Gravitation and Cosmology

—The paper is devoted to some of the difficulties which the Wheeler-DeWitt quantum geometrodynamics encountered, in particular, a strong mathematical proof that this theory is gauge-invariant, the definition of the wave function of the Universe through a path integral and the illegality of asymptotic boundary conditions in quantum gravity, the derivation of the Wheeler-DeWitt equation from the path integral and the equivalence of the Dirac quantization scheme with other approaches, the problem of definition of physical states in quantum gravity, possible realizations of the Everett concept of “relative states.” These problems are rarely discussed in the literature. They are related to the guiding idea that quantum theory of gravity must be gauge-invariant. It will lead to the question if it is possible to achieve this goal in a mathematically consistent way.

This study links the fractal structure of physical space-time to quantum-mechanical laws. It is shown that primitive distortions of the pregeometric surface, a fractal cell of 3D space, gives birth to a condition eliminating the metric defect while providing “eternal validity” of the exclusive algebras (of real, complex, and quaternion numbers). Written in the physical units typical for the micro-world entities, this condition acquires the precise form of the Pauli equation describing mechanics of the quantum electron with spin.

The paper studies the influence of periodically changing boundary conditions on the motion of a test null string “inside” an axially symmetric null string domain that has a layered structure and preserves its size. It is shown that the action of the gravitational field of the null string domain on a test null string whose size (radius) is smaller than the size of the domain, under any initial conditions, leads to oscillations of the test null string inside a limited spatial region. The motion (drift) of the region where the test null string oscillations take place, depends on the relationship between the initial parameters characterizing the test null string and the null string domain. These regions can be considered as particles with an effective nonzero rest mass, localized in space. It is noted that the properties of these particles should be determined by the trajectories of null strings moving within a limited spatial volume, and under the influence of changing external conditions, one sort of particles can pass into another one.

An anthropic explanation of the evident smallness of the dark energy (DE) density value implies the existence of a time-dependent component of the scalar field, serving, together with a negative-valued cosmological constant, as one of two components to the overall DE density. The observers (i.e. us) might then only evolve in those regions of the universe where the sum of those two components (the positive and a negative ones) is sufficiently close to zero. However, according to Vilenkin and Garriga, the scalar field component has to slowly but surely diminish in time. In about a trillion years, this process will put a cap to the now-observable accelerated expansion of the universe, leading to a subsequent phase of impending collapse. However, the vanishing scalar field might also produce some rather unexpected singularities with a finite nonzero scale factor. We analyze this possibility using a particular example of a Sudden Future Singularities (SFS) and come to a startling conclusion that the time required for an SFS to arise must be “comparable” to the lifetime of the observable universe.

The article discusses and substantiates a self-consistent approach to the macroscopic description of systems with gravitational interaction. Corrections to the equation of state of the fluid are found, based on the macroscopic Einstein equations which were obtained by averaging over microscopic spherically symmetric metric fluctuations created by primordial black holes in a fluid medium. It is shown that these corrections are effectively equivalent to addition to the system of a fluid with the equation of state p = −ε. In addition, it is shown that, in this case, the mass of black holes can grow at a modern stage of evolution to very large values.

We analyze two branches of five-dimensional theories from a methodological point of view: Kaluza’s theory and the Klein-Fock-Rumer (KFR) 5-optics, aimed at a geometric description of electromagnetism and masses of elementary particles, respectively. These two theories have a number of problematic points such as the Planckian masses in Kaluza’s theory, or appearing of a configuration space in the KFR scheme. We propose a simple six-dimensional toy model of Kaluza-Klein type with compactification on a two-dimensional torus \(\mathbb{T}^2\), which demonstrates a possible way to overcome these difficulties in quite a simple manner. The key features of our approach are: merging of the above two 5D theories into a unique 6D theory; using the signature (+-) of extra space allowing one to renormalize the mass spectrum; and a special kind of truncation of the full isometry group to the electromagnetic gauge group U(1). The latter feature allows a geometrical interpretation in terms of an effective 5D theory on a hypersurface \(\Sigma\subset\mathbb{T}^2\) being a leaf of a linear foliation of the 2-torus. Possible consequences and open questions arising in such a scheme are briefly discussed.

In the framework of the Stueckelberg-Wheeler-Feynman concept of a “one-electron Universe” we consider a world line implicitly defined by a system of algebraic (precisely, polynomial) equations. A collection of pointlike “particles” of two kinds on the world line (or its complex extension) is defined by the real (complex conjugate) roots of the polynomial system and is detected then by an external inertial observer through light cone connections. Then the observed collective dynamics of the particle ensemble is, generally, subject to a number of Lorentz-invariant conservation laws. Remarkably, this property follows from the Vieta formulas for roots of the generating polynomial system. At some discrete instants of the observer’s proper time, mergers and subsequent transmutations of a pair of particles-roots take place, thus simulating the processes of annihilation/creation of a particle/antiparticle pair.

Using variable gravitational and cosmological constants, the mechanism of particle creation in the universe is studied for Friedmann–Robertson–Walker (FRW) space-times at high dimensions to explain the early deceleration and present accelerating phases. To investigate the dynamics of these phases, we consider two scale factor of the forms \(a(t)=\sqrt{t^\alpha{e^t}}\) and \(a(t)=\sqrt[m]{{\rm{sin}}h(kt)}\), which yield two different time-dependent deceleration parameters. Firstly, we modify the d-dimensional time-dependent field equations, including the general formulation of particle creation and entropy generation mechanisms. Then, we investigate the time dependence of a few quantities such as the particle creation rate ψ, the entropy S, the gravitational constant G, the cosmological constant Λ, the energy density ρ, the deceleration parameter q, etc. We show that all constants and quantities, except the gravitational constant G and the entropy S, characteristically decrease with time for two kinds of scale factors in all dimensions. However, the gravitational constant G and the entropy S increase with time. Additionally, we show that the cosmological constant Λ is unexpectedly independent of the particle creation mechanism, unlike G.