arXiv subjects

P. P. Fiziev

Publications and source records attributed to P. P. Fiziev.

At least 19 recordsLinked to original sources

arXiv2016

A new approach to the connection problem for local solutions to the general Heun equation

We present new solution of the the connection problem for local solutions to the general Heun equation. Our approach is based on the symmetric form of the Heun's differential equation \cite{Fiziev14,Fiziev16} with four different regular singular points $z_{1,2,3,4}$. The four special regular points in the complex plane: $Z_{123},Z_{234},Z_{341},Z_{412}$ are the centers of the circles, defined by the different triplets $\{z_k,z_l,z_m\}$ with corresponding different indexes and play fundamental role, since the coefficients of the connection matrix can be expressed using the values of local solutions of the general Heun's equation at these points. A special case when all coefficients can be calculated using only one of the points $Z_{klm}$ is also considered.

math-ph↗

arXiv2012

The (2+1)-dim Axial Universes -- Solutions to the Einstein Equations, Dimensional Reduction Points, and Klein-Fock-Gordon Waves

The paper presents a generalization and further development of our recent publications where solutions of the Klein-Fock-Gordon equation defined on a few particular $D=(2+1)$-dim static space-time manifolds were considered. The latter involve toy models of 2-dim spaces with axial symmetry, including dimension reduction to the 1-dim space as a singular limiting case. Here the non-static models of space geometry with axial symmetry are under consideration. To make these models closer to physical reality, we define a set of "admissible" shape functions $ρ(t,z)$ as the $(2+1)$-dim Einstein equations solutions in the vacuum space-time, in the presence of the $Λ$-term, and for the space-time filled with the standard "dust". It is curious that in the last case the Einstein equations reduce to the well-known Monge-Ampère equation, thus enabling one to obtain the general solution of the Cauchy problem, as well as a set of other specific solutions involving one arbitrary function. A few explicit solutions of the Klein-Fock-Gordon equation in this set are given. An interesting qualitative feature of these solutions relates to the dimension reduction points, their classification, and time behavior. In particular, these new entities could provide us with novel insight into the nature of P- and T-violation, and of Big Bang. A short comparison with other attempts to utilize dimensional reduction of the space-time is given.

gr-qc↗

arXiv2011

Solutions of the Klein-Gordon equation on manifolds with variable geometry including dimensional reduction

We develop the recent proposal to use dimensional reduction from the four-dimensional space-time D=(1+3) to the variant with a smaller number of space dimensions D=(1+d), d < 3, at sufficiently small distances to construct a renormalizable quantum field theory. We study the Klein-Gordon equation on a few toy examples ("educational toys") of a space-time with variable special geometry, including a transition to a dimensional reduction. The examples considered contain a combination of two regions with a simple geometry (two-dimensional cylindrical surfaces with different radii) connected by a transition region. The new technique of transforming the study of solutions of the Klein-Gordon problem on a space with variable geometry into solution of a one-dimensional stationary Schrödinger-type equation with potential generated by this variation is useful. We draw the following conclusions: (1) The signal related to the degree of freedom specific to the higher-dimensional part does not penetrate into the smaller-dimensional part because of an inertial force inevitably arising in the transition region (this is the centrifugal force in our models). (2) The specific spectrum of scalar excitations resembles the spectrum of the real particles; it reflects the geometry of the transition region and represents its "fingerprints". (3) The parity violation due to the asymmetric character of the construction of our models could be related to violation of the CP symmetry.

hep-th↗

arXiv2009

Toward a New Model of the Central Engine of GRB

We present new developments of the simple model of the central engine of GRB, proposed recently. The model is based on minimal assumptions: some rotating compact relativistic object at the center and stable perturbations of its rotating gravitational field, described by Teukolsky Master Equation. We show that using nonstandard polynomial solutions to the angular Teukolsky equation we can describe the formation of collimated jets of various forms. Appearance of imaginary part of the superradiance-like frequency is established for the first time for pure vacuum black hole jet solutions of Teukolsky equation.

