This comment is devoted to the recalculation of the Casimir energy of a
massless scalar field in the Kerr black hole surrounded by quintessence
derived in [B. Toshmatov, Z. Stuchlík and B. Ahmedov, Eur. Phys. J. Plus
132, 98 (2017)] and its comparison with the results recently obtained in
[V. B. Bezerra, M. S. Cunha, L. F. F. Freitas and C. R. Muniz, Mod.
Phys. Lett. A 32, 1750005 (2017)] in the spacetime [S. G. Ghosh, Eur.
Phys. J. C 76, 222 (2016)]. We have shown that in the more realistic
spacetime which does not have the failures illustrated here, the Casimir
energy is significantly bigger than that derived in [V. B. Bezerra, M.
S. Cunha, L. F. F. Freitas and C. R. Muniz, Mod. Phys. Lett. A 32,
1750005 (2017)], and the difference becomes crucial especially in the
regions of near horizons of the spacetime.
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The model is constructed to describe the Schwarzschild-like black hole
enclosed in the dust cosmological background. It is an exact solution of
Einstein equations for spherically symmetric dust distribution, and is a
special case of Lemaitre-Tolman-Bondi solutions. The motion of the test
particle in the model is investigated in comoving coordinate frame.
Observable velocity of the particle is found from geodesic equations. It
is shown that chosen reference system does not allow to solve the
problem of 'all or nothing' behavior.
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We study behaviour of gravitational waves in the recently introduced
general relativistic polytropic spheres containing a region of trapped
null geodesics extended around radius of the stable null circular
geodesic that can exist for the polytropic index N > 2.138 and the
relativistic parameter, giving ratio of the central pressure
pc to the central energy density ρc, higher than σ
= 0.677. In the trapping zones of such polytropes, the effective
potential of the axial gravitational wave perturbations resembles those
related to the ultracompact uniform density objects, giving thus similar
long-lived axial gravitational modes. These long-lived linear
perturbations are related to the stable circular null geodesic and due
to additional non-linear phenomena could lead to conversion of the
trapping zone to a black hole. We give in the eikonal limit examples of
the long-lived gravitational modes, their oscillatory frequencies and
slow damping rates, for the trapping zones of the polytropes with N in
(2.138,4). However, in the trapping polytropes the long-lived damped
modes exist only for very large values of the multipole number l >
50, while for smaller values of l the numerical calculations indicate
existence of fast growing unstable axial gravitational modes. We
demonstrate that for polytropes with N >= 3.78, the trapping region
is by many orders smaller than extension of the polytrope, and the mass
contained in the trapping zone is about 10-3 of the total
mass of the polytrope. Therefore, the gravitational instability of such
trapping zones could serve as a model explaining creation of central
supermassive black holes in galactic halos or galaxy clusters.
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We investigate ringed accretion disks composed of two tori (rings)
orbiting on the equatorial plane of a central supermassive Kerr black
hole. We discuss the emergence of the instability phases of each ring of
the macro-configuration (ringed disk) according to the Paczynski
violation of mechanical equilibrium. In the full general relativistic
treatment, we consider the effects of the geometry of the Kerr
spacetimes relevant to the characterization of the evolution of these
configurations. The discussion of ring stability in different spacetimes
enables us to identify particular classes of central Kerr attractors
depending on their dimensionless spin. As a result of this analysis, we
set constraints on the evolutionary schemes of the ringed disks relative
to the torus morphology and on their rotation relative to the central
black hole and to each other. The dynamics of the unstable phases of
this system is significant for the high-energy phenomena related to
accretion onto supermassive black holes in active galactic nuclei and
the extremely energetic phenomena in quasars, which could be observed in
their X-ray emission.
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We construct regular rotating black hole and no-horizon spacetimes based
on the recently introduced spherically symmetric generic regular black
hole spacetimes related to electric or magnetic charge under nonlinear
electrodynamics coupled to general relativity that for special values of
the spacetime parameters reduce to the Bardeen and Hayward spacetimes.
We show that the weak and strong energy conditions are violated inside
the Cauchy horizons of these generic rotating black holes. We give the
boundary between the rotating black hole and no-horizon spacetimes and
determine the black hole horizons and the boundary of the ergosphere. We
introduce the separated Carter equations for the geodesic motion in
these rotating spacetimes. For the most interesting new class of the
regular spacetimes, corresponding for magnetic charges to the Maxwell
field in the weak field limit of the nonlinear electrodynamics, we
determine the structure of the circular geodesics and discuss their
properties. We study the epicyclic motion of a neutral particle moving
along the stable circular orbits around the "Maxwellian" rotating
regular black holes. We show that epicyclic frequencies measured by the
distant observers and related to the oscillatory motion of the neutral
test particle along the stable circular orbits around the rotating
singular and regular Maxwellian black holes are always smaller than ones
in the Kerr spacetime.
