Jiří Podolský

A NEW LOOK AT THE PLEBAŃSKI–DEMIAŃSKI FAMILY OF SOLUTIONS

The Plebanski-Demianski metric, and those that can be obtained from it by tak- ing coordinate transformations in certain limits, include the complete family of space-times of type D with an aligned electromagnetic field and a possibly non-zero cosmological constant. Starting with a new form of the line element which is better suited both for physical interpretation and for identifying different subfamilies, we review this entire family of solutions. Our metric for the expanding case explicitly includes two parameters which represent the acceleration of the sources and the twist of the repeated principal null congruences, the twist being directly related to both the angular velocity of the sources and their NUT-like properties. The non- expanding type D solutions are also identified. All special cases are derived in a simple and transparent way.

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.

Gravitational waves in vacuum spacetimes with cosmological constant. I. Classification and geometrical properties of nontwisting type N solutions

All nontwisting Petrov-type N solutions of vacuum Einstein field equations with cosmological constant Λ are summarized. They are shown to belong either to the nonexpanding Kundt class or to the expanding Robinson–Trautman class. Invariant subclasses of each class are defined and the corresponding metrics are given explicitly in suitable canonical coordinates. Relations between the subclasses and their geometrical properties are analyzed. In the subsequent paper these solutions are interpreted as exact gravitational waves propagating in de Sitter or anti-de Sitter spacetimes.

General Kundt spacetimes in higher dimensions

We investigate a general metric of the Kundt class of spacetimes in higher dimensions. Geometrically, it admits a non-twisting, non-shearing and non-expanding geodesic null congruence. We calculate all components of the curvature and Ricci tensors, without assuming any specific matter content, and discuss algebraic types and main geometric constraints imposed by general Einsteins field equations. We explicitly derive Einstein–Maxwell equations, including an arbitrary cosmological constant, in the case of vacuum or possibly an aligned electromagnetic field. Finally, we introduce canonical subclasses of the Kundt family and we identify the most important special cases, namely generalized pp-waves, VSI or CSI spacetimes, and gyratons.

Interpreting the C-metric

The basic properties of the C-metric are well known. It describes a pair of causally separated black holes which accelerate in opposite directions under the action of forces represented by conical singularities. However, these properties can be demonstrated much more transparently by making use of recently developed coordinate systems for which the metric functions have a simple factor structure. These enable us to obtain explicit Kruskal–Szekeres-type extensions through the horizons and construct two-dimensional conformal Penrose diagrams. We then combine these into a three-dimensional picture which illustrates the global causal structure of the spacetime outside the black hole horizons. Using both the weak field limit and some invariant quantities, we give a direct physical interpretation of the parameters which appear in the new form of the metric. For completeness, relations to other familiar coordinate systems are also discussed.

Accelerating and rotating black holes

An exact solution of Einsteins equations which represents a pair of accelerating and rotating black holes (a generalized form of the spinning C-metric) is presented. The starting point is a form of the Plebanski–Demianski metric which, in addition to the usual parameters, explicitly includes parameters which describe the acceleration and angular velocity of the sources. This is transformed to a form which explicitly contains the known special cases for either rotating or accelerating black holes. Electromagnetic charges and a NUT parameter are included, the relation between the NUT parameter l and the Plebanski–Demianski parameter n is given, and the physical meaning of all parameters is clarified. The possibility of finding an accelerating NUT solution is also discussed.

Gravitational waves in vacuum spacetimes with cosmological constant. II. Deviation of geodesics and interpretation of nontwisting type N solutions

In a suitably chosen essentially unique frame tied to a given observer in a general spacetime, the equation of geodesic deviation can be decomposed into a sum of terms describing specific effects: isotropic (background) motions associated with the cosmological constant, transverse motions corresponding to the effects of gravitational waves, longitudinal motions and Coulomb-type effects. Conditions under which the frame is parallelly transported along a geodesic are discussed. Suitable coordinates are introduced and an explicit coordinate form of the frame is determined for spacetimes admitting a nontwisting null congruence. Specific properties of all nontwisting type N vacuum solutions with cosmological constant Λ (nonexpanding Kundt class and expanding Robinson–Trautman class) are then analyzed. It is demonstrated that these spacetimes can be understood as exact transverse gravitational waves of two polarization modes “+” and “×,” shifted by π/4, which propagate “on” Minkowski, de Sitter or anti-de Sitter backgrounds. It is also shown that the solutions with Λ>0 may serve as exact demonstrations of the cosmic “no-hair” conjecture in radiative spacetimes with no symmetry.

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.

Generalized Kundt waves and their physical interpretation

We present the complete family of spacetimes with a non-expanding, shear-free, twist-free, geodesic principal null congruence (Kundt waves) that are of algebraic type III and for which the cosmological constant (Λc) is non-zero. The possible presence of an aligned pure radiation field is also assumed. These spacetimes generalize the known vacuum solutions of type N with arbitrary Λc and type III with Λc = 0. It is shown that there are two, one and three distinct classes of solutions when Λc is respectively zero, positive and negative. The wave surfaces are plane, spherical or hyperboloidal in Minkowski, de Sitter or anti-de Sitter backgrounds respectively, and the structure of the family of wave surfaces in the background spacetime is described. The weak singularities which occur in these spacetimes are interpreted in terms of envelopes of the wave surfaces.

Boosted static multipole particles as sources of impulsive gravitational waves

It is shown that the known solutions for nonexpanding impulsive gravitational waves generated by null particles of arbitrary multipole structure can be obtained by boosting the Weyl solutions describing static sources with arbitrary multipole moments, at least in a Minkowski background. We also discuss the possibility of boosting static sources in (anti\char21{})de Sitter backgrounds, for which exact solutions are not known, to obtain the known solutions for null multipole particles in these backgrounds.