Alena Pravdova

Classification of the Weyl tensor in higher dimensions

PCT No. PCT/GB88/00981 Sec. 371 Date May 1, 1990 Sec. 102(e) Date May 1, 1990 PCT Filed Nov. 14, 1988 PCT Pub. No. WO89/04245 PCT Pub. Date May 18, 1989.Flexible reinforced polymeric material (30), typically a tape, comprises two elongate flexible support layers (31, 39) and a plurality of lengthwise extending cords (36) secured thereto. The respective edges of the support layers (31, 39) are transversely offset to result in a pair of edge regions and some cords (36) are sandwiched between the support layers while at least one cord (32, 33, 40) and (37, 38, 41) is secured to one of the edge regions. Preferably the two edge regions have corresponding cord arrangements (32, 33, 40) and (37, 38, 41) provided such that the cords (32, 33, 40) of one edge region may be caused to interlock with those (37, 38, 41) at the other edge region.

Vanishing scalar invariant spacetimes in higher dimensions

We study manifolds with Lorentzian signature and prove that all scalar curvature invariants of all orders vanish in a higher dimensional Lorentzian spacetime if and only if there exists an aligned non-expanding, non-twisting, geodesic null direction along which the Riemann tensor has negative boost order.

Bianchi identities in higher dimensions

A higher dimensional frame formalism is developed in order to study implications of the Bianchi identities for the Weyl tensor in vacuum spacetimes of the algebraic types III and N in arbitrary dimension n. It follows that the principal null congruence is geodesic and expands isotropically in two dimensions and does not expand in n − 4 spacelike dimensions or does not expand at all. It is shown that the existence of such principal geodesic null congruence in vacuum (together with an additional condition on twist) implies an algebraically special spacetime. We also use the Myers–Perry metric as an explicit example of a vacuum type D spacetime to show that principal geodesic null congruences in vacuum type D spacetimes do not share this property.

Generalization of the Geroch?Held?Penrose formalism to higher dimensions

Geroch, Held and Penrose invented a formalism for studying spacetimes admitting one or two preferred null directions. This approach is very useful for studying algebraically special spacetimes and their perturbations. In the present paper, the formalism is generalized to higher-dimensional spacetimes. This new formalism leads to equations that are considerably simpler than those of the higher-dimensional Newman-Penrose formalism employed previously. The dynamics of p-form test fields is analyzed using the new formalism and some results concerning algebraically special p-form fields are proved.

All spacetimes with vanishing curvature invariants

All Lorentzian spacetimes with vanishing invariants constructed from the Riemann tensor and its covariant derivatives are determined. A subclass of the Kundt spacetimes results and we display the corresponding metrics in local coordinates. Some potential applications of these spacetimes are discussed.

Higher dimensional Kerr–Schild spacetimes

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 ifthenull KS vector k isgeodetic (or, equivalently, if Tabk 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 thattheycanonlybeofthetypeIIorD,andwecharacterizeopticalpropertiesof 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.

Generalizations of pp-wave spacetimes in higher dimensions

We shall investigate

Algebraic classification of higher dimensional spacetimes based on null alignment

D

WANDs of the black ring

-dimensional Lorentzian spacetimes in which all of the scalar invariants constructed from the Riemann tensor and its covariant derivatives are zero. These spacetimes are higher-dimensional generalizations of

The Newman–Penrose formalism in higher dimensions: vacuum spacetimes with a non-twisting geodetic multiple Weyl aligned null direction

D