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Quintessential dark energy with density ρ and pressure p is governed by
an equation of state of the form p=ωqρ with the
quintessential parameter ω_qin (-1;-1/3). We derive the geometry of
quintessential rotating black holes, generalizing thus the Kerr
spacetimes. Then we study the quintessential rotating black hole
spacetimes with the special value of ωq = -2/3 when the
resulting formulae are simple and easily tractable. We show that such
special spacetimes can exist for the dimensionless quintessential
parameter c < 1/6 and determine the critical rotational parameter
a0 separating the black hole and naked singularity spacetime
in dependence on the quintessential parameter c . For the spacetimes
with ωq = -2/3 we give all the black hole characteristics and
demonstrate local thermodynamical stability. We present the integrated
geodesic equations in separated form and study in details the circular
geodetical orbits. We give radii and parameters of the photon circular
orbits, marginally bound and marginally stable orbits. We stress that
the outer boundary on the existence of circular geodesics, given by the
so-called static radius where the gravitational attraction of the black
hole is balanced by the cosmic repulsion, does not depend on the
dimensionless spin of the rotating black hole, similarly to the case of
the Kerr-de Sitter spacetimes with vacuum dark energy. We also give
restrictions on the dimensionless parameters c and a of the spacetimes
allowing for existence of stable circular geodesics. Finally, using
numerical methods we generalize the discussion of the circular geodesics
to the black holes with arbitrary quintessential parameter
ωq.
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We demonstrate that in the framework of standard general relativity,
polytropic spheres with properly fixed polytropic index n and
relativistic parameter σ , giving a ratio of the central pressure
pc to the central energy density ρc , can contain
a region of trapped null geodesics. Such trapping polytropes can exist
for n >2.138 , and they are generally much more extended and massive
than the observed neutron stars. We show that in the n - σ parameter
space, the region of allowed trapping increases with the polytropic
index for intervals of physical interest, 2.138 <n <4 . Space
extension of the region of trapped null geodesics increases with both
increasing n and σ >0.677 from the allowed region. In order to relate
the trapping phenomenon to astrophysically relevant situations, we
restrict the validity of the polytropic configurations to their
extension rextr corresponding to the gravitational mass M ̃2
M☉ of the most massive observed neutron stars. Then, for the
central density ρc̃1 015 g cm-3 , the
trapped regions are outside rextr for all values of 2.138
<n <4 ; for the central density ρc̃5 ×1 015
g cm-3 , the whole trapped regions are located inside
rextr for 2.138 <n <3.1 ; while for ρc̃1
016 g cm-3 , the whole trapped regions are inside
rextr for all values of 2.138 <n <4 , guaranteeing
astrophysically plausible trapping for all considered polytropes. The
region of trapped null geodesics is located close to the polytrope
center and could have a relevant influence on the cooling of such
polytropes or binding of gravitational waves in their interior.
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In the weak field approximation, we study the gravitational lensing near
the regular Bardeen, Hayward and Ayon-Beato-Garcia (ABG) black holes
surrounded by plasma. The exact expressions for the deflection angle of
the photons due to the effect of the gravitational field and the plasma
have been obtained. The analysis of the image source brightness
magnification in the background spacetimes of (i) Bardeen, (ii) Hayward
and (iii) ABG regular black holes have shown that the increase of the
corresponding charge of regular black hole causes the increase in the
magnification of the source image. In addition to the primary ring, one
may observe the secondary ring with smaller magnification. The influence
of the plasma with (i) constant and (ii) radial power law electron
density to the magnification of the source image has been studied.
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A new intrinsically-relativistic kinetic mechanism for generation of
nonisotropic relativistic kinetic equilibria in collisionless N-body
systems is pointed out. The theory is developed in the framework of the
covariant Vlasov statistical description. The new effect is based on the
constraints placed by the conservation laws of neutral single-particle
dynamics in prescribed background curved-spacetimes demonstrating
existence of Killing tensors. As an illustration, the particular case of
the Kerr spacetime admitting the so-called Carter constant for the
particle geodesic motion is considered. The general functional form of
the equilibrium kinetic distribution function (KDF) is determined and an
explicit realization in terms of Gaussian-like distributions is
provided. It is shown that, due to the Carter constant, these
equilibrium KDFs exhibit an anisotropic phase-space functional
dependence in terms of the single-particle 4-velocity components, giving
rise to corresponding nonisotropic continuum fluid fields. The
qualitative properties of the equilibrium stress-energy tensor
associated with these systems are discussed, with a particular emphasis
on the related occurrence of temperature anisotropy effects. The theory
is susceptible of astrophysical applications, including in particular
the statistical properties of dark matter (DM) halos around stellar-mass
or galactic-center black holes.
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We study the shadow of the rotating black hole with quintessential
energy (i) in vacuum, (ii) in the presence of plasma with radial power-
law density. For the vacuum case, the quintessential field parameter of
the rotating black hole significantly changes the shape of the shadow.
With increasing quintessential field parameter, the radius of the shadow
also increases. With the increase of the radius of the shadow of the
rotating black hole, the quintessential field parameter causes decrease
of the distortion of the shadow shape: in the presence of the
quintessential field parameter, the shadow of the fast rotating black
hole becomes too close to the circle. We assume the distant observer of
the black hole shadow to be located near the so-called static radius
where the gravitational attraction of the black hole is just balanced by
the cosmic repulsion. The shape and size of the shadow of quintessential
rotating black hole surrounded by plasma depends on (i) plasma
parameters, (ii) black hole spin and (iii) quintessential field
parameter. With the increase of the plasma refraction index, the
apparent radius of the shadow increases. However, for the large values
of the quintessential field parameter, the change of the black hole
shadow shape due to the presence of plasma is not significant, i.e. the
effect of the quintessential field parameter dominates over the plasma
effect.
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