Oldřich Semerák

Gravitating discs around a Schwarzschild black hole II

A sequence of exact spacetimes is obtained describing the fields of a Schwarzschild black hole surrounded by stable static axisymmetric thin discs having their inner rim at the least possible radius. In the previous paper we only required stability with respect to perturbations in the disc plane, while it turns out that for discs with relative mass >0.23 the perturbations in perpendicular direction are more dangerous. The discs of the resulting sequence have their inner rims just on, or very close to, circular geodesics marginally stable with respect to either of the perturbations. Redshift from static and Keplerian observers in the disc is computed. The inverted first Morgan-Morgan counter-rotating disc, used in superpositions, has a number of satisfactory physical properties, but it has turned out to have a curvature singularity at the inner rim. However, this is only a consequence of a too steep radial start of density, not present in (inverted) “higher” Morgan-Morgan solutions.

Expulsion of external fields from extreme horizons: Example of an external gravitational field

In analogy with the Meissner effect in (super) conductors, black holes expel stationary axisymmetric external fields when approaching the extreme state. This has been studied on magnetic fields in the literature. Using the recently obtained solution describing a rotating black hole surrounded by an axisymmetric thin annular disc, we show that the effect also applies to an external gravitational field.

Towards gravitating discs around stationary black holes

This article outlines the search for an exact general relativistic description of the exterior (vacuum) gravitational field of a rotating spheroidal black hole surrounded by a realistic axially symmetric disc of matter. The problem of multi-body stationary spacetimes is first exposed from the perspective of the relativity theory and astrophysics, listing the basic methods employed and results obtained. Then, basic formulas for stationary axisymmetric solutions are summarized. Remaining sections review what we have learnt with Miroslav Zacek and Tomas Zellerin about certain static and stationary situations recently. Although the survey part is quite general, the list of references cannot be complete. Our main desideratum was the informative value rather than originality -- novelties have been preferred, mainly reviews and those with detailed introductions.

Forces at the Kerr singularity

In a classical-like language introduced in a previous paper, the fact that the ring singularity of the Kerr spacetime cannot be left/reached by timelike geodesic which is not bound to the equatorial plane is interpreted as a result of the interplay of the gravitational and – mainly – the centrifugal forces.

Some aspects of motion in the Kerr field

We find that the Kerr field, the simplest relativistic gravitational field of a rotating source, may help in (pre-)collimation of cosmic jets emanating from many active galactic nuclei. Solving the geodesic motion in the Kerr field, we use in part a tetrad approach — the description of quantities by their physically measured components in the locally Cartesian frame of a local observer (rather than by their coordinate components). The stationary frames, the most important local frames in the Kerr spacetime, are reviewed and their properties discussed. We use the Newtonian “force” language to interpret the occurrence of the “rotospheres”, the toroidal regions that circle round the Kerr sources; in these regions, the 4-acceleration of a stationary observer depends on his orbital angular velocity in a way going against common intuition. We give the rotospheres in various Kerr spacetimes and introduce a new class of privileged observers (“extremally accelerated observers”). We then generalize this approach to any motion in any spacetime. In the Kerr spaces, we construct the cones generated by directions along which photons can escape to infinity from a given point. The results are consistent with the expected influence of rotational dragging for black holes, whereas they are rather complicated for naked singularities.

Parameters of Black Holes in Sources with Periodic Variability

We propose a simple and unambiguous way to deduce the parameters of black holes which may reside in AGNs and some types of X-ray binaries. The black-hole mass and angular momentum are determined in physical units. The method is applicable to the sources with periodic components of variability, provided one can assume the following: (i) Variability is due to a star or a stellar-mass compact object orbiting the central black hole and passing periodically through an equatorial accretion disk (variability time-scale is given by the orbital period). (ii) The star orbits almost freely, deviation of its trajectory due to passages through the disk being very weak (secular); the effect of the star on the disk, on the other hand, is strong enough to yield observable photometric and spectroscopic features. (iii) The gravitational field within the nucleus is that of the (Kerr) black hole, the star and the disk contribute negligibly.

On Geodesic Dynamics in Deformed Black-Hole Fields

“Almost all” seems to be known about isolated stationary black holes in asymptotically flat space-times and about the behaviour of test matter and fields in their backgrounds. The black holes likely present in galactic nuclei and in some X-ray binaries are commonly being represented by the Kerr metric, but actually they are not isolated (they are detected only thanks to a strong interaction with the surroundings), they are not stationary (black-hole sources are rather strongly variable) and they also probably do not live in an asymptotically flat universe. Such “perturbations” may query the classical black-hole theorems (how robust are the latter against them?) and certainly affect particles and fields around, which can have observational consequences. In the present contribution we examine how the geodesic structure of the static and axially symmetric black-hole space-time responds to the presence of an additional matter in the form of a thin disc or ring. We use several different methods to show that geodesic motion may become chaotic, to reveal the strength and type of this irregularity and its dependence on parameters. The relevance of such an analysis for galactic nuclei is briefly commented on.

Geodesic Chaos in Perturbed Black-Hole Fields

Dynamics of time-like geodesics in the static and axially symmetric field of a black hole surrounded by a thin disc is studied by two recurrence methods, the recurrence plots (RPs) and the average of directional vectors (ADVs). Their results supplement the information obtained before from Poincare surfaces of section and from phase-variable evolutions and the corresponding power spectra. The occurrence of chaos due to the presence of ambient matter may be important for evolution and appearance of astrophysical black-hole systems.

Exact solutions for discs around stationary black holes

Black holes surrounded by axisymmetric structures prosper in some of the most interesting sources in the universe. However, a consistent exact description of the gravitational field of these systems is still lacking. In a static case, the task reduces to Laplace equation and the fields of multiple sources follow by mere superposition. In a rotating case, non-linearity of the Einstein equations resists simple grasp, but even then the theory of completely integrable systems seems to verge on satisfactory solutions. It seeks them in terms of θ-functions on special manifolds connected – symptomatically – with the names of Riemann and Hilbert.