Non-uniqueness of the metric in Lorentzian manifolds
This paper generalises the work of the reviewer [Gen. Relativ. Gravitation 15, 581-589 (1983; Zbl 0514.53018) 20, No. 4, 399-406 (1988; Zbl 0642.53027) and J. Math. Phys. 32, No. 1, 181-187 (1991; Zbl 0731.53024)] and involves the question of how many metrics \(g'\) on a manifold \(M\) are compatible with a Levi-Civita connection obtained from a metric \(g\) on \(M\). The above three papers were essentially concerned with the 4-dimensional Lorentzian signature of general relativity and the present paper involves the general \(n\)-dimensional Lorentz case. The main result obtained here shows that the 4-dimensional case is typical. It is couched in holonomy terms and involves Wu's generalisation of the de Rham (positive definite) decomposition. This generalisation distinguishes between the cases when a non-null (or non-degenerate) holonomy invariant subspace (a de Rham decomposition) exists and when it doesn't. The present paper shows that if \(M\) is simply connected and admits a Lorentz metric \(g\) with Levi-Civita connection \(\Gamma\) then an alternative metric \(g'\) compatible with \(\Gamma\) and not a constant multiple of \(g\) exists if either a de Rham decomposition exists, or if \(M\) admits a covariantly constant (parallel) null vector field. The forms of alternative metrics are discussed and are essentially those of the earlier references above extended to higher dimensions. The proof involves the idea of a submersive connection.
scientific article; zbMATH DE number 509101