Type III and N Einstein spacetimes in higher dimensions: General properties (original) (raw)
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Classical and Quantum Gravity, 2008
Vacuum spacetimes admitting a non-twisting geodetic multiple Weyl aligned null direction (WAND) are analyzed in arbitrary dimension using recently developed higher-dimensional Newman-Penrose (NP) formalism. We determine dependence of the metric and of the Weyl tensor on the affine parameter r along null geodesics generated by the WAND for type III and N spacetimes and for a special class of type II and D spacetimes, containing e.g. Schwarzschild-Tangherlini black holes and black strings and branes.
Spacetimes of Weyl and Ricci type N in higher dimensions
Classical and Quantum Gravity, 2016
We study the geometrical properties of null congruences generated by an aligned null direction of the Weyl tensor (WAND) in spacetimes of the Weyl and Ricci type N (possibly with a non-vanishing cosmological constant) in an arbitrary dimension. We prove that a type N Ricci tensor and a type III or N Weyl tensor have to be aligned. In such spacetimes, the multiple WAND has to be geodetic. For spacetimes with type N aligned Weyl and Ricci tensors, the canonical form of the optical matrix in the twisting and non-twisting case is derived and the dependence of the Weyl and the Ricci tensors and Ricci rotation coefficients on the affine parameter of the geodetic null congruence generated by the WAND is obtained.
Newman-Penrose formalism in higher dimensions: vacuum spacetimes with a non-twisting multiple WAND
2008
Vacuum spacetimes admitting a non-twisting geodetic multiple Weyl aligned null direction (WAND) are analyzed in arbitrary dimension using recently developed higher-dimensional Newman-Penrose (NP) formalism. We determine dependence of the metric and of the Weyl tensor on the affine parameter r along null geodesics generated by the WAND for type III and N spacetimes and for a special class of type II and D spacetimes, containing e.g. Schwarzschild-Tangherlini black holes and black strings and branes.
On higher dimensional Einstein spacetimes with a warped extra dimension
Classical and Quantum Gravity, 2011
We study higher dimensional Einstein spacetimes that can be mapped conformally on other Einstein spacetimes. These admit a simple warped line element (with one extra dimension) that was originally introduced by Brinkmann and that has subsequently appeared in various contexts to describe, e.g., different braneworld models or warped black strings. After clarifying the relation between the general Brinkmann metric and other more specific coordinate systems, we analyze the algebraic type of the Weyl tensor of the solutions. In particular, we describe the relation between Weyl aligned null directions (WANDs) of the lower dimensional Einstein slices and of the full spacetime, which in some cases can be algebraically more special. Possible spacetime singularities introduced by the warp factor are determined via a study of scalar curvature invariants and of Weyl components measured by geodetic observers. Finally, we illustrate how Brinkmann's metric can be employed to generate new solutions by presenting the metric of spinning and accelerating black strings in five dimensional anti-de Sitter
Type D Einstein spacetimes in higher dimensions
Classical and Quantum Gravity, 2007
We show that all static spacetimes in higher dimensions n > 4 are necessarily of Weyl types G, I i , D or O. This also applies to stationary spacetimes provided additional conditions are fulfilled, as for most known black hole/ring solutions. (The conclusions change when the Killing generator becomes null, such as at Killing horizons, on which we briefly comment.) Next we demonstrate that the same Weyl types characterize warped product spacetimes with a onedimensional Lorentzian (timelike) factor, whereas warped spacetimes with a two-dimensional Lorentzian factor are restricted to the types D or O. By exploring algebraic consequences of the Bianchi identities, we then analyze the simplest non-trivial case from the above classes-type D vacuum spacetimes, possibly with a cosmological constant, dropping, however, the assumptions that the spacetime is static, stationary or warped. It is shown that for 'generic' type D vacuum spacetimes (as defined in the text) the corresponding principal null directions are geodetic in arbitrary dimension (this in fact also applies to type II spacetimes). For n 5, however, there may exist particular cases of type D vacuum spacetimes which admit non-geodetic multiple principal null directions and we explicitly present such examples in any n 7. Further studies are restricted to five dimensions, where the type D Weyl tensor is fully described by a 3 × 3 real matrix ij . In the case with 'twistfree' (A ij = 0) principal null geodesics we show that in a 'generic' case ij is symmetric and eigenvectors of ij coincide with eigenvectors of the expansion matrix S ij providing us thus in general with three preferred spacelike directions of the spacetime. Similar results are also obtained when relaxing the twistfree condition and assuming instead that ij is symmetric. The five-dimensional Myers-Perry black hole and Kerr-NUT-AdS metrics in arbitrary dimension are also briefly studied as specific illustrative examples of type D vacuum spacetimes.
