Gravitational Field of Massive Point Particle in General Relativity

Utilizing various gauges of the radial coordinate we give a description of static spherically symmetric space-times with point singularity at the center and vacuum outside the singularity. We show that in general relativity (GR) there exist a two-parameters family of such solutions to the Einstein equations which are physically distinguishable but only some of them describe the gravitational field of a single massive point particle with nonzero bare mass $M_0$. In particular, we show that the widespread Hilbert's form of Schwarzschild solution, which depends only on the Keplerian mass $M<M_0$, does not solve the Einstein equations with a massive point particle's stress-energy tensor as a source. Novel normal coordinates for the field and a new physical class of gauges are proposed, in this way achieving a correct description of a point mass source in GR. We also introduce a gravitational mass defect of a point particle and determine the dependence of the solutions on this mass defect. The result can be described as a change of the Newton potential $\phi_{{}_N}=-G_{{}_N}M/r$ to a modified one: $\phi_{{}_G}=-G_{{}_N}M/ (r+G_{{}_N} M/c^2\ln{{M_0}\over M})$ and a corresponding modification of the four-interval. In addition we give invariant characteristics of the physically and geometrically different classes of spherically symmetric static space-times created by one point mass. These space-times are analytic manifolds with a definite singularity at the place of the matter particle.Comment: 16 pages, no figures, latex file, changes in the Abstract,typos correcte

On the Exact Solutions of the Regge-Wheeler Equation in the Schwarzschild Black Hole Interior

We solve exactly the Regge-Wheeler equation for axial perturbations of the Schwarzschild metric in the black hole interior in terms of Heun functions and give a description of the spectrum and the eigenfunctions of the interior problem. The phenomenon of attraction and repulsion of the discrete eigenvalues of gravitational waves is discovered.Comment: 12 pages, latex file, 5 figures, changes in the title, new section, new comments, new references and new acknowledgments adde

Novel representation of the general Heun's functions

In the present article we introduce and study a novel type of solutions of the general Heun's equation. Our approach is based on the symmetric form of the Heun's differential equation yielded by development of the Felix Klein symmetric form of the Fuchsian equations with an arbitrary number $N\geq 4$ of regular singular points. We derive the symmetry group of these equations which turns to be a proper extension of the Mobius group. We also introduce and study new series solution of symmetric form of the general Heun's differential equation (N=4) which treats simultaneously and on an equal footing all singular points. Hopefully, this new form will simplify the resolution of the existing open problems in the theory of general Heun's functions and can be used for development of new effective computational methods.Comment: 11 pages LaTex file, amendments in the text and formulas, new acknowledgments added, typos correcte

Basic Principles of 4D Dilatonic Gravity and Some of Their Consequences for Cosmology, Astrophysics and Cosmological Constant Problem

We present a class of simple scalar-tensor models of gravity with one scalar field (dilaton $\Phi$) and only one unknown function (cosmological potential $U(\Phi)$). These models might be considered as a stringy inspired ones with broken SUSY. They have the following basic properties: 1) Positive dilaton mass, $m_\Phi$, and positive cosmological constant $\Lambda$, define two extremely different scales. The models under consideration are consistent with the known experimental facts if $m_\Phi > 10^{-3} eV$ and $\Lambda=\Lambda^{obs}\sim 10^{-56} cm^{-2}$. 2) Einstein week equivalence principle is strictly satisfied and extended to scalar-tensor theories of gravity using a novel form of principle of "constancy of fundamental constants". 3) The dilaton plays simultaneously role of inflation field and quintessence field and yields a sequential hyper-inflation with graceful exit to asymptotic de Sitter space-time which is an attractor, and is approached as $\exp(-\sqrt{3\Lambda^{obs}} ct/2)$. The time duration of inflation is $\Delta t_{infl} \sim m_\Phi^{-1}$. 4) Ultra-high frequency ($\omega_\Phi \sim m_\Phi$) dilatonic oscillations take place in asymptotic regime. 5) No fine tuning. (The Robertson-Walker solutions of general type have the above properties.) 6) A novel adjustment mechanism for cosmological constant problem seems to be possible: the huge value of cosmological constant in the stringy frame is re-scaled to its observed value by dilaton after transition to phenomenological frame.Comment: 34 pages, 5 figures, LaTeX file, added references, corrected typo

Novel representation of the general Heun's functions. Back to the beginning

We study a novel type of solutions of the general Heun's equation, based on its symmetric form. We derive the symmetry group of this equation which is a proper extension of the Mobius group. The new series solution treat simultaneously and on an equal footing all singular points.Comment: 5 pages, tex file, no figures, Proceedings of the International Conference DAYS on DIFFRACTION 201

