Jiří Bičák

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.

Disc sources for conformastationary metrics

Conformastationary metrics - those of the form have been derived by Perjes and by Israel and Wilson as source-free solutions of the Einstein-Maxwell equations. By analogy with the conformastatic metrics which have charged dust sources it was assumed that conformastationary metrics would be the external metrics of charged dust in steady motion. However, for axially symmetric conformastationary metrics we show that, as well as moving dust, hoop tensions are always necessary to balance the centrifugal forces induced by the motion. Exact examples of conformastationary metrics with disc sources are worked out in full. Generalizations to non-axially symmetric conformastationary metrics are indicated.

Symmetries of asymptotically flat electrovacuum space–times and radiation

Symmetries compatible with asymptotic flatness and admitting gravitational and electromagnetic radiation are studied by using the Bondi–Sachs–van der Burg formalism. It is shown that in axially symmetric electrovacuum space–times in which at least locally a smooth null infinity in the sense of Penrose exists, the only second allowable symmetry is either the translational symmetry or the boost symmetry. Translationally invariant space–times with, in general, a straight “cosmic string” along the axis of symmetry are nonradiative although they can have a nonvanishing news function. The boost-rotation symmetric space–times are radiative. They describe “uniformly accelerated charged particles” or black holes which in general may also be rotating—the axial and an additional Killing vector are not assumed to be hypersurface orthogonal. The general functional forms of both gravitational and electromagnetic news functions, and of the mass aspect and total mass of asymptotically flat boost-rotation symmetric space–...

Relativistic closed-form Hamiltonian for many-body gravitating systems in the post-Minkowskian approximation.

The Hamiltonian for a system of relativistic bodies interacting by their gravitational field is found in the post-Minkowskian approximation, including all terms linear in the gravitational constant. It is given in a surprisingly simple closed form as a function of canonical variables describing the bodies only. The field is eliminated by solving inhomogeneous wave equations, applying transverse-traceless projections, and using the Routh functional. By including all special relativistic effects our Hamiltonian extends the results described in classical textbooks of theoretical physics. As an application, the scattering of relativistic objects is considered.

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.

Stationary electromagnetic fields around black holes. II. General solutions and the fields of some special sources near a Kerr black hole

The electromagnetic field of a general stationary source, occurring in the vicinity of a rotating (Kerr) black hole, is obtained by solving the Maxwell and Teukolsky equations. The field is expressed both outside and inside the radius at which the source is located. As examples the fields of point charges, charged rings, current loops, and magnetic dipoles not necessarily located in axisymmetric positions are calculated. The electromagnetic field occurring when a Kerr black hole is placed in an originally uniform magnetic field is derived without assuming the alignment of the direction of the magnetic field and the axis of symmetry of the black hole.

Gauge invariant Hamiltonian formalism for spherically symmetric gravitating shells

The dynamics of a spherically symmetric thin shell with arbitrary rest mass and surface tension interacting with a central black hole is studied. A careful investigation of all classical solutions reveals that the value of the radius of the shell and of the radial velocity as an initial datum does not determine the motion of the shell; another configuration space must, therefore, be found. A different problem is that the shell Hamiltonians used in the literature are complicated functions of momenta (nonlocal) and they are gauge dependent. To solve these problems, the existence is proved of a gauge-invariant super-Hamiltonian that is quadratic in momenta and that generates the shell equations of motion. The true Hamiltonians are shown to follow from the super-Hamiltonian by a reduction procedure including a choice of gauge and solution of constraint; one important step in the proof is a lemma stating that the true Hamiltonians are uniquely determined (up to a canonical transformation) by the equations of motion of the shell, the value of the total energy of the system, and the choice of time coordinate along the shell. As an example, the Kraus-Wilczek Hamiltonian is rederived from the super-Hamiltonian. The super-Hamiltonian coincides with that of a fictitious particle moving in a fixed two-dimensional Kruskal spacetime under the influence of two effective potentials. The pair consisting of a point of this spacetime and a unit timelike vector at the point, considered as an initial datum, determines a unique motion of the shell.

On asymptotically flat solutions of Einstein's equations periodic in time: II. Spacetimes with scalar-field sources

We extend the work in our earlier paper (Biˇ´ ak J et al 2010 Class. Quantum Grav. 27 055007 ) to show that time-periodic, asymptotically flat solutions of the Einstein equations analytic at I, whose source is one of a range of scalarfield models, are necessarily stationary. We also show that, for some of these scalar-field sources, in stationary, asymptotically flat solutions analytic at I, the scalar field necessarily inherits the symmetry. To prove these results we investigate miscellaneous properties of massless and conformal scalar fields coupled to gravity, in particular Bondi mass and its loss.

Boost-rotation symmetric gravitational null cone data

The potential role of boost-rotation symmetric vacuum spacetimes as test beds for numerical studies of gravitational radiation is discussed. For application to null cone evolution codes, these spacetimes are analyzed in terms of their data on the preferred null cone left invariant by the symmetry group. On this cone, an explicit solution of the Bondi hypersurface and evolution equations is found. This solution has a smooth vertex, a smooth interior, and, except for polar singularities, admits a well-definedℐ+.

On the asymptotic structure of axisymmetric radiative spacetimes

The asymptotic structure of axisymmetric radiative spacetimes which need not admit global null infinity is discussed. It is shown that from the asymptotic symmetry group a unique translation subgroup can locally be singled out and the news tensor can be obtained. The asymptotic structure of axisymmetric spacetimes with an infinite cosmic string and some of the exact radiative solutions with boost-rotation symmetry are treated as examples.