Weyl Aligned Null Direction Research Articles

On the Goldberg–Sachs theorem in six dimensions

We study the “shear-free” part of the Goldberg–Sachs theorem for type II and D six-dimensional Einstein spacetimes. A spacetime is of the type II or D if it admits a double Weyl aligned null direction (WAND). The Goldberg–Sachs theorem then restricts geometric properties of double WANDs. This restriction can be expressed in terms of constraints on the form of the “optical matrix” defining the expansion, rotation, and shear of the double WAND. We show that a rank-four and rank-three optical matrix is orthogonally similar to one of three canonical forms, reducing the number of free parameters of the optical matrix from ten to five.

On higher dimensional Einstein spacetimes with a non-degenerate double Weyl aligned null direction

We prove that higher dimensional Einstein spacetimes which possess a geodesic, non-degenerate double Weyl aligned null direction (WAND) must additionally possess a second double WAND (thus being of type D) if either: (a) the Weyl tensor obeys (, i.e. the Weyl type is II(abd)); (b) is twistfree. Some comments about an extension of the Goldberg–Sachs theorem to six dimensions are also made.

Algebraic structure of Robinson–Trautman and Kundt geometries in arbitrary dimension

We investigate the Weyl tensor algebraic structure of a fully general family of D-dimensional geometries that admit a non-twisting and shear-free null vector field . From the coordinate components of the curvature tensor we explicitly derive all Weyl scalars of various boost weights. This enables us to give a complete algebraic classification of the metrics in the case when the optically privileged null direction is a (multiple) Weyl aligned null direction (WAND). No field equations are applied, so the results are valid not only in Einsteinʼs gravity, including its extension to higher dimensions, but also in any metric gravitation theory that admits non-twisting and shear-free spacetimes. We prove that all such geometries are of type I(b), or more special, and we derive surprisingly simple necessary and sufficient conditions under which is a double, triple or quadruple WAND. All possible algebraically special types, including the refinement to subtypes, are thus identified, namely II(a), II(b), II(c), II(d), III(a), III(b), N, O, IIi, IIIi, D, D(a), D(b), D(c), D(d), and their combinations. Some conditions are identically satisfied in four dimensions. We discuss both important subclasses, namely the Kundt family of geometries with the vanishing expansion () and the Robinson–Trautman family (, and in particular ). Finally, we apply Einsteinʼs field equations and obtain a classification of all Robinson–Trautman vacuum spacetimes. This reveals fundamental algebraic differences in the and cases, namely that in higher dimensions there only exist such spacetimes of types D(a) ≡ D(abd), D(c) ≡ D(bcd) and O.

Explicit algebraic classification of Kundt geometries in any dimension

We present an algebraic classification, based on the null alignment properties of the Weyl tensor, of the general Kundt class of spacetimes in arbitrary dimension D for which the non-expanding, non-twisting, shear-free null direction k is a (multiple) Weyl aligned null direction (WAND). No field equations are used, so that the results apply not only to Einstein’s gravity and its direct extension to higher dimensions, but also to any metric theory of gravity which admits the Kundt spacetimes. By an explicit evaluation of the Weyl tensor in a natural null frame we demonstrate that all Kundt geometries are of type I(b) or more special, and we derive simple necessary and sufficient conditions under which k becomes a double, triple or quadruple WAND. All possible algebraically special types, including the refinement to subtypes, are identified, namely II(a), II(b), II(c), II(d), III(a), III(b), N, O, IIi, IIIi, D(a), D(b), D(c) and D(d). The corresponding conditions are surprisingly clear and expressed in an invariant geometric form. Some of them are always satisfied in four dimensions. To illustrate our classification scheme, we apply it to the most important subfamilies of the Kundt class, namely the pp-waves, the VSI spacetimes, and the generalization of the Bertotti–Robinson, Nariai, and Plebański–Hacyan direct-product spacetimes of any dimension.

On the Goldberg–Sachs theorem in higher dimensions in the non-twisting case

We study a generalization of the ‘shearfree part’ of the Goldberg–Sachs theorem for Einstein spacetimes admitting a non-twisting multiple Weyl aligned null direction (WAND) ℓ in n ⩾ 6 spacetime dimensions. The form of the corresponding optical matrix ρ is restricted by the algebraically special property in terms of the degeneracy of its eigenvalues. In particular, there necessarily exists at least one multiple eigenvalue, and further constraints arise in various special cases. For example, when ρ is non-degenerate and certain (boost weight zero) Weyl components do not vanish, all eigenvalues of ρ coincide and such spacetimes thus correspond to the Robinson–Trautman class. On the other hand, in certain degenerate cases all non-zero eigenvalues can be distinct. We also present explicit examples of Einstein spacetimes admitting some of the permitted forms of ρ, including examples violating the ‘optical constraint’. The obtained restrictions on ρ are, however, in general not sufficient for ℓ to be a multiple WAND, as demonstrated by a few ‘counterexamples’. We also discuss the geometrical meaning of these restrictions in terms of integrability properties of certain totally null distributions. Finally, we specialize our analysis to the six-dimensional case, where all the permitted forms of ρ are given in terms of just two parameters. In the appendices, some examples are given and certain results pertaining to (possibly) twisting multiple WANDs of Einstein spacetimes are presented.

