of a gravitational wave vanish ?

Details Why do all the curvature invariants of a gravitational wave vanish ? Why do all the curvature invariants of a gravitational wave vanish ?

1994-04-19

arxiv.org/abs/gr-qc/9404037v1

G. Sardanashvily (Ed.) New frontiers in gravitation, Hadronic Press (1996) 337-344

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

9 pages

Scientific paper

We prove the theorem valid for (Pseudo)-Riemannian manifolds $V_n$: "Let $x \in V_n$ 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 $$ds \sp 2 \quad = \quad 2 \, du \, dv \, + \, a\sp 2 (u) \, dw \sp 2 \, + \, b\sp 2 (u) \, dz \sp 2 $$ 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.

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