Vojtech Pravda

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.

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.

Generalizations of pp-wave spacetimes in higher dimensions

We shall investigate

WANDs of the black ring

D

Curvature invariants in type-N spacetimes

-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

On a Five-Dimensional Version of the Goldberg-Sachs Theorem

-dimensional pp-wave spacetimes, which have been of interest recently in the context of string theory in curved backgrounds in higher dimensions.

Exact solutions to quadratic gravity

Necessary conditions for various algebraic types of the Weyl tensor in higher dimensions are determined. These conditions are then used to find Weyl aligned null directions for the black ring solution. It is shown that the black ring solution is algebraically special, of type Ii, while locally on the horizon the type is II. One exceptional subclass – the Myers-Perry solution – is of type D.