Perfect fluid tori with uniform distribution of the specific angular
momentum orbiting the Kerr-de Sitter black holes or naked singularities
are studied. Closed equipotential surfaces corresponding to stationary
toroidal discs are allowed only in the spacetimes admitting stable
circular geodesics. The last closed surface crosses itself in the
cusp(s) enabling outflow(s) of matter from the torus due to the
violation of hydrostatic equilibrium. The repulsive cosmological
constant, Λ > 0, implies the existence of the outer cusp (with
a stabilizing effect on the tori because of excretion, i.e., outflow of
matter from the torus into the outer space) and the strong collimation
of open equipotential surfaces along the rotational axis. Both the
effects take place nearby so-called static radius where the
gravitational attraction is just balanced by the cosmic repulsion. The
plus-family discs (which are always corotating in the black-hole
backgrounds but can be counterrotating, even with negative energy of the
fluid elements, in some naked-singularity backgrounds) are thicker and
more extended than the minus-family ones (which are always
counterrotating in all backgrounds). If parameters of the
naked-singularity spacetimes are very close to the parameters of extreme
black-hole spacetimes, the family of possible disc-like configurations
includes members with two isolated discs where the inner one is always a
counterrotating accretion disc. Mass estimates for tori with
nonrelativistic adiabatic equation of state give limits on their central
mass-density, for which the approximation of test fluid is adequate.
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We show that in the equatorial plane of marginally stable thick discs
(with uniformly distributed specific angular momentum l(r, θ) =
const) the orbital velocity relative to the LNRF has a positive radial
gradient in the vicinity of black holes with a > 0.99979. The change
of sign of the velocity gradient occurs just above the center of the
thick toroidal discs, in the region where stable circular geodesics of
the Kerr spacetime are allowed. The global character of the phenomenon
is given in terms of topology changes of the von Zeipel surfaces
(equivalent to equivelocity surfaces in the tori with l(r, θ) =
const). Toroidal von Zeipel surfaces exist around the circle
corresponding to the minimum of the equatorial LNRF velocity profile,
indicating a possibility of development of some vertical instabilities
in those parts of marginally stable tori with positive gradient of the
LNRF velocity. Eventual oscillatory frequencies connected with the
phenomenon are given in a coordinate-independent form.
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Kilohertz Quasi-Periodic Oscillations (QPOs) have been detected in many
accreting X-ray binaries. It has been suggested that the highest QPO
frequencies observed in the modulation of the X-ray flux reflect a
non-linear resonance between two modes of accreting disk oscillation.
This hypothesis implies certain very general predictions, several of
which have been borne out by observations. Some of these follow from
properties of non-linear oscillators, while the others are specific to
oscillations of fluid in strong gravity. A 3:2 resonant ratio of
frequencies can be clearly recognized in the black-hole as well as in
the neutron-star QPO data.
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It is hard to imagine curved spacetimes of General Relativity. A simple
but powerful way how to achieve this is visualizing them via embedding
diagrams of both ordinary geometry and optical reference geometry. They
facilitate to gain an intuitive insight into the gravitational field
rendered into a curved spacetime, and to assess the influence of
parameters like electric charge and spin of a black hole, magnetic field
or cosmological constant. Optical reference geometry and related
inertial forces and their relationship to embedding diagrams are
particularly useful for investigation of test particles motion.
Embedding diagrams of static and spherically symmetric, or stationary
and axially symmetric black-hole and naked-singularity spacetimes thus
present a useful concept for intuitive understanding of these
spacetimes' nature. We concentrate on general way of embedding into
3-dimensional Euclidean space, and give a set of illustrative examples.
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The influence of the observed relict vacuum energy on the fluctuations
of CMBR going through cosmological matter condensations is studied in
the framework of the Einstein-Strauss-de Sitter vakuola model. It is
shown that refraction of light at the matching surface of the vakuola
and the expanding Friedman universe can be very important during
accelerated expansion of the universe, when the velocity of the matching
surface relative to static Schwarzchildian observers becomes
relativistic. Relevance of the refraction effect for the temperature
fluctuations of CMBR is given in terms of the redshift and the angular
extension of the fluctuating region.
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We determine the influence of a nonzero cosmological constant on dynamical stability of spherically symmetric perfect fluid configurations. The equations governing small radial oscillations and the related Sturm-Liouville eigenvalue equation for eigenmodes of the oscillations are generalised for the case of nonzero cosmological constant. The Sturm-Liouville equation is then applied in the case of polytropic equation of state. Based on this approach, the dependence of the critical value of adiabatic index on relativistic parameter (defined as ratio of central pressure to central energy density) is established for values of polytropic index 3/2 and 3 and for both repulsive and attractive cosmological constant using numerical computations. It is shown that a repulsive cosmological constant rises the critical adiabatic index and decreases the critical radius under which the dynamical instability occurs.
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We determine the influence of a nonzero cosmological constant on
dynamical stability of spherically symmetric perfect fluid
configurations. The equations governing small radial oscillations and
the related Sturm-Liouville eigenvalue equation for eigenmodes of the
oscillations are generalised for the case of nonzero cosmological
constant. The Sturm-Liouville equation is then applied in the case of
polytropic equation of state. Based on this approach, the dependence of
the critical value of adiabatic index on relativistic parameter (defined
as ratio of central pressure to central energy density) is established
for values of polytropic index 3/2 and 3 and for both repulsive and
attractive cosmological constant using numerical computations. It is
shown that a repulsive cosmological constant rises the critical
adiabatic index and decreases the critical radius under which the
dynamical instability occurs.
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Equilibrium conditions and spin dynamics of spinning test particles are discussed in the stationary and axially symmetric Kerr de Sitter black-hole or naked-singularity spacetimes. The general equilibrium conditions are established, but due to their great complexity, the detailed discussion of the equilibrium conditions and spin dynamics is presented only in the simple and most relevant cases of equilibrium positions in the equatorial plane and on the symmetry axis of the spacetimes. It is shown that due to the combined effect of the rotation of the source and the cosmic repulsion the equilibrium is spin dependent in contrast to the spherically symmetric spacetimes. In the equatorial plane, it is possible at the so-called static radius, where the gravitational attraction is balanced by the cosmic repulsion, for the spinless particles as well as for spinning particles with arbitrarily large phