Penrose limits of homogeneous spaces

Philip, Simon

doi:10.1016/j.geomphys.2005.08.002

Cited by 30 publications

(36 citation statements)

References 24 publications

(42 reference statements)

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“…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 homogeneity of a Lorentzian manifold is inherited by its Penrose limit were studied extensively by Figueroa-O'Farrill, Meessen and Philip [15,23]. Moreover, having linear Einstein equations and a large number of parallel spinor fields, higher-dimensional plane waves and pp-waves recently appeared as supergravity backgrounds, e.g.…”

Section: Background and Main Resultsmentioning

confidence: 99%

Locally homogeneous pp-waves

Globke

Leistner

2016

Journal of Geometry and Physics

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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...

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Section: Background and Main Resultsmentioning

confidence: 99%

Locally homogeneous pp-waves

Globke

Leistner

2016

Journal of Geometry and Physics

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“…Kajzer [8] proved this result for Lie groups endowed with a left-invariant metric). A corresponding result also holds in the Lorentzian case, provided the space is reductive homogeneous [21]. In the framework of pseudo-Riemannian geometry, homogeneous geodesics aquire a new interest.…”

Section: Introductionmentioning

confidence: 84%

Homogeneous Geodesics of Three-Dimensional Unimodular Lorentzian Lie Groups

Calvaruso

Marinosci

2006

MedJM

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“…In fact, symmetric spaces are particular examples of geodesic orbit spaces, which are manifolds whose geodesics coincide with the orbits of one-parameter subgroups of isometries. In particular, the geodesics are homogeneous, for which there is a well-developed theory of plane-wave limits [15,16] including an explicit and easily implemented Lie-theoretic formula relating the choice of geodesic to the geometry of the plane wave.…”

Section: The Case D =mentioning

confidence: 99%