Locally homogeneous pp-waves

Abstract

Abstract. We show that every n-dimensional locally homogeneous pp-wave is a plane wave, provided it is indecomposable and its curvature operator, when acting on 2-forms, has rank greater than one. As a consequence we obtain that indecomposable, Ricci-flat locally homogeneous pp-waves are plane waves. This generalises a classical result by Jordan, Ehlers and Kundt in dimension 4. Several examples show that our assumptions on indecomposability and the rank of the curvature are essential. Background and main resultsA semi-Riemannian manifold (M, g) is homogeneous if it admits a transitive action by a group of isometries. This means that for each pair of points p and q in M there is an isometry of (M, g) that maps p to q. In the spirit of Felix Klein's Erlanger Programm to characterise geometries by their symmetry group, homogeneous manifolds are fundamental building blocks in geometry. Homogeneity is strongly tied to the geometry and the curvature of a manifold. For example, homogeneous Riemannian manifolds are geodescially complete, and, as an example for the link to curvature, we recall the celebrated result that any Ricci-flat homogeneous Riemannian manifold is flat [3]. A weaker version of homogeneity which still guarantees that the manifold looks the same everywhere is local homogeneity: a semi-Riemannian manifold is locally homogeneous if for each pair of points p and q in M there is an isometry defined on a neighbourhood of p that maps p to q.Here we will study local homogeneity for a certain class of Lorentzian manifolds, the so-called pp-waves and the plane waves. Locally, an (n + 2)-dimensional pp-wave admits coordinates (x − , x 1 , . . . , x n , x + ) such that (1.1) g := 2dxwhere H = H(x 1 , . . . , x n , x + ) is a function not depending on x − . For a plane wave, this function is required to be quadratic in the x i 's with x + -dependent coefficients. In general, they are not homogeneous, but they admit a parallel null (i.e. non-zero and light-like) vector field. An invariant definition of pp-waves and plane waves is given as follows: A Lorentzian manifold (M, g) is a pp-wave if it admits a parallel null vector field V ∈ Γ(T M), i.e., V = 0, g(V, V ) = 0 and ∇V = 0, and if its curvature endomorphism R : Λ 2 T M → Λ 2 T M is non-zero and satisfies where V ⊥ := {X ∈ T M | g(X, V ) = 0}. A plane wave is a pp-wave with the additional conditionFour-dimensional pp-waves were discovered in a mathematical context by Brinkmann [7] as one class of Einstein spaces that can be mapped conformally onto each other. In physics, plane waves and pp-waves appeared in general relativity [11], where they continue to play an important role (see for example [6,17] for more references) as metrics for which the Einstein equations become linear and, when they solve these equations, describe the propagation of gravitational waves with flat surfaces as wave fronts. Later Penrose discovered that when "zooming in on null geodesics" every space-times has a plane wave as limit [22]. More recently, the conditions under which the hom...

Exact Parallel Waves in General Relativity

“…If we wish to demand that the wavefront be flat, then this is most succinctly described (see [13]) by considering the Riemann tensor as a map on bivectors (antisymmetric 2-tensors) in Z ⊥ ∧ Z ⊥ , in which case the flatness condition for the wavefront becomes…”

“…24.39] a pp-wave is defined as a Lorentzian manifold admitting a covariantly constant, null vector field (that is, our definition of a parallel wave). Of course such a spacetime would not have planar wavefront, and most articles [12], [13], [15] include also the curvature condition Eq. 12, as is done here.…”

“…Such symmetries can be viewed as generated by vector fields, and the explicit form of these generators is given in [16, Eq. 2.12], for a wave constructed in such a way that it has a finite wave profile 13 . Note also that the gravitational plane wave of Definition 4 above is in fact a special case of our pp-wave spacetimes (Definition 3), and corresponds to the "plane wave" mentioned in the comparison to the linear theory.…”

Optical geometries

“…Homogeneous plane waves and pp-waves have been studied and classified in [9] and [34]. The geodesic completeness of compact pp-waves is proved in [50].…”

Parallel spinors on globally hyperbolic Lorentzian four-manifolds

“…Einstein field equations is well-posed [23,21]. A natural geometric condition to impose on a globally hyperbolic spin four-manifold (M, g) is the existence of a spinor parallel with respect to the Levi-Civita associated to g. Despite the fact that the local structure of Lorentzian fourmanifolds admitting a parallel spinor is well-known since the early days of mathematical general relativity and supergravity [46], see also [12], the more refined global differential geometric and topological aspects of such Lorentzian manifolds have been addressed in the literature only recently [5,8,33,34,35], see also [4,9,14,26,43] for related global problems in Lorentzian geometry. The global differential geometric and topological study of globally hyperbolic Lorentzian manifolds of special holonomy, of which manifolds admitting a parallel irreducible spinor constitute a particular class, was indeed proposed in [33] as a long term research program in the study of Lorentzian manifolds of special geometric type.…”