Counting the number of Killing vectors in a 3D spacetime

Abstract

We devise an algorithm which allows one to count the number of Killing vectors for a Lorentzian manifold of dimension 3. Our algorithm relies on the principal traces of powers of the Ricci tensor and branches intricately according to the values of differential invariants arising from the compatibility conditions of the Killing equation. As illustrating examples, we classify the Lifshitz and pp-wave spacetimes into a hierarchy based on their level of symmetry. A complete classification of spacetimes admitting 4 Killing vectors is also presented.

All the three dimensional Lorentzian metrics admitting three Killing vectors

“…In terms of the Ricci rotation coefficients {κ i , η i , τ i } (i = u, v, e) defined in (A13) and frame components of the Ricci tensor (A16), this only occurs for (see eqs. (3.19), (3.22) and (3.25) in [5])…”

“…From the class 3 condition, the eigenvalues (α, β) are constants. The second obstruction matrix reads (4.59) in [5], which vanishes provided {κ 3 , η 1 , η 2 , η 3 , τ 2 , τ 3 } = constants . (9.3)…”

“…this paper, since this epithet represents how the isometry group acts on the manifold. The classification of class 3 in [5] relies on the property of Segre classification for the traceless Ricci tensor S a b . Our result obtained in this paper may be generalized into higher dimensions, if we properly find out the obstruction matrices.…”

“…Ricci [9] and Bianchi [10] have proved that the next maximum number of KVs is four, and have carried out the complete classification of all possible local forms of g with four KVs. The pursuit for the Lorentzian counterpart has been implemented by Kručkovič [11] and recently refined in [5] by making use of obstruction criteria for KVs (for the underlying geometric analysis of these geometries, see [12]). This is another intriguing opportunity offered by recent developments in [4,5].…”

“…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 k describes the two-dimensional maximally symmetric space with a constant curvature k = 0, ±1, and d Σ2 k corresponds to its Lorentzian counterpart.…”