Том 22, № 3 (2016)

The fractal equations of mechanics (quantum and classical) are clearly demonstrated to be definitions of an arbitrary potential on a fractal complex number valued surface. The developed approach helps us to show that a translational motion of any rotating compact object (point-like particle) can be equivalently represented by a specific rotation of a virtual ring described in terms of a fractal “wave function”, the model endowing the particle with a set of quantum characteristics including quantization of the ring’s space translation.

We report on the experimental results of testing a new physical method of determination of the gravitational potential differences and orthometric heights by measuring the relativistic effect of gravitational redshift of frequency by means of atomic clocks. The experiment was performed in the Altai Mountains between two geodetic stations, Shebalino and Sieminski Pass, separated by about 850 meters in altitude. The measured mean value of the frequency shift caused by the change in the gravitational potential between the two stations is (δf/f₀)_grav = 7.980 × 10⁻¹⁴, with the dispersion σ_f = 7.27 × 10⁻¹⁵ referred to the time interval of the experiment.

Propagation of gravitational disturbances at the speed of light is one of the key predictions of the General Theory of Relativity. This result is now backed indirectly by the observations of the behavior of the ephemeris of binary pulsar systems. These new results have increased the interest in the mathematical theory of gravitational waves in the last decades, and severalmathematical approaches have been developed for a better understanding of the solutions. In this paper we develop a modal series expansion technique in which solutions can be built for plane waves from a seed integrable function. The convergence of these series is proven by the Raabe-Duhamel criteria, and we show that these solutions are characterized by a well-defined and finite curvature tensor and also a finite energy content.

By averaging the Einstein equations over transverse gravitational perturbations we obtain a closed set of two ordinary differential equations describing the macroscopic cosmological evolution of the isotropic spatially flat Universe filled with gravitational radiation. We have found an asymptotic solution of the evolution equation for the gravitational perturbation amplitude. Substituting this solution to the Einstein equations averaged over gravitational perturbations, we obtain a single nonlinear ordinary differential second-order evolution equation with respect to the macroscopic scale factor. We have also found a solution of the evolution equation for the scale factor in the WKB approximation which analytically describes the process of transformation from the ultrarelativistic mode of cosmological expansion to an inflationary one.

We consider static, spherically symmetric configurations of electrically charged dust in Einstein-Maxwell-dilaton gravity where the Lagrangian contains the interaction term P(χ)F_μνF^μν with an arbitrary function P(χ). Unlike previous studies, we assume that the scalar field χ (which can be canonical or phantom) does not have a charge of its own and exists only due to this interaction. We discuss possible solutions to the field equations, in particular, those describing black holes (BHs) and quasi-black holes (QBHs). The latter are globally regular configurations whose size is very close to that of a BH of the same mass and which therefore are almost indistinguishable from BHs for a distant observer. Some general features are revealed, a family of explicitly integrable models is found, and some explicit examples are presented.

We endeavor to take into account the extra-relativistic effects in the mass-energy relation due to Einstein. We consider a modified mass-energy relation and consequently find a strict bound on the Chandrasekhar limit for a white dwarf star. We find that the said limit gets modified to a certain extent, and we have been able to find the maximum mass of a stable white dwarf star.