A realistic model of a neutron star in minimal dilatonic gravity

We present a derivation of the basic equations and boundary conditions for relativistic static spherically symmetric stars (SSSS) in the model of minimal dilatonic gravity (MDG) which offers an alternative and simultaneous description of the effects of dark matter (DM) and dark energy (DE) using one dilaton field $\Phi$. The numerical results for a realistic equation of state (EOS) MPA1 of neutron matter are presented for the first time. The three very different scales, the Compton length of the scalar field $\lambda_\Phi$, the star's radius $r^*$, and the finite radius of the MDG Universe $r_{U}$ are a source of numerical difficulties. Owing to the introduction of a new dark scalar field $\varphi=\ln(1+\ln\Phi)$, we have been able to study numerically an unprecedentedly large interval of $\lambda_\Phi$ and have discovered the existence of $\lambda_\Phi^{crit}\approx 2.1$\ km for a neutron star with MPA1 EOS. This is related to the bifurcation of the physical domain in the phase space of the system. Some novel physical consequences are discussed.Comment: 10 pages, Latex file, 10 figures, English correcte

On gravitational waves from classical three body problem

Using an effective one body approach we describe in detail gravitational waves from classical three body problem on a non-rotating straight line and derive their basic physical characteristics. Special attention is paid to the irregular motions of such systems and to the significance of double and triple collisions. The conclusive role of the collinear solutions is also discussed in short. It is shown that the residuals may contain information about irregular motion of the source of gravitational waves.Comment: Latex file, 12 figures, English corrected, short comments added in the text and in the abstrac

Withholding Potentials, Absence of Ghosts and Relationship between Minimal Dilatonic Gravity and f(R) Theories

We study the relation between Minimal Dilatonic Gravity (MDG) and f(R) theories of gravity and establish strict conditions for their {\em global} equivalence. Such equivalence takes place only for a certain class of cosmological potentials, dubbed here {\em withholding potentials}, since they prevent change of the sign of dilaton $\Phi$. The withholding property ensures the attractive character of gravity, as well as absence of ghosts and a tachyon in the gravi-dilaton sector and yields certain asymptotic of the admissible functions $f(R)$. Large classes of withholding cosmological potentials and functions $f(R)$ are found and described in detail. It is shown that the popular choices of $f(R)$ functions are not withholding ones. The particle content of the gravi-dilaton sector is found using perturbation theory around de Sitter vacuum of MDG. The graviton remains massless, since it obeys conformal invariant field equation in the de Sitter space-time. The $R/6$ term in the conformal invariant wave operator introduces a very small mass scale $m_{R} \approx 1.5 \times 10^{-38} m_e$, $m_e$ being the mass of the electron. The mass of the dilaton is much larger: $m\gtrapprox 10^{29} m_{R}$. Two new phenomena: scalaron waves and induction of gravitational waves by the scalaron field are discussed using the derived wave equations for scalaron and graviton. The MDG and f(R) theories are shown to predict physical deviations from GR. Seemingly, the MDG and f(R) theories, when globally equivalent, offer a unified description of dark energy and dark matter.Comment: LaTeX file, 13 pages, 4 figures. Typos corrected. A slightly shortened version: the examples of f(R) models which are not equivalent to the MDG removed together with the corresponding figures (See version 2.). Some remarks removed or replaced by new ones. To appear in Physical Review

Numerical stability of the electromagnetic quasinormal and quasibound modes of Kerr black holes

The proper understanding of the electromagnetic counterpart of gravity-waves emitters is of serious interest to the multimessenger astronomy. In this article, we study the numerical stability of the quasinormal modes (QNM) and quasibound modes (QBM) obtained as solutions of the Teukolsky Angular Equation and the Teukolsky Radial Equation with appropriate boundary conditions. We use the epsilon-method for the system featuring the confluent Heun functions to study the stability of the spectra with respect to changes in the radial variable. We find that the QNM and QBM are stable in certain regions of the complex plane, just as expected, while the third "spurious" spectrum was found to be numerically unstable and thus unphysical. This analysis shows the importance of understanding the numerical results in the framework of the theory of the confluent Heun functions, in order to be able to distinguish the physical spectra from the numerical artifacts.Comment: 19 pages, 6 figures, final published versio

New results for electromagnetic quasinormal and quasibound modes of Kerr black holes

The perturbations of the Kerr metric and the miracle of their exact solutions play a critical role in the comparison of predictions of general relativity with astrophysical observations of compact massive objects. The differential equations governing the late-time ring-down of the perturbations of the Kerr metric, the Teukolsky Angular Equation and the Teukolsky Radial Equation, can be solved analytically in terms of confluent Heun functions. In this article, we solve numerically the spectral system formed by those exact solutions and we obtain the electromagnetic (EM) spectra of the Kerr black hole. Because of the novel direct way of imposing the boundary conditions, one is able to discern three different types of spectra: the well-known quasinormal modes (QNM), the symmetric with respect to the real axis quasibound modes (QBM) and a spurious spectrum who is radially unstable. This approach allows clearer justification of the term "spurious" spectrum, which may be important considering the recent interest in the spectra of the electromagnetic counterparts of events producing gravitational waves.Comment: 12 pages, 8 figures, some sections reworked and new comments added, final published version. arXiv admin note: substantial text overlap with arXiv:1112.031