Gravitation: Following the Prague Inspiration

The following sections are included:

• PREFACE

• List of publications of Jiří Bičák

• Contents

In this yet another contribution to the still continuing Machian discussion we first summarize the history and meaning of Mach's principle. Then a specific model allowing to discuss Mach's ideas in a cosmological context is treated: we consider a spacetime consisting of an empty void separated from an almost Friedmann-Lemaître-Robertson-Walker (FLRW) dust universe by a spherically symmetric slowly rotating shell which is comoving with the cosmic dust. We treat in a unified manner all types of the FLRW universes. The metric is expressed in terms of a constant characterizing the angular momentum of the shell and parametrized by the comoving radius of the shell. Assuming the rotation to be a first order perturbation, we compute the dragging of inertial frames as well as the apparent motion of distant stars within the void. Finally, we discuss the Machian features of the model in terms of quantities that are measurable in principle.

The role of a nonzero cosmological constant is discussed for black-hole or naked-singularity spacetimes, Einstein–Strauss vacuola spacetimes, and general relativistic polytropic fluid spheres. In case of the black-hole spacetimes, motion of test particles and photons and equilibrium configurations of test perfect fluid are considered. Their relevance to astrophysically very important accretion processes in disk regime is established.

Structure of a stellar cluster is examined assuming that it is embedded in the gravitational field of a supermassive black hole. Members of the cluster experience orbital decay via collisions with an accretion disc and several other dissipative processes. The attention is concentrated on the region of gravitational dominance of the central body. It is shown how mutual interaction between stars and the surrounding environment establishes a non-spherical shape and anisotropy of the cluster. In some cases, the stellar sub-system acquires ring-type geometry.

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 (section 1) and astrophysics (section 2), listing the basic methods employed and results obtained. Then (in section 3) basic formulas for stationary axisymmetric solutions are summarized. Sections 4 and 5 review what we have learnt with Miroslav Žáček and Tomáš Zellerin about certain static and stationary situations recently. Concluding remarks are given in section 6. 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.

Optical reference geometry and related inertial forces are discussed for the static, spherically or axially symmetric black-hole, and naked-singularity spacetimes as well as Kerr–Newman spacetimes. Properties of the optical reference geometry are illustrated using embedding diagrams of the equatorial plane of the spacetimes considered.

We study the influence of the strong gravitational field of neutron stars on the light seen by distant observers. The gravitational field is approximated by the Kerr metric including thus the dragging effect caused by stellar rotation. Using this approximation, we construct pictures of compact rotating neutron stars as they would look like if light of these stars could avoid being disturbed by the environment surrounding the stars.

Exact solutions exist which describe impulsive gravitational waves propagating in Minkowski, de Sitter, or anti-de Sitter universes. These may be either nonexpanding or expanding. Both cases in each background are reviewed here from a unified point of view. All the main methods for their construction are described systematically: the Penrose "cut and paste" method, explicit construction of continuous coordinates, distributional limits of sandwich waves, embedding from higher dimensions, and boosts of sources or limits of infinite accelerations. Attention is concentrated on the most interesting specific solutions. In particular, the nonexpanding impulsive waves that are generated by null multipole particles are described. These generalize the well-known Aichelburg–Sexl and Hotta–Tanaka monopole solutions. Also described are the expanding spherical impulses that are generated by snapping and colliding strings. Geodesics and some other properties of impulsive wave spacetimes are also summarized.

Physical interpretation of some stationary and non-stationary regions of the spinning C-metric is presented. They represent different spacetime regions of a uniformly accelerated Kerr black hole. Stability of geodesics corresponding to equilibrium points in a general stationary spacetime with an additional symmetry is also studied and results are then applied to the spinning C-metric.

Lorentzian conical surfaces of two kinds are introduced: surfaces where an angle is removed from or added to the light cone and surfaces where an angle is removed from a region ranging from the past to the future light cone. Surfaces of the first kind represent a spacetime with collapsing or exploding inertial frames and those of the second kind represent a spacetime with nonorientable time. If a surface of the second kind is glued together in a Möbius-like way along a segment of the removed angle, which is only possible if it contains a ridge, the segment acts as a time machine. All the surfaces proposed here are physically unacceptable because energy conditions are violated; the time machine, in addition, is unacceptable because it produces inconsistent histories.

The semiclassical Einstein equations are reviewed with special attention to the stress-energy tensor appearing in them. A functional approach to the renormalization of the stress energy tensor is suggested.

A reformulation of a physical theory in which measurements at the initial and final moments of time are treated independently is discussed, both on the classical and quantum levels. Methods of the standard quantum mechanics are used to quantize boundary phase space to obtain boundary quantum mechanics — a theory that does not depend on the distinction between the initial and final moments of time, a theory that can be formulated without reference to the causal structure. As a supplementary material, the geometrical description of quantization of a general (e.g. curved) configuration space is presented.