Type III and N Einstein spacetimes in higher dimensions: General properties

Physical Review D (Impact Factor: 4.86). 05/2010; 82:064043. DOI: 10.1103/PhysRevD.82.064043
Source: arXiv


The Sachs equations governing the evolution of the optical matrix of geodetic WANDs (Weyl aligned null directions) are explicitly solved in n-dimensions in several cases which are of interest in potential applications. This is then used to study Einstein spacetimes of type III and N in the higher dimensional Newman-Penrose formalism, considering both Kundt and expanding (possibly twisting) solutions. In particular, the general dependence of the metric and of the Weyl tensor on an affine parameter r is obtained in a closed form. This allows us to characterize the peeling behaviour of the Weyl "physical" components for large values of r, and thus to discuss, e.g., how the presence of twist affects polarization modes, and qualitative differences between four and higher dimensions. Further, the r-dependence of certain non-zero scalar curvature invariants of expanding spacetimes is used to demonstrate that curvature singularities may generically be present. As an illustration, several explicit type N/III spacetimes that solve Einstein's vacuum equations (with a possible cosmological constant) in higher dimensions are finally presented. Comment: 19 pages

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    • "Scalar curvature invariants of vacuum Einstein equations were also studied [10]. There are also attempts to generalise vacuum Einstein solutions of type N space-times to higher dimensions [11]. "
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    ABSTRACT: In this work we use Newman Penrose two-spinor formalism to derive decoupled equations for vector fields in Petrov type N space-times. We first evaluate vector wave equation by representing four-vectors by one complex and two real scalars. This approach leads to a decoupled second order differential equation for one of the real scalars if and only if the space-time is of type N. The solution for this scalar can --in principle-- be used to derive decoupled equations for the other scalars. We directly apply the results for the vector wave equation to Proca equation for massive vector fields. We evaluate Maxwell equations in terms of Newman Penrose complex scalars of electromagnetism. We derive a decoupled second order differential equation for $\phi_0$, valid in type N space-times. Substituting any solution for $\phi_0$ in Maxwell equations, leads to two first order differential equations for $\phi_1$. We show that these first order equations identically satisfy integrability conditions. Thus, any solution for $\phi_0$ guarantees the existence of a solution for $\phi_1$, via either of the first order differential equations.
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    • "Moreover, the relations (B6) are equivalent to the constraints (15), (16). Also, in [62] [63] [76] [83] the notation "
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    ABSTRACT: We present a general method which can be used for geometrical and physical interpretation of an arbitrary spacetime in four or any higher number of dimensions. It is based on the systematic analysis of relative motion of free test particles. We demonstrate that local effect of the gravitational field on particles, as described by equation of geodesic deviation with respect to a natural orthonormal frame, can always be decomposed into a canonical set of transverse, longitudinal and Newton-Coulomb-type components, isotropic influence of a cosmological constant, and contributions arising from specific matter content of the universe. In particular, exact gravitational waves in Einstein's theory always exhibit themselves via purely transverse effects with D(D-3)/2 independent polarization states. To illustrate the utility of this approach we study the family of pp-wave spacetimes in higher dimensions and discuss specific measurable effects on a detector located in four spacetime dimensions. For example, the corresponding deformations caused by a generic higher-dimensional gravitational waves observed in such physical subspace, need not be tracefree.
    Physical review D: Particles and fields 01/2012; 85(4). DOI:10.1103/PhysRevD.85.044057 · 4.86 Impact Factor
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    • "that U ⊂ KCSI. ppIII RF are examples of spacetimes which are KCSI (and VSI) but not U. Notice however that QG E ⊂ CSI since examples of QG E metrics with non-vanishing expansion mentioned in this section have in general non-trivial curvature invariants [21] "
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    ABSTRACT: We study exact vacuum solutions to quadratic gravity (QG) of the Weyl types N and III. We show that in an arbitrary dimension all Einstein spacetimes of the Weyl type N with an appropriately chosen effective cosmological constant Λ are exact solutions to QG and we refer to explicitly known metrics within this class. For type III Einstein spacetimes, an additional constraint follows from the field equations of QG and examples of spacetimes obeying such constraint are given. However, type III pp waves do not satisfy this constraint and thus do not solve QG. For type N, we also study a wider class of spacetimes admitting a pure radiation term in the Ricci tensor. In contrast to the Einstein case, the field equations of generic QG determine optical properties of the geometry and restrict such exact solutions to the Kundt class. We provide examples of these metrics.
    Physical review D: Particles and fields 07/2011; 84(2). DOI:10.1103/PhysRevD.84.024047 · 4.86 Impact Factor
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