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Penrose hypothesis and instability of naked singularities in static spherically symmetric systems with scalar fields
Authors:
A. V. Tugay,
V. I. Zhdanov,
Yu. V. Taistra
Abstract:
General relativistic static spherically symmetric (SSS) asymptotically flat configurations with scalar fields typically contain naked singularities at the center. We consider minimally coupled scalar fields with power-law potentials leading to the Coulomb asymptotic of the field $φ(r)\approx Q/r$ for large values of the radial variable r. The configurations are uniquely defined by total mass and a…
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General relativistic static spherically symmetric (SSS) asymptotically flat configurations with scalar fields typically contain naked singularities at the center. We consider minimally coupled scalar fields with power-law potentials leading to the Coulomb asymptotic of the field $φ(r)\approx Q/r$ for large values of the radial variable r. The configurations are uniquely defined by total mass and a Q-parameter characterizing the strength of the scalar field at spatial infinity. The focus is on the linear stability against radial (monopole) perturbations of the SSS configurations satisfying conditions of asymptotic flatness. Our numerical investigations show the existence of divergent modes of small perturbations against the static background, at least for sufficiently small values of Q. This means instability of the configurations, confirming the well-known Penrose conjecture about the nonexistence of naked singularities - in this particular case. On the other hand, we have not found divergent modes of linear radial perturbations for sufficiently large Q.
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Submitted 13 July, 2025;
originally announced July 2025.
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Static spherically symmetric solutions of the f(R) gravity
Authors:
V. I. Zhdanov
Abstract:
Static spherically symmetric (SSS) solutions of f(R) gravity are studied in the Einstein frame. The solutions involve SSS configuration mass M and scalaron mass $μ$ (in geometrized units); for typical astrophysical masses, the dimensionless parameter $Mμ$ has very large value. We found analytic solutions on a finite interval for $Mμ\to \infty$ in case of a family of scalaron potentials. The asympt…
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Static spherically symmetric (SSS) solutions of f(R) gravity are studied in the Einstein frame. The solutions involve SSS configuration mass M and scalaron mass $μ$ (in geometrized units); for typical astrophysical masses, the dimensionless parameter $Mμ$ has very large value. We found analytic solutions on a finite interval for $Mμ\to \infty$ in case of a family of scalaron potentials. The asymptotically flat solutions on $(0,\infty)$ have been studied numerically for $Mμ$ up to $10^{20}$ in case of the quadratic f(R) model.
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Submitted 25 July, 2025; v1 submitted 13 March, 2025;
originally announced March 2025.
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Universal structure of spherically symmetric astrophysical objects in f(R) gravity
Authors:
Valery I. Zhdanov
Abstract:
$f(R)$ gravity is a well-known modification of General Relativity, that can be reduced to a scalar-tensor theory by a conformal transformation (Einstein frame). We study static spherically symmetric (SSS) asymptotically flat vacuum configurations of the $f(R)…
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$f(R)$ gravity is a well-known modification of General Relativity, that can be reduced to a scalar-tensor theory by a conformal transformation (Einstein frame). We study static spherically symmetric (SSS) asymptotically flat vacuum configurations of the $f(R)$ gravity in the Einstein frame for three known scalaron potentials. The main attention is paid to solutions in case of astrophysically relevant configuration masses and scalaron mass $μ$ larger than several $meV$. Analytical and numerical analysis reveals remarkably similar properties of some elements of the SSS solutions for different $M$, $μ$ and sizes of the scalarization region $r_0$. In particular, the scalaron field has universal behavior regardless of the configurations mass and $r_0>100 r_g$ in case of each of the models considered. Moreover, some elements of the solutions are practically the same for the different models. Asymptotic parameters of the metric near the naked singularity at the center of the SSS configuration are obtained for all the models.
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Submitted 25 July, 2025; v1 submitted 4 December, 2024;
originally announced December 2024.
