arXiv subjects
Publications and source records attributed to V. N. Melnikov.
The evolution of 4-dimensional and (4+d)-dimensional (d=1,2) cosmological models based on the integrable Weyl geometry are considered numerically both for empty space-time and for scalar field with non minimal coupling with gravity.
Exact static, spherically symmetric solutions to the Einstein-Maxwell-scalar equations, with a dilatonic-type scalar-vector coupling, in $D$-dimensional gravity with a chain of $n$ Ricci-flat internal spaces are considered. Their properties and special cases are discussed. A family of multidimensional dilatonic black-hole solutions is singled out, depending on two integration constants (related to black hole mass and charge) and three free parameters of the theory (the coordinate sphere, internal space dimensions, and the coupling constant). The behaviour of the solutions under small perturbations preserving spherical symmetry, is studied. It is shown that the black-hole solutions without a dilaton field are stable, while other solutions, possessing naked singularities, are catastrophically unstable.
A multidimensional cosmological model with space-time consisting of $n (n \ge 2)$ Einstein spaces $M_{i}$ is investigated in the presence of a cosmological constant $Λ$ and a homogeneous minimally coupled scalar field $φ(t)$ as a matter source. Classical and quantum wormhole solutions are obtained for $Λ< 0$ and all $M_{i}$ being Ricci-flat. Classical wormhole solutions are also found for $Λ< 0$ and only one of the $M_{i}$ being Ricci-flat for the case of spontaneous compactification of the internal dimensions with fine tuning of parameters.
The recently suggested SEE (Satellite Energy Exchange) method of measuring the gravitational constant $G$, possible equivalence principle violation (measured by the Eötvös parameter $η$) and the hypothetic 5th force parameters $α$ and $λ$ on board a drag-free Earth's satellite is discussed and further developed. Various particle trajectories near a heavy ball are numerically simulated. Some basic sources of error are analysed. The $G$ measurement procedure is modelled by noise insertion to a ``true'' trajectory. It is concluded that the present knowledge of $G, α$ (for $λ\geq 1$ m) and $η$ can be improved by at least two orders of magnitude.
The validity conditions for the extended Birkhoff theorem in multidimensional gravity with $n$ internal spaces are formulated, with no restriction on space-time dimensionality and signature. Examples of matter sources and geometries for which the theorem is valid are given. Further generalization of the theorem is discussed.
A cosmological model describing the evolution of $n$ Einstein spaces $(n>1)$ with $m$-component perfect-fluid matter is considered. When all spaces are Ricci-flat and for any $α$-th component the pressures in all spaces are proportional to the density: $p_{i}^{(α)} = (1- h_{i}^{(α)}) ρ^{(α)}$, $h_{i}^{(α)}$ = const, the Einstein and Wheeler-DeWitt equations are integrated in the cases: i) $m=1$, for all $h_{i}^{(α)}$; ii) $m > 1$, for some special sets of $h_{i}^{(α)}$. For $m=1$ the quantum wormhole solutions are also obtained. \\