Marcello Ortaggio

Robinson?Trautman spacetimes in higher dimensions

As an extension of the Robinson?Trautman solutions of D = 4 general relativity, we investigate higher dimensional spacetimes which admit a hypersurface orthogonal, non-shearing and expanding geodesic null congruence. Einsteins field equations with an arbitrary cosmological constant and possibly an aligned pure radiation are fully integrated so that the complete family is presented in a closed explicit form. As a distinctive feature of higher dimensions, the transverse spatial part of the general line element must be a Riemannian Einstein space, but it is otherwise arbitrary. On the other hand, the remaining part of the metric is?perhaps surprisingly?not so rich as in the standard D = 4 case, and the corresponding Weyl tensor is necessarily of algebraic type D. While the general family contains (generalized) static Schwarzschild?Kottler?Tangherlini black holes and extensions of the Vaidya metric, there is no analogue of important solutions such as the C-metric.

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.

Algebraic classification of higher dimensional spacetimes based on null alignment

We review recent developments and applications of the classification of the Weyl tensor in higher dimensional Lorentzian geometries. First, we discuss the general setup, i.e. main definitions and methods for the classification, some refinements and the generalized Newman–Penrose and Geroch–Held–Penrose formalisms. Next, we summarize general results, such as a partial extension of the Goldberg–Sachs theorem, characterization of spacetimes with vanishing (or constant) curvature invariants and the peeling behaviour in asymptotically flat spacetimes. Finally, we discuss certain invariantly defined families of metrics and their relation with the Weyl tensor classification, including Kundt and Robinson–Trautman spacetimes; the Kerr–Schild ansatz in a constant-curvature background; purely electric and purely magnetic spacetimes; direct and (some) warped products; and geometries with certain symmetries. To conclude, some applications to quadratic gravity are also overviewed.

Explicit Kundt type II and N solutions as gravitational waves in various type D and O universes

A particular yet large class of non-diverging solutions which admits a cosmological constant, electromagnetic field, pure radiation and/or general non-null matter component is explicitly presented. These spacetimes represent exact gravitational waves of arbitrary profiles which propagate in background universes such as Minkowski, conformally flat (anti-)de Sitter, Edgar–Ludwig, Bertotti–Robinson and type D (anti-)Nariai or Plebanski–Hacyan spaces, and their generalizations. All possibilities are discussed and interpreted using a unifying simple metric form. Sandwich and impulsive waves propagating in the above background spaces with different geometries and matter content can easily be constructed. New solutions are identified, e.g. type D pure radiation or explicit type II electrovacuum waves in the (anti-)Nariai universe. It is also shown that, in general, there are no conformally flat Einstein–Maxwell fields with a non-vanishing cosmological constant.

Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension

We consider time reversal transformations to obtain twofold orthogonal splittings of any tensor on a Lorentzian space of arbitrary dimension n. Applied to the Weyl tensor of a spacetime, this leads to a definition of its electric and magnetic parts relative to an observer (defined by a unit timelike vector field u), in any dimension. We study the cases where one of these parts vanishes in detail, i.e., purely electric (PE) or magnetic (PM) spacetimes. We generalize several results from four to higher dimensions and discuss new features of higher dimensions. For instance, we prove that the only permitted Weyl types are G, Ii and D, and discuss the possible relation of u with the Weyl aligned null directions (WANDs); we provide invariant conditions that characterize PE/PM spacetimes, such as Bel–Debever-like criteria, or constraints on scalar invariants, and connect the PE/PM parts to the kinematic quantities of u; we present conditions under which direct product spacetimes (and certain warps) are PE/PM, which enables us to construct explicit examples. In particular, it is also shown that all static spacetimes are necessarily PE, while stationary spacetimes (such as spinning black holes) are in general neither PE nor PM. Whereas ample classes of PE spacetimes exist, PM solutions are elusive; specifically, we prove that PM Einstein spacetimes of type D do not exist, in any dimension. Finally, we derive corresponding results for the electric/magnetic parts of the Riemann tensor, which is useful when considering spacetimes with matter fields, and moreover leads to first examples of PM spacetimes in higher dimensions. We also note in passing that PE/PM Weyl (or Riemann) tensors provide examples of minimal tensors, and we make the connection hereof with the recently proved alignment theorem (Hervik 2011 Class. Quantum Grav. 28 215009). This in turn sheds new light on the classification of the Weyl tensors based on null alignment, providing a further invariant characterization that distinguishes the (minimal) types G/I/D from the (non-minimal) types II/III/N.

Radiation from accelerated black holes in an anti-de Sitter universe

Radiative properties of gravitational and electromagnetic fields generated by uniformly accelerated charged black holes in asymptotically de Sitter spacetime are studied by analyzing the C-metric exact solution of the Einstein-Maxwell equations with a positive cosmological constant. Its global structure and physical properties are thoroughly discussed. We explicitly find and describe the specific pattern of radiation which exhibits the dependence of the fields on a null direction along which the (spacelike) conformal infinity is approached. This directional characteristic of radiation supplements the peeling behavior of the fields near the infinity. The interpretation of the solution is achieved by means of various coordinate systems, and suitable tetrads. Relation to the Robinson-Trautman framework is also presented.

Impulsive waves in the Nariai universe

A new class of exact solutions is presented which describes impulsive waves propagating in the Nariai universe. It is constructed using a six-dimensional embedding formalism adapted to the background. Due to the topology of the latter, the wave front consists of two non-expanding spheres. Special sub-classes representing pure gravitational waves (generated by null particles with an arbitrary multipole structure) or shells of null dust are analyzed in detail. Smooth isometries of the metrics are briefly discussed. Furthermore, it is shown that the considered solutions are impulsive members of a more general family of radiative Kundt spacetimes of type-II. A straightforward generalization to impulsive waves in the anti-Nariai and Bertotti-Robinson backgrounds is described. For a vanishing cosmological constant and electromagnetic field, results for well known impulsive pp-waves are recovered.

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 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.

Impulsive waves in electrovac direct product spacetimes with Λ

A complete family of non-expanding impulsive waves in spacetimes which are the direct product of two 2-spaces of constant curvature is presented. In addition to previously investigated impulses in Minkowski, (anti-)Nariai and Bertotti–Robinson universes, a new explicit class of impulsive waves which propagate in the exceptional electrovac Plebanski–Hacyan spacetimes with a cosmological constant Λ is constructed. In particular, pure gravitational waves generated by null particles with an arbitrary multipole structure are described. The metrics are impulsive members of a more general family of the Kundt spacetimes of type II. The well-known pp-waves are recovered for Λ = 0.