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Spherically symmetric configurations in the quadratic $f(R)$ gravity
Authors:
V. I. Zhdanov,
O. S. Stashko,
Yu. V. Shtanov
Abstract:
We study spherically symmetric configurations of the quadratic $f(R)$ gravity in the Einstein frame. In case of a purely gravitational system, we have determined the global qualitative behavior of the metric and the scalaron field for all static solutions satisfying the conditions of asymptotic flatness. These solutions are proved to be regular everywhere except for a naked singularity at the cent…
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We study spherically symmetric configurations of the quadratic $f(R)$ gravity in the Einstein frame. In case of a purely gravitational system, we have determined the global qualitative behavior of the metric and the scalaron field for all static solutions satisfying the conditions of asymptotic flatness. These solutions are proved to be regular everywhere except for a naked singularity at the center; they are uniquely determined by the total mass $\mathfrak{M}$ and the "scalar charge" $Q$ characterizing the strength of the scalaron field at spatial infinity. The case $Q=0$ yields the Schwarzschild solution, but an arbitrarily small $Q\ne 0$ leads to the appearance of a central naked singularity having a significant effect on the neighboring region, even when the space-time metric in the outer region is practically insensitive to the scalaron field. Approximation procedures are developed to derive asymptotic relations near the naked singularity and at spatial infinity, and the leading terms of the solutions are presented. We investigate the linear stability of the static solutions with respect to radial perturbations satisfying the null Dirichlet boundary condition at the center and numerically estimate the range of parameters corresponding to stable/unstable configurations. In particular, the configurations with sufficiently small $Q$ turn out to be linearly unstable.
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Submitted 27 July, 2024; v1 submitted 25 March, 2024;
originally announced March 2024.
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Discreteness effects in $N$-body simulations of warm dark matter
Authors:
Yuri Shtanov,
Valery I. Zhdanov
Abstract:
In cosmological $N$-body simulations of warm dark matter, thermal velocities of dark-matter particles are sometimes taken into account by adding random initial velocities to the particles of simulation. However, a particle in the $N$-body system represents a huge collection of dark-matter particles, whose average thermal velocity is very close to zero. We consider justification of the procedure of…
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In cosmological $N$-body simulations of warm dark matter, thermal velocities of dark-matter particles are sometimes taken into account by adding random initial velocities to the particles of simulation. However, a particle in the $N$-body system represents a huge collection of dark-matter particles, whose average thermal velocity is very close to zero. We consider justification of the procedure of adding thermal velocities in $N$-body simulations and build a simple model of their influence on the power spectrum. Our model captures the physical effect of suppression of the power spectrum at small wave numbers and also explains its artificial enhancement at large wave numbers, observed in numerical simulations with added thermal velocities. The cause of this enhancement is the disturbance of the growth rate of the density profile introduced when adding random initial thermal velocities. Specifically, the model predicts a turnover in the behavior of the simulated power spectrum at a certain wave number $k_*$, beyond which it grows as $P (k) \propto k^2$. Our treatment is generalized to a system consisting of several matter components with different thermal velocity dispersion. We also estimate the effects of discreteness related to the bulk velocity field and establish the conditions under which these effects dominate over those of thermal velocities.
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Submitted 27 March, 2024; v1 submitted 15 July, 2023;
originally announced July 2023.
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Quasi-normal modes of naked singularities in presence of non-linear scalar fields
Authors:
O. S. Stashko,
O. V. Savchuk,
V. I. Zhdanov
Abstract:
We study linear perturbations against static spherically symmetric background configurations of General Relativity with a real scalar field (SF), which is minimally coupled with gravity; it is non-linear due to the presence of the self-action potential. The background solutions have a naked singularity at the center of the configuration. The focus is on the stability of the background and fundamen…
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We study linear perturbations against static spherically symmetric background configurations of General Relativity with a real scalar field (SF), which is minimally coupled with gravity; it is non-linear due to the presence of the self-action potential. The background solutions have a naked singularity at the center of the configuration. The focus is on the stability of the background and fundamental frequencies of the quasi-normal modes (QNM) of the axial perturbations in the Regge-Wheeler gauge. The problem is reduced to one hyperbolic master equation with an effective potential $W_{\rm eff}$, which turns out to be positive for a general non-negative SF potential; this ensures the linear stability with respect to this kind of perturbations.
For numerical simulations, the SF potential was chosen in the power-law form $V(φ)\simφ^{2n}$ with $2<n\le 40$. We extracted the fundamental frequencies of QNM for different $n$ and various sets of the background configuration parameters. The results show that even for a small background SF, there is a significant difference between the fundamental frequencies and ones in case of the Schwarzschild background. The results are also compared with the case of the Fisher-Janis-Newman-Winicour background dealing with a massless linear scalar field.