astro-ph.HE↗

arXiv2009

Classes of Exact Solutions to Regge-Wheeler and Teukolsky Equations

The Regge-Wheeler equation describes axial perturbations of Schwarzschild metric in linear approximation. Teukolsky Master Equation describes perturbations of Kerr metric in the same approximation. We present here unified description of all classes of exact solutions to these equations in terms of the confluent Heun's functions. Special attention is paid to the polynomial solutions, which yield novel applications of Teukolsky Master Equation for description of relativistic jets and astrophysical explosions.

gr-qc↗

arXiv2006

Exact Solutions of Regge-Wheeler Equation and Quasi-Normal Modes of Compact Objects

The well-known Regge-Wheeler equation describes the axial perturbations of Schwarzschild metric in the linear approximation. From a mathematical point of view it presents a particular case of the confluent Heun equation and can be solved exactly, due to recent mathematical developments. We present the basic properties of its general solution. A novel analytical approach and numerical techniques for study the boundary problems which correspond to quasi-normal modes of black holes and other simple models of compact objects are developed.

gr-qc↗

arXiv2005

Novel Properties of Bound States of Klein-Gordon Equation in Gravitational Field of Massive Point

We consider for the first time the solutions of Klein-Gordon equation in gravitational field of {\em a massive} point source in GR. We examine numerically the basic bounded quantum state and the next few states in the discrete spectrum for different values of the orbital momentum. A novel feature of the solutions under consideration is the essential dependence if their physical properties on the gravitational mass defect of the point source, even not introduced up to recently. It yields a repulsion or an attraction of the quantum levels up to their quasi-crossing.

gr-qc↗

arXiv2004

On the Solutions of Einstein Equations with Massive Point Source

We show that Einstein equations are compatible with the presence of massive point particles and find corresponding two parameter family of their solutions which depends on the bare mechanical mass $M_0>0$ and the Keplerian mass $M<M_0$ of the point source of gravity. The global analytical properties of these solutions in the complex plane define a unique preferable radial variable of the problem.

gr-qc↗

arXiv2004

Novel Geometrical Models of Relativistic Stars III. The Point Particle Idealization

We describe a novel class of geometrical models of relativistic stars. Our approach to the static spherically symmetric solutions of Einstein equations is based on a careful physical analysis of radial gauge conditions. It brings us to a two parameter family of relativistic stars without stiff functional dependence between the stelar radius and stelar mass. As a result, a point particle idealization -- a limiting case of bodies with finite dimension, becomes possible in GR, much like in Newtonian gravity. We devote this article to detailed mathematical study of this limit.

astro-ph↗

arXiv2004

Novel Geometrical Models of Relativistic Stars. II. Incompressible Stars and Heavy Black Dwarfs

In a series of articles we describe a novel class of geometrical models of relativistic stars. Our approach to the static spherically symmetric solutions of Einstein equations is based on a careful physical analysis of radial gauge conditions. It turns out that there exist heavy black dwarfs: relativistic stars with arbitrary large mass, which are to have arbitrary small radius and arbitrary small luminosity. In the present article we mathematically prove this new phenomena, using a detailed consideration of incompressible GR stars. We study the whole two parameter family of solutions of extended TOV equations for incompressible stars. This example is used to illustrate most of the basic features of the new geometrical models of relativistic stars. Comparison with newest observational data is discussed

astro-ph↗

arXiv2004

Novel Geometrical Models of Relativistic Stars. I. The General Scheme

In a series of articles we describe a novel class of geometrical models of relativistic stars. Our approach to the static spherically symmetric solutions of Einstein equations is based on a careful physical analysis of radial gauge conditions. It brings us to a two parameter family of relativistic stars without stiff functional dependence between the stelar radius and stelar mass. It turns out that within this family there do exist relativistic stars with arbitrary large mass, which are to have arbitrary small radius and arbitrary small luminosity. In addition, point particle idealization, as a limiting case of bodies with finite dimension, becomes possible in GR, much like in Newton gravity.

