Vol 23, No 2 (2017)

The paper outlines some aspects of the theory of gravity and electromagnetism based on a topology-geometric interpretation of matter and its properties as manifestations of a non-Euclidean geometry of the physical hypersurface of dimension 3 embedded in a Euclidean space of dimension 4. We derive the basic equations of the theory, leading to equations similar to those of the classical theory of gravity and electromagnetism as well as those of quantum theory. It is shown that in this theory the observed effect of a hidden mass, or dark matter, is explained in a natural way by effects of the geometry of the physical hypersurface.

We propose an approximate theory describing electromagnetism and gravity as a single direct particle interaction. A new element in this theory is the assumption on real simultaneous (combined) existence of retarded and advanced interactions. Another essential principle of this theory is Mach’s principle, according to which the interaction of particles in a certain local region is inextricably connected with the dynamics of all particles in the Universe. A consequence ofMach’s principle in this theory is that in terrestrial experiments the advanced electromagnetic interaction is many orders of magnitude smaller than the retarded one (but is not precisely zero). We consider a possible mechanism of emergence of particle masses due to electromagnetic interaction. The proposed theory is relativistic but non-quantum. In this regard, we consider only three types of particles (electrons, protons, and neutrons), and the difference of particle masses is explained only at a qualitative level. The resulting equation of motion for particles is studied in the nonrelativistic approximation. Along with the Lorentz force and the radiative friction force, it contains terms describing the gravitational interaction. Their form is similar to those known in gravielectromagnetism, which is an approximation to general relativity.

On the basis of a qualitative analysis of the set of differential equations of the standard cosmological model it is shown that in the case of zero cosmological constant Λ this set has a stable center corresponding to zero values of the potential and its derivative at infinity. Thus the model based on a single massive classical scalar field would give a flat Universe in the infinite future. A numerical simulation of the dynamic system corresponding to the set of Einstein-Klein-Gordon equations has shown that at late times of the evolution the invariant cosmological acceleration has an oscillating nature and changes from −2 (braking), to +1 (acceleration). The average value of the cosmological acceleration is negative and is equal to −1/2. Oscillations of the cosmological acceleration happen in the background of a rapidly falling Hubble parameter. In the case of a nonzero value of Λ, depending on its value, three various qualitative behavior types of the dynamic system on the 2D plane (Φ, \(\mathop \Phi \limits^ \cdot \)) are possible, which correspond either to a zero attractive focus or to a stable attractive knot with zero values of the potential and its derivative. Herewith, the system asymptotically enters a secondary inflation. Numerical simulations have shown that with Λ < 3 × 10⁻⁸ m², the macroscopic value of the cosmological acceleration behaves similarly to the case Λ = 0, i.e. in the course of the cosmological evolution there appears a lasting stage on which this value is close to −1/2, which corresponds to a non-relativistic equation of state.

Quantum-mechanical motion of a spin-half particle is examined in the axially symmetric fields of static naked singularities formed by a mass distribution with a quadrupole moment (q-metric). The analysis is performed by means of the method of effective potentials of the Dirac equation generalized to the case where radial and angular variables are not separated. If −1 < q < qlim, |qlim| ≪ 1, where q is the quadrupolemoment in proper units, the naked singularities do not exclude the existence of stationary bound states of Dirac particles for a prolate mass distribution in the q-metric along the axial axis. For an oblate mass distribution, the naked singularities of the q-metric are separated from a Dirac particle by infinitely large repulsive barriers followed by a potential well which deepens while moving apart from the equator (from θ = θ_min or θ = π − θ_min) toward the poles. The poles make an exception, and at 0 < q < q*, there are some points θ_i for particle states with j ≥ 3/2.

We review a possible nonminimal (dilatonic) coupling of a scalar field (axion-like particle) to electromagnetism, through experimental and observational constraints. Such a coupling is motivated from recent quasar spectrum observations that indicate a possible spatial and/or temporal variation of the fine-structure constant. We consider a dilatonic coupling of the form B_F (ϕ) = 1+gϕ. The strongest bound on the coupling parameter g is derived from the weak equivalence principle tests, which impose g < 1.6 × 10⁻¹⁷ GeV⁻¹. This constraint is strong enough to rule out this class of models as a cause for an observable cosmological variation of the fine structure constant unless a chameleon mechanism is implemented. Also, we argue that a similar coupling occurs in chameleon cosmology, another candidate dark mater particle, and we estimate the cosmological consequences by both effects. It should be clarified that this class of models is not necessarily ruled out in the presence of a chameleon mechanism which can freeze the dynamics of the scalar field in high density laboratory regions.

The paper deals with spatially homogeneous and anisotropic Kantowski-Sachs and Bianchi universes with a general non-canonical scalar field with the Lagrangian L = F(X) − Ω(ϕ), where \(X = \frac{1}{2}{\phi _i}{\phi ^i}\). We discuss a general non-canonical scalar field in three different cosmologies: (i) cosmology with a constant potential, Ω(ϕ) = Ω₀ = const, (ii) cosmology with a constant equation-of-state parameter, i.e., γ_ϕ = const, and (iii) cosmology with a constant speed of sound, i.e., c_s² = const. For a constant potential, we have shown that the k-essence Lagrangian and the Lagrangian of the present model are equivalent. Dissipation of anisotropy, when the universe is filled with a general non-canonical scalar field, is investigated. The existence of an average bounce in Kantowski-Sachs and locally rotationally symmetric Bianchi-I and Bianchi-III models is discussed in detail.