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Submitted 17 December, 2023; v1 submitted 9 July, 2023;
originally announced July 2023.
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Pion stars embedded in neutrino clouds
Authors:
O. S. Stashko,
O. V. Savchuk,
L. M. Satarov,
I. N. Mishustin,
M. I. Gorenstein,
V. I. Zhdanov
Abstract:
We study self-gravitating multi-pion systems (pion stars) in a state of the Bose condensate. To ensure stability of such stars, it is assumed that they are immersed in the lepton background. Two different phenomenological equations of state (EoS) for the pion matter are used, some of them having the first order phase transition. The model parameters are chosen to reproduce the recent lattice QCD d…
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We study self-gravitating multi-pion systems (pion stars) in a state of the Bose condensate. To ensure stability of such stars, it is assumed that they are immersed in the lepton background. Two different phenomenological equations of state (EoS) for the pion matter are used, some of them having the first order phase transition. The model parameters are chosen to reproduce the recent lattice QCD data at zero temperature and large isospin chemical potential. It is shown that the mass-radius diagrams of pion stars obtained with phenomenological EoS are close to ones calculated in the ideal gas model. We analyze properties of neutrino clouds which are necessary for stabilizing the pion stars.
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Submitted 10 March, 2023;
originally announced March 2023.
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Circular orbits of test particles interacting with massless linear scalar field of the naked singularity
Authors:
O. S. Stashko,
V. I. Zhdanov
Abstract:
We study effects of the particles coupling with scalar field (SF) on the distribution of stable circular orbits (SCO) around the naked singularity described by the well-known Fisher-Janis-Newman-Winicour solution. The power-law and exponential models of the particle--SF interaction are analyzed. The focus is on the non-connected SCO distributions. We show that coupling between particles and SF can…
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We study effects of the particles coupling with scalar field (SF) on the distribution of stable circular orbits (SCO) around the naked singularity described by the well-known Fisher-Janis-Newman-Winicour solution. The power-law and exponential models of the particle--SF interaction are analyzed. The focus is on the non-connected SCO distributions. We show that coupling between particles and SF can essentially complicate the topology of the SCO distributions. In particular, it can lead to new non-overlapping SCO regions, which are separated by unstable orbits and/or by regions where the circular orbits do not exist.
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Submitted 21 October, 2022; v1 submitted 31 August, 2022;
originally announced September 2022.
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New Horizons for Fundamental Physics with LISA
Authors:
K. G. Arun,
Enis Belgacem,
Robert Benkel,
Laura Bernard,
Emanuele Berti,
Gianfranco Bertone,
Marc Besancon,
Diego Blas,
Christian G. Böhmer,
Richard Brito,
Gianluca Calcagni,
Alejandro Cardenas-Avendaño,
Katy Clough,
Marco Crisostomi,
Valerio De Luca,
Daniela Doneva,
Stephanie Escoffier,
Jose Maria Ezquiaga,
Pedro G. Ferreira,
Pierre Fleury,
Stefano Foffa,
Gabriele Franciolini,
Noemi Frusciante,
Juan García-Bellido,
Carlos Herdeiro
, et al. (116 additional authors not shown)
Abstract:
The Laser Interferometer Space Antenna (LISA) has the potential to reveal wonders about the fundamental theory of nature at play in the extreme gravity regime, where the gravitational interaction is both strong and dynamical. In this white paper, the Fundamental Physics Working Group of the LISA Consortium summarizes the current topics in fundamental physics where LISA observations of GWs can be e…
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The Laser Interferometer Space Antenna (LISA) has the potential to reveal wonders about the fundamental theory of nature at play in the extreme gravity regime, where the gravitational interaction is both strong and dynamical. In this white paper, the Fundamental Physics Working Group of the LISA Consortium summarizes the current topics in fundamental physics where LISA observations of GWs can be expected to provide key input. We provide the briefest of reviews to then delineate avenues for future research directions and to discuss connections between this working group, other working groups and the consortium work package teams. These connections must be developed for LISA to live up to its science potential in these areas.
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Submitted 3 May, 2022;
originally announced May 2022.
