Why do all the curvature invariants of a gravitational wave vanish ?
| dc.creator | Schmidt, Hans - Jürgen | |
| dc.date | 1994-04-19 | |
| dc.date.accessioned | 2026-07-07T03:30:19Z | |
| dc.date.available | 2026-07-07T03:30:19Z | |
| dc.description | 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. | |
| dc.description | 9 pages | |
| dc.identifier | https://arxiv.org/abs/gr-qc/9404037 | |
| dc.identifier | http://arxiv.org/abs/gr-qc/9404037 | |
| dc.identifier | G. Sardanashvily (Ed.) New frontiers in gravitation, Hadronic Press (1996) 337-344 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/35499 | |
| dc.subject | General Relativity and Quantum Cosmology | |
| dc.title | Why do all the curvature invariants of a gravitational wave vanish ? | |
| dc.type | text |