The generic condition is generic
The authors study the usual genericity condition in general relativity for vectors at a given point of a Lorentz manifold. A vector \(X \neq 0\) is called generic if \(X^ cX^ dX_{[a}R_{b]cd[e}X_{f]}\neq 0\). This condition holds in an open, dense set of vectors. Their main result is that in a 4-dimensional Lorentz manifold the Riemann tensor vanishes (at the given point) if there are 5 non-null vectors in general position which are all non-generic. In the case that the Ricci tensor satisfies \(\text{Ric} = \lambda g\) for some \(\lambda\in\mathbb{R}\) at this point 3 linearly independent, timelike vectors (or 5 distinct non-zero null vectors) which are non-generic are sufficient to conclude that the Riemann tensor vanishes at the given point. These (and related results) are proved with the methods of linear algebra.
scientific article; zbMATH DE number 465601
scientific article