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Singularities in static spherically symmetric configurations of General Relativity with strongly nonlinear scalar fields
Authors:
O. S. Stashko,
V. I. Zhdanov
Abstract:
There are a number of publications on relativistic objects dealing either with black holes or naked singularities in the center. Here we show that there exist static spherically symmetric solutions of Einstein equations with a strongly nonlinear scalar field, which allow the appearance of singularities of a new type (``spherical singularities'') outside the center of curvature coordinates. As the…
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There are a number of publications on relativistic objects dealing either with black holes or naked singularities in the center. Here we show that there exist static spherically symmetric solutions of Einstein equations with a strongly nonlinear scalar field, which allow the appearance of singularities of a new type (``spherical singularities'') outside the center of curvature coordinates. As the example, we consider a scalar field potential $\sim$$\sinh(φ^{2n}),\,n>2$, which grows rapidly for large field values. The space-time is assumed to be asymptotically flat. We fulfill a numerical investigation of solutions with different $n$ for different parameters, which define asymptotic properties at spatial infinity. Depending on the configuration parameters, we show that the distribution of the stable circular orbits of test bodies around the configuration is either similar to that in the case of the Schwarzschild solution (thus mimicking an ordinary black hole), or it contains additional rings of unstable orbits.
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Submitted 3 October, 2021; v1 submitted 4 September, 2021;
originally announced September 2021.
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Thin accretion discs around spherically symmetric configurations with nonlinear scalar fields
Authors:
O. S. Stashko,
V. I. Zhdanov,
A. N. Alexandrov
Abstract:
We study stable circular orbits (SCO) around static spherically symmetric configuration of General Relativity with a non-linear scalar field (SF). The configurations are described by solutions of the Einstein-SF equations with monomial SF potential $V(φ)=|φ|^{2n}$, $n>2$, under the conditions of the asymptotic flatness and behavior of SF $φ\sim 1/r$ at spatial infinity. We proved that under these…
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We study stable circular orbits (SCO) around static spherically symmetric configuration of General Relativity with a non-linear scalar field (SF). The configurations are described by solutions of the Einstein-SF equations with monomial SF potential $V(φ)=|φ|^{2n}$, $n>2$, under the conditions of the asymptotic flatness and behavior of SF $φ\sim 1/r$ at spatial infinity. We proved that under these conditions the solution exists and is uniquely defined by the configuration mass $M>0$ and scalar "charge" $Q$. The solutions and the space-time geodesics have been investigated numerically in the range $n\le40$, $|Q|\le 60$, $M\le60$. We focus on how nonlinearity of the field affects properties of SCO distributions (SCOD), which in turn affect topological form of the thin accretion disk around the configuration. Maps are presented showing the location of possible SCOD types for different $M,Q,n$. We found many differences from the Fisher-Janis-Newman-Winicour metric (FJNW) dealing with the linear SF, though basic qualitative properties of the configurations have much in common with the FJNW case. For some values of $n$, a topologically new SCOD type was discovered that is not available for the FJNW metric. All images of accretion disks have a dark spot in the center (mimicking an ordinary black hole), either because there is no SCO near the center or because of the strong deflection of photon trajectories near the singularity.
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Submitted 11 July, 2021;
originally announced July 2021.
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Verification of Einstein's formula for gravitational deflection of light using observations of Galactic microlensing
Authors:
A. N. Alexandrov,
V. M. Sliusar,
V. I. Zhdanov
Abstract:
The potential of the gravitational microlensing inside our Galaxy for testing the Einstein formula for the gravitational light deflection is discussed. For this purpose, the lens mapping is modified by introducing parameter eps, which characterizes the deviation from this formula. An example of such deviation described by a simple power law is analyzed. We formed a sample of 100 microlensing light…
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The potential of the gravitational microlensing inside our Galaxy for testing the Einstein formula for the gravitational light deflection is discussed. For this purpose, the lens mapping is modified by introducing parameter eps, which characterizes the deviation from this formula. An example of such deviation described by a simple power law is analyzed. We formed a sample of 100 microlensing light curves using the data of the Optical Gravitational Lensing Experiment (OGLE) for 2018. The resulting eps value does not contradict the General Relativity within 1 percent errors.
