Structure of second-order symmetric Lorentzian manifolds

Abstract

Abstract. Second-order symmetric Lorentzian spaces, that is to say, Lorentzian manifolds with vanishing second derivative ∇∇R ≡ 0 of the curvature tensor R, are characterized by several geometric properties, and explicitly presented. Locally, they are a product M = M 1 × M 2 where each factor is uniquely determined as follows: M 2 is a Riemannian symmetric space and M 1 is either a constant-curvature Lorentzian space or a definite type of plane wave generalizing the Cahen-Wallach family. In the proper case (i.e., ∇R = 0 at some point), the curvature tensor turns out to be described by some local affine function which characterizes a globally defined parallel lightlike direction. As a consequence, the corresponding global classification is obtained, namely: any complete second-order symmetric space admits as universal covering such a product M 1 × M 2 . From the technical point of view, a direct analysis of the second-symmetry partial differential equations is carried out leading to several results of independent interest relative to spaces with a parallel lightlike vector field -the so-called Brinkmann spaces.

On the (non-)uniqueness of the Levi-Civita solution in the Einstein–Hilbert–Palatini formalism

“…These connections appear only when there exists a parallel Pµ and, then, one can write locally Pµ = dx and Bµ = dx/(1−x). When Pµ is not lightlike then the spacetime decomposes as a product N ×R and x can be regarded as a natural coordinate on R. When it is lightlike, then the spacetime becomes a Brinkmann space (see [34]) and x can be regarded as a natural lightcone coordinate u. This shows the exceptionality of these connections "physically indistinguishable to Levi-Civita, but not Palatini.…”

Curvature properties of Gödel metric

“…We refer to [49,50] and references therein for results on manifolds satisfying (7). A semi-Riemannian manifold (M, g), n ≥ 3, is said to be pseudosymmetric [51] if the tensor R · R and the Tachibana tensor Q(g, R) are linearly dependent at every point of M .…”

Rigidity of geodesic completeness in the Brinkmann class of gravitational wave spacetimes

“…For each k ∈ N, we may use Lemma 4.4 with F = −Ĥ and pick a piecewise smooth curve z k : t ∈ [0, 1] → B R k (0) ⊂ C ≡ R 2 such that (i) z k (0) = z k (1) = (0, 0) and z(t k ) = p k for some t k ∈ (0, 1), (ii) -1 0Ĥ (z k (t))dt ≥ − 1 5Ĥ (p k ), and (iii) ż k (t) 2 dt ≤ 50π 2 R 2 k . Using these curves, we may define for each k ∈ N, the piecewise curve Z k : t ∈ [0, 1] → C given by Z k (t) = e iα∆t z k (t) for each t ∈ [0, 1], where α is defined in (6) and it is constant due the autonomous character of (M, g). Observe that, from construction,H(α∆t, Z k (t)) =Ĥ(z k (t)).…”

“…After finishing this paper, we became aware of another, much more general context in which the assumptions in Theorem 1.2 are natural 2 , namely in a local classification scheme, recently carried out by M. Mars and J.M.M. Senovilla [20] (see also [6]), for a class of algebraically special spacetimes which includes both Kerr and Brinkmann spacetimes (as well as generalizations of these). More specifically, in [20] the authors investigate 4-dimensional Einstein spacetimes (M, g) endowed with a Killing vector field Y ∈ Γ(T M ) and satisfying a certain "alignment" relation between the Weyl tensor and the curl of Y.…”