Lorentzian symmetric spaces

Cahen, Michel; Wallach, Nolan R.

doi:10.1090/s0002-9904-1970-12448-x

Cited by 136 publications

(172 citation statements)

References 2 publications

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“…(For constant A ij , this is a Lorentzian symmetric space [40,6].) In this case, 12) and the tidal force is attractive for positive eigenvalues of A ij and repulsive for negative eigenvalues.…”

Section: The Spacetimementioning

confidence: 99%

“…As far as we are aware, this is the first study of classical stability for non-static, though possibly stationary, spacetimes. Although we are only able to explicitly solve the relevant equations for the plane waves with constant profiles (the so-called Cahen-Wallach spaces [40]), this class of spacetimes certainly includes cases for which the (mass) 2 of the worldsheet scalars is negative. We will show that this class of plane waves does not exhibit classical instabilities.…”

Section: Introductionmentioning

confidence: 99%

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String theory and the classical stability of plane waves

2003

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The presence of fields with negative mass-squared typically leads to some form of instability in standard field theories. The observation that, at least in the light-cone gauge, strings propagating in plane wave spacetimes can have worldsheet scalars with such tachyon-like masses suggests that the supergravity background may itself be unstable. To address this issue, we perform a perturbative analysis around the type IIB vacuum plane wave, the solution which most obviously generates worldsheet scalars with negative mass-squared. We argue that this background is perturbatively stable.

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Section: The Spacetimementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

String theory and the classical stability of plane waves

2003

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“…In this case the metric describes a Lorentzian symmetric space G/K with K = R 9 and G a (solvable) Lie group depending on A [22,21]. Solutions of this kind are space-times with a null (F 2 4 = 0) homogeneous flux and were referred to as Hpp-waves in [21].…”

Section: Pp-waves In Supergravitymentioning

confidence: 99%

Strings in plane wave backgrounds

Pankiewicz

2003

Fortschritte der Physik

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I review aspects of string theory on plane wave backgrounds emphasising the connection to gauge theory given by the BMN correspondence. Topics covered include the Penrose limit and its role in deriving the BMN duality from AdS/CFT, light-cone string field theory in the maximally supersymmetric plane wave and extensions of the correspondence to less supersymmetric backgrounds.a Based on the author's PhD thesis,

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“…Plane waves satisfy the vacuum Einstein equations, i.e., are Ricci-flat if and only if S(x + ) is traceless for all x + . A subclass of plane waves are the solvable Lorentzian symmetric spaces, called Cahen-Wallach spaces after [8]. As symmetric spaces, they satisfy ∇R = 0 which forces the matrix S to be constant.…”

Section: Pp-waves and Plane Wavesmentioning

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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Lorentzian symmetric spaces

Cited by 136 publications

References 2 publications

String theory and the classical stability of plane waves

String theory and the classical stability of plane waves

Strings in plane wave backgrounds

Locally homogeneous pp-waves

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