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Submitted 16 April, 2020;
originally announced April 2020.
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Static spherically symmetric configurations with N non-linear scalar fields: global and asymptotic properties
Authors:
V. I. Zhdanov,
O. S. Stashko
Abstract:
In case of a spherically symmetric non-linear scalar field (SF) in flat space, besides singularity at the center, spherical singularities can occur for non-zero values of radial variable $r>0$. We show that in the General Relativity the gravitational field suppresses the occurrence of the spherical singularities under some generic conditions. Our consideration deals with asymptotically flat space-…
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In case of a spherically symmetric non-linear scalar field (SF) in flat space, besides singularity at the center, spherical singularities can occur for non-zero values of radial variable $r>0$. We show that in the General Relativity the gravitational field suppresses the occurrence of the spherical singularities under some generic conditions. Our consideration deals with asymptotically flat space-times around static spherically symmetric configurations in presence of $N$ non-linear SFs, which are minimally coupled to gravity. Constraints are imposed on the SF potentials, which guarantee a monotonicity of the fields as functions of radial variable; also the potentials are assumed to be exponentially bounded. We give direct proof that solutions of the joint system of Einstein -- SF equations satisfying the conditions of asymptotic flatness are regular for all values of $r$, except for naked singularities in the center $r=0$ in the Schwarzschild (curvature) coordinates. Asymptotic relations for SF and metric near the center are derived, which appear to be remarkably similar to the case of the Fisher solution for free SF. These relations determine two main types of the corresponding geodesic structure when photons can be captured by the singularity or not depending on the existence of the photon sphere. To illustrate, the case of one SF with monomial potential is analyzed in detail numerically. We show that the image of the accretion disk around the singularity, observed from infinity, can take the form of a bright ring with a dark spot in the center, like the case of an ordinary black hole.
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Submitted 10 March, 2020; v1 submitted 1 December, 2019;
originally announced December 2019.
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Scalar field vs hydrodynamic models in the homogeneous isotropic cosmology
Authors:
V. I. Zhdanov,
S. S. Dylda
Abstract:
We study relations between hydrodynamical (H) and scalar field (SF) models of the dark energy in the early Universe. Main attention is paid to SF described by the canonical Lagrangian within the homogeneous isotropic spatially flat cosmology. We analyze requirements that guarantee the same cosmological history for the SF and H-models at least for solutions with specially chosen initial conditions…
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We study relations between hydrodynamical (H) and scalar field (SF) models of the dark energy in the early Universe. Main attention is paid to SF described by the canonical Lagrangian within the homogeneous isotropic spatially flat cosmology. We analyze requirements that guarantee the same cosmological history for the SF and H-models at least for solutions with specially chosen initial conditions and we present a differential equation for the SF potential that ensures such a restricted equivalence of the SF and H-models. Also, we derived a condition that guarantees an approximate equivalence when there is a small difference between energy momentum tensors of the models. The "equivalent" scalar field potentials for linear equations of state (EOS) are found in an explicit form, we also present an examples with more complicated EOS.
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Submitted 11 June, 2018; v1 submitted 6 January, 2018;
originally announced January 2018.
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Qualitative types of cosmological evolution in hydrodynamic models with barotropic equation of state
Authors:
V. I. Zhdanov,
S. S. Dylda
Abstract:
We study solutions of the Friedmann equations in case of the homogeneous isotropic Universe filled with a perfect fluid. The main points concern the monotony properties of the solutions, the possibility to extend the solutions on all times and occurrence of singularities. We present a qualitative classification of all possible solutions in case of the general smooth barotropic equation of state of…
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We study solutions of the Friedmann equations in case of the homogeneous isotropic Universe filled with a perfect fluid. The main points concern the monotony properties of the solutions, the possibility to extend the solutions on all times and occurrence of singularities. We present a qualitative classification of all possible solutions in case of the general smooth barotropic equation of state of the fluid, provided the speed of sound is finite. The list of possible scenarios includes analogs of the "Big Rip" in the future and/or in the past as well as singularity free solutions and oscillating Universes. Extensions of the results to the multicomponent fluids are discussed.
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Submitted 12 July, 2017;
originally announced July 2017.
