All the three dimensional Lorentzian metrics admitting three Killing vectors

Abstract

We obtain all the three-dimensional Lorentzian metrics which admit three Killing vectors. The classification has been done with the aid of the formalism which exploits the obstruction criteria for the Killing equations recently developed by present authors. The current classification method does not rely on the transitivity property of the isometry group. It turns out that the Lorentzian manifold harbors a much richer spectrum of metrics with various Segre types, compared to the Riemannian case. PACS numbers:Theorem Let (M, g) be a three-dimensional Lorentzian manifold. If it admits exactly three linearly independent Killing vectors, the manifold is locally isometric to any one of the metrics described in table I.An inspection of the explicit second obstruction matrices in [5] shows that the three KVs appear for "class 2" and "class 3" in the terminology therein. We shall moniker class 2 as "inhomogeneous" and class 3 as "homogeneous" in

An alternative to the Simon tensor

“…The analysis of [55] follows in part the spirit of the Cartan-Karlhede program, according to which the equivalence problem and the isometry group of a given metric can be addressed by the Riemann tensor and its covariant derivatives. Recently, this Cartan-Karlhede program has been streamlined substantially into a practical form by [56][57][58] in three dimensions. It is then a promising route to examine generalizations of these algorithms into 3 + 1 dimensions, for the labeling the Kerr-NUT solution under more relaxed conditions.…”

An alternative to the Simon tensor

“…The analysis of [51] follows in part the spirit of the Cartan-Karlhede program, according to which the equivalence problem and the isometry group of a given metric can be addressed by the Riemann tensor and its covariant derivatives. Recently, this Cartan-Karlhede program has been streamlined substantially into a practical form by [52][53][54] in three dimensions. It is then a promising route to examine generalizations of these algorithms into 3 + 1 dimensions, for the labeling the Kerr-NUT solution under more relaxed conditions.…”

Homogeneous three-dimensional Lorentzian spaces

“…Consequently, we obtain: Theorem 10. Let g be a three-dimensional Lorentzian metric admitting a group G 4 of isometries with S 2 = 0, and H, r, and ρ the Ricci concomitants defined in (10), (11) and (12). Then: , where τ ± = (1 − ρ) ± ρ 2 − 4ρ.…”

“…If Ω = 0, then r = 0 and ∇ = ζ ⊗ . In this case the Ricci tensor and its derivative define just a constant scalar ρ = ζ 2 that can be computed as (10) in terms of the Ricci tensor and its covariant derivative. We summarize this situation in the following.…”

“…Recently, a great effort has been made to deepen our knowledge of these spaces. Note the paper by Nozawa and Tomoda [10] (see also references therein) where all the (not necessarily homogeneous) three-dimensional Lorentzian metrics admitting three Killing vectors have been explicitly obtained. In a previous paper [11] the same authors develop an algorithm based on the obstruction criteria for the Killing equations to determine the dimension of the isometry group admitted by a three-dimensional metric in both the Lorentzian and the Riemannian cases.…”