On algebraically special vacuum spacetimes in five dimensions

Vacuum solutions admitting a hypersurface-orthogonal repeated principal null direction are an important class of 4D algebraically special spacetimes. We investigate the 5D analogues of such solutions: vacuum spacetimes admitting a hypersurface-orthogonal multiple Weyl aligned null direction (WAND). Such spacetimes fall into four families determined by the rank of the 3 × 3 matrix that defines the expansion and shear of the multiple WAND. The rank 3 and rank 0 cases have been studied previously. We investigate the two remaining families. We show how to define coordinates which lead to a considerable simplification of the Einstein equation with cosmological constant. The rank 2 case gives warped product and Kaluza–Klein versions of the 4D Robinson–Trautman solutions as well as some new solutions. The rank 1 case gives product, or analytically continued Schwarzschild, spacetimes.

On a five-dimensional version of the Goldberg–Sachs theorem

Previous work has found a higher dimensional generalization of the ‘geodesic part’ of the Goldberg–Sachs theorem. We investigate the generalization of the ‘shear-free part’ of the theorem. A spacetime is defined to be algebraically special if it admits a multiple Weyl aligned null direction (WAND). The algebraically special property restricts the form of the ‘optical matrix’ that defines the expansion, rotation and shear of the multiple WAND. After working out some general constraints that hold in arbitrary dimensions, we determine necessary algebraic conditions on the optical matrix of a multiple WAND in a five-dimensional Einstein spacetime. We prove that one can choose an orthonormal basis to bring the 3 × 3 optical matrix to one of three canonical forms, each involving two parameters, and we discuss the existence of an ‘optical structure’ within these classes. Examples of solutions corresponding to each form are given. We give an example which demonstrates that our necessary algebraic conditions are not sufficient for a null vector field to be a multiple WAND, in contrast with the 4D result.

A higher dimensional generalization of the geodesic part of the Goldberg–Sachs theorem

In more than four spacetime dimensions, a multiple Weyl-aligned null direction (WAND) need not be geodesic. It is proved that any higher dimensional Einstein spacetime admitting a non-geodesic multiple WAND also admits a geodesic multiple WAND. All five-dimensional Einstein spacetimes admitting a non-geodesic multiple WAND are determined.

Bel–Debever criteria for the classification of the Weyl tensor in higher dimensions

An extension to higher dimensions of the Bel–Debever characterization of the Weyl tensor is considered. This provides algebraic conditions that uniquely determine the multiplicity of a Weyl aligned null direction (WAND), and thus the principal Weyl type, in a frame-independent way. The specification of several ‘subtypes’ is also encompassed by the criteria. We further comment on a Cartan-like geometrical interpretation of WANDs in terms of their invariance properties under parallel transport around infinitesimal loops. As a result, restrictions on the algebraic types permitted in spacetimes that admit a recurrent/covariantly constant vector field are outlined.

Type II Einstein spacetimes in higher dimensions

This paper shows that many of the results derived by Pravda et al (2007 Class. Quantum Grav. 24 4407–28) for higher-dimensional type D Einstein spacetimes can be generalized to all Einstein spacetimes admitting a multiple Weyl-aligned null direction (WAND), the main new result being the extension to include the type II case. Examples of type D Einstein spacetimes admitting non-geodesic multiple WANDs are given in all dimensions greater than 4.

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 if the null KS vector k is geodetic (or, equivalently, if Tabkakb = 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.

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

Vacuum spacetimes admitting a non-twisting geodetic multiple Weyl aligned null direction (WAND) are analysed in arbitrary dimension using a recently developed higher-dimensional Newman–Penrose (NP) formalism. We determine the 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. For types III and N, all metric components are at most quadratic polynomials in r while for types II and D the r-dependence of the metric as well as of the Weyl tensor is determined by an integer m corresponding to the rank of the expansion matrix Sij. It is shown that for non-vanishing expansion, all these spacetimes contain a curvature singularity. As an illustrative example, a shearing expanding type N five-dimensional vacuum solution is also re-derived using higher-dimensional NP formalism. This solution can be, however, identified with a direct product of a known four-dimensional type N metric with an extra dimension.