Higher dimensional Kerr–Schild spacetimes
Classical and Quantum Gravity, 2009
We investigate general properties of Kerr-Schild (KS) metrics in n > 4 spacetime dimensions. First, we show that the Weyl tensor is of type II or more special if the null KS vector k is geodetic (or, equivalently, if T ab k a k b = 0). We subsequently specialize to vacuum KS solutions, which naturally split into two families of non-expanding and expanding metrics. After demonstrating that non-expanding solutions are equivalent to the known class of vacuum Kundt solutions of Weyl type N, we analyze expanding solutions in detail. We show that they can only be of the type II or D, and we characterize optical properties of the multiple Weyl aligned null direction (WAND) k. In general, k has caustics corresponding to curvature singularities. In addition, it is generically shearing. Nevertheless, we arrive at a possible 'weak' n > 4 extension of the Goldberg-Sachs theorem, limited to the KS class, which matches previous conclusions for general type III/N solutions. In passing, properties of Myers-Perry black holes and black rings related to our results are also briefly discussed.
Einstein–Weyl spaces and near-horizon geometry
Classical and Quantum Gravity, 2017
We show that a class of solutions of minimal supergravity in five dimensions is given by lifts of three-dimensional Einstein-Weyl structures of hyper-CR type. We characterise this class as most general near-horizon limits of supersymmetric solutions to the fivedimensional theory. In particular we deduce that a compact spatial section of a horizon can only be a Berger sphere, a product metric on S 1 × S 2 or a flat three-torus. We then consider the problem of reconstructing all supersymmetric solutions from a given near-horizon geometry. By exploiting the ellipticity of the linearised field equations we demonstrate that the moduli space of transverse infinitesimal deformations of a nearhorizon geometry is finite-dimensional.
Refinements of the Weyl tensor classification in five dimensions
Classical and Quantum Gravity, 2012
We refine the null alignment classification of the Weyl tensor of a five-dimensional spacetime. The paper focusses on the algebraically special alignment types N, III, II and D, while types I and G are briefly discussed. A first refinement is provided by the notion of spin type of the components of highest boost weight. Second, we analyze the Segre types of the Weyl operator acting on bivector space and examine the intersection with the spin type classification. We present a full treatment for types N and III, and illustrate the classification from different viewpoints (Segre type, rank, spin type) for types II and D, paying particular attention to possible nilpotence, which is a new feature of higher dimensions. We also point out other essential differences with the four-dimensional case. In passing, we exemplify the refined classification by mentioning the special subtypes associated to certain important spacetimes, such as Myers-Perry black holes, black strings, Robinson-Trautman spacetimes, and purely electric/magnetic type D spacetimes.
Weyl covariant theories of gravity in 3-dimensional Riemann–Cartan–Weyl space-times
Classical and Quantum Gravity, 2019
We discuss locally Weyl (scale) covariant generalisation of quadratic curvature gravity theory in three dimensions using Riemann-Cartan-Weyl space-times. We show that this procedure of Weyl gauging yields a consistent generalisation for a particular class of quadratic curvature gravity theories which includes the New Massive Gravity theory.
Weyl type N solutions with null electromagnetic fields in the Einstein-Maxwell p-form theory
General Relativity and Gravitation, 2017
We consider d-dimensional solutions to the electrovacuum Einstein-Maxwell equations with the Weyl tensor of type N and a null Maxwell (p + 1)-form field. We prove that such spacetimes are necessarily aligned, i.e. the Weyl tensor of the corresponding spacetime and the electromagnetic field share the same aligned null direction (AND). Moreover, this AND is geodetic, shear-free, non-expanding and non-twisting and hence Einstein-Maxwell equations imply that Weyl type N spacetimes with a null Maxwell (p + 1)-form field belong to the Kundt class. Moreover, these Kundt spacetimes are necessarily CSI and the (p + 1) form is V SI. Finally, a general coordinate form of solutions and a reduction of the field equations are discussed.