Vol 22, No 4 (2016)

Primitive mapping of 2D fractal spaces yields a formulation of the Schro¨ dinger equation and endows its solutions and the respective 3D objects with specific geometric images. In particular, it is shown that the simplest 1D-box solution comprising no parameters of particles motion can be interpreted as a 2D inhomogeneous string oscillating on a real-imaginary fractal surface or as a 3D static spindle with a harmonically distributed mass spectrum. The description of an inertially moving similar object is obtained using a Bargmann-type theorem applied to the Bohm equations, and, as their exact solution, a fractal function containing explicit kinematic terms.

We study the properties of an effective potential for the scale factor of extra dimensions in a Kaluza-Klein-type model with a spherical extra factor space, including a function of the scalar curvature and other quadratic curvature invariants, taking into account the Casimir energy of massless scalar fields. We demonstrate the existence of a minimum of the potential, able to induce a physically reasonable value of the effective cosmological constant in our space-time. Under the adopted assumptions, it is shown that the huge Casimir energy density can be compensated by the fine-tuned contribution of the curvature-nonlinear terms in the original action.

The SEE (Satellite Energy Exchange) mission was suggested and designed under NASA Project in 1992–2002 to make extremely accurate measurements on fundamental gravitation by observing the orbital perturbation of unconstrained orbiting test bodies in a femto-g to atto-g environment. The mission uses novel and original test body dynamics. Its broad objective is to support the development of gravity theory and its unification with other interactions by carrying out sensitive gravitational tests capable of discriminating among alternative theories. The SEE mission introduced and utilized new technology for near-zero-g environment creation, passive cryogenic temperature control, passive station-keeping capability, and non-contact sub-micron-accuracy distance measurements, all of which promise to have a wide variety of applications.

The Bazanski approach to deriving paths is applied to Finsler geometry. The approach is generalized and applied to a new developed geometry called “Absolute parallelism with Finslerian Flavor” (FAP). A set of path equations is derived for the FAP. It is a horizontal (h) set. A striking feature in this set is that the coefficient of the torsion term jumps by a step of one-half from one equation to the other. It is tempting to believe that the h-set admits some quantum features. Comparisons with the corresponding sets in other geometries are given. Conditions for reducing the set of path equations obtained to well-known path equations in some geometries are summarized in a schematic diagram.

The derivation of the “baroenergetic” equation of state is reconsidered in rather a general form following Bernoulli’s kinetic approach but with Lorentz energy-momentum dependence. It is shown that uniform versions of the equation of state generally used in cosmology are valid only in extreme limits: an ultrarelativistic photon gas (electromagnetic radiation) and a nonrelativistic gas of massive particles. Possible ways of generalization of these results are pointed out and briefly discussed.

We consider a scalar-tensor theory of gravitation with the scalar source being the trace of the stress-energy tensor of the scalar field itself andmatter. We obtain an example of a numerical solution of the cosmological equations which shows that under some special choice of the scalar parameters, there exists a slow-roll regime in which the modern values of the Hubble and deceleration parameters may be obtained.

The existence of a minimal observable length is the common prediction of different theories of quantum gravity such as string theory, loop quantum gravity and black hole physics. The ordinary Heisenberg uncertainty principle (HUP) is not compatible with this prediction, so the HUP was generalized to the Generalized Uncertainty Principle (GUP) in literature. The consequences of this generalization are the generalized commutation relations between operators and so the generalized Hamiltonian for systems. In this paper, we represent the solutions of a harmonic oscillator and a particle in a box in the sense of a generalized commutation relation. We show that using of the Hellmann-Feynman (HF) theorem leads to the approximate energy spectrum of a harmonic oscillator and the exact energy spectrum of a particle in a box. In these cases, we do not solve the corresponding generalized Schro¨ dinger equation directly, but the HF theorem exhibit the effect of GUP on the energy spectrum. Also we obtain the uncertainties in both position and momentum for a harmonic oscillator and derive the GUP.

We examine an inflationary scenario in Bianchi Type V space-time for a barotropic fluid distribution with variable bulk viscosity and decaying vacuum energy density. We observe that the matter density ρ, the coefficient of bulk viscosity ζ and the expansion θ all diverge at τ = 0. The spatial volume increases with time, representing an inflationary scenario. The deceleration parameter q < 0 for barotropic, dust and radiation dominated models representing an accelerated universe, while for a stiff fluid distribution q > 0 corresponding to a decelerated universe. The vacuum energy density Λ decreases with time. The entropy per unit volume is proportional to the absolute temperature. The energy conditions (weak, dominant and strong) are discussed for the model. The reality condition ρ + p ≥ 0 is violated for the inflationary model due to the presence of a scalar field (φ). We also discuss the importance of Bianchi Type V model where the anisotropy dies away during the inflationary era. We also calculate the inflationary parameters and compare the results with the Planck data and discuss their compatibility with anisotropy and BAO estimates. The cosmological constant Λ is a function of time without break general covariance. We also discuss the bounds of the model, how the model isotropizes, where the fluid goes after inflation and how viscosity may realize a graceful exit from inflation to a radiation dominated era.