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Spherically symmetric configurations of General Relativity in presence of scalar field: separation of test body circular orbits
Authors:
O. S. Stashko,
V. I. Zhdanov
Abstract:
We study test-body orbits in the gravitational field of a static spherically symmetric object in presence of a minimally coupled nonlinear scalar field. We generated a two-parametric family of scalar field potentials, which allow finding solutions of Einstein's equations in an analytic form. The results are presented by means of hypergeometric functions; they describe either a naked singularity (N…
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We study test-body orbits in the gravitational field of a static spherically symmetric object in presence of a minimally coupled nonlinear scalar field. We generated a two-parametric family of scalar field potentials, which allow finding solutions of Einstein's equations in an analytic form. The results are presented by means of hypergeometric functions; they describe either a naked singularity (NS) or a black hole (BH). Our numerical investigation shows that in both cases the stable circular orbits can form separated (non-connected) regions around the configuration. We found existence conditions for such separated regions and present examples for some family parameters in case of NS and BH. The results may be of interest for testing models of the dynamical dark energy.
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Submitted 13 July, 2017; v1 submitted 9 February, 2017;
originally announced February 2017.
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Astrophysical and cosmological problems of invisible mass and dark energy in the Universe
Authors:
P. Belli,
L. A. Berdina,
R. Bernabei,
A. Bogdan,
R. S. Boiko,
A. Yu. Burgazli,
F. Cappella,
R. Cerulli,
D. M. Chernyak,
F. A. Danevich,
A. d'Angelo,
M. V. Eingorn,
S. H. Fakhr,
E. Fedorova,
E. N. Galashov,
A. Giuliani,
B. I. Hnatyk,
A. Incicchitti,
G. Ivashchenko,
V. V. Kobychev,
O. O. Kobzar,
H. Kraus,
B. N. Kropivyansky,
A. V. Kudinova,
Yu. A. Kulinich
, et al. (31 additional authors not shown)
Abstract:
The Workshop on results of the Project Kosmomikrofizyka-2 (Astroparticle Physics) of the National Academy of Sciences (NAS) of Ukraine "Astrophysical and cosmological problems of invisible mass and dark energy in the Universe" was held on November 21-22, 2012 in the Institute for Nuclear Research, Kyiv, Ukraine (http://lpd.kinr.kiev.ua/kmf12). This Project was carried out during three years (2010-…
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The Workshop on results of the Project Kosmomikrofizyka-2 (Astroparticle Physics) of the National Academy of Sciences (NAS) of Ukraine "Astrophysical and cosmological problems of invisible mass and dark energy in the Universe" was held on November 21-22, 2012 in the Institute for Nuclear Research, Kyiv, Ukraine (http://lpd.kinr.kiev.ua/kmf12). This Project was carried out during three years (2010-2012) by scientists from various universities and institutes of the National Academy of Sciences of Ukraine; it was a logical continuation of the previous scientific program of the NAS of Ukraine "Researches of structure and composition of the Universe, hidden mass and dark energy (Kosmomikrofizyka)" in 2007-2009. These programs were devoted to theoretical and experimental investigations in astronomy, astrophysics, cosmology, physics of atomic nuclei and particle physics, which are related with the problems of dark matter and dark energy in the Universe.
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Submitted 16 April, 2013;
originally announced April 2013.
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Non-metric Generalizations of Relativistic Gravitational Theory and Observational Data Interpretation
Authors:
A. N. Alexandrov,
I. B. Vavilova,
V. I. Zhdanov
Abstract:
We discuss theoretical formalisms concerning with experimental verification of General Relativity (GR). Non-metric generalizations of GR are considered and a system of postulates is formulated for metric-affine and Finsler gravitational theories. We consider local observer reference frames to be a proper tool for comparing predictions of alternative theories with each other and with the observat…
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We discuss theoretical formalisms concerning with experimental verification of General Relativity (GR). Non-metric generalizations of GR are considered and a system of postulates is formulated for metric-affine and Finsler gravitational theories. We consider local observer reference frames to be a proper tool for comparing predictions of alternative theories with each other and with the observational data. Integral formula for geodesic deviation due to the deformation of connection is obtained. This formula can be applied for calculations of such effects as the bending of light and time-delay in presence of non-metrical effects.
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Submitted 30 October, 2006;
originally announced October 2006.