astro-ph↗

arXiv2004

Point Electric Charge in General Relativity

Using a proper gauge condition the static spherically symmetric solutions of Einstein-Maxwell equations with charged point source at the center are derived. It is shown that the solutions of the field equations are a three-parameter family depending on the Keplerian mass $M$, the charge $Q$ and the bare mass $M_0$. The result can be interpreted as a correction to Newton's gravitational potential and Coulomb's electric potential which are both regular at the centre where the massive point is placed. A correction to Gauss theorem is derived based on the nontrivial topology of the corresponding spacetime.

hep-th↗

arXiv2004

Applications of Lobachevsky Geometry to the Relativistic Two-Body Problem

In this talk we consider the geometrical basis for the reduction of the relativistic 2-body problem, much like the non-relativistic one, to describing the motion of an effective particle in an external field. It is shown that this possibility is deeply related with the Lobachevsky geometry. The concept of relativistic reduced mass and effective relativistic particle is discussed using this geometry. Different recent examples for application of relativistic effective particle are described in short.

hep-th↗

arXiv2003

Numerical Modeling of Charged Black Holes with Massive Dilaton

In this paper the static, spherically symmetric and electrically charged black hole solutions in Einstein-Born-Infeld gravity with massive dilaton are investigated numerically. The Continuous Analog of Newton Method (CANM) is used to solve the corresponding nonlinear multipoint boundary value problems (BVPs). The linearized BVPs are solved numerically by means of collocation scheme of fourth order. A special class of solutions are the extremal ones. We show that the extremal horizons within the framework of the model satisfy some nonlinear system of algebraic equations. Depending on the charge $q$ and dilaton mass $γ$, the black holes can have no more than three horizons. This allows us to construct some Hermite polynomial of third order. Its real roots describe the number, the type and other characteristics of the horizons.

gr-qc↗

arXiv2003

Inflation and Oscillations of Universe in 4D Dilatonic Gravity

We investigate the inflation of Universe in a model of four dimensional dilatonic gravity with a massive dilaton field $Φ$. The dilaton plays simultaneously the roles of an inflation field and a quintessence field. It yields a sequential {\em hyper}-inflation with a graceful exit to asymptotic de Sitter space-time, which is an attractor, and is approached as $\exp(-\sqrt{3Λ^{obs}} ct/2)$. The time duration of the inflation is reciprocal to the the mass of the dilaton: $Δt_{infl}\sim m_{{}_Φ}^{-1}$. The typical number of e-folds in the simplest model of this type is shown to be realistic without fine tuning.

gr-qc↗

arXiv2001

Effective one-body approach to the relativistic two-body problem

The relativistic 2-body problem, much like the non-relativistic one, is reduced to describing the motion of an effective particle in an external field. The concept of a relativistic reduced mass and effective particle energy introduced some 30 years ago to compute relativistic corrections to the Balmer formula in quantum electrodynamics, is shown to work equally well for classical electromagnetic and gravitational interaction. The results for the gravitational 2-body problem have more than academic interest since they apply to the study of binary pulsars that provide precision tests for general relativity. They are compared with recent results derived by other methods.

gr-qc↗

arXiv2000

A Free Boundary Problem in the Theory of the Stars

We investigate numerically models of the static spherically symmetric boson-fermion stars in the scalar-tensor theory of gravity with massive dilaton field. The proper mathematical model of such stars is interpreted as a nonlinear two-parametric eigenvalue problem with unknown internal boundary. To solve this problem the Continuous Analogue of Newton Method is used.

astro-ph↗

arXiv2000

A Minimal Model for Dilatonic Gravity

We study a new minimal scalar-tensor model of gravity with Brans-Dicke factor $ω(Φ)\equiv 0$ and cosmological factor $Π(Φ)$. The constraints on $Π(Φ)$ from known gravitational experiments are derived. We show that almost any time evolution of the scale factor in a homogeneous isotropic Universe can be obtained via properly chosen $Π(Φ)$ and discuss the general properties of models of this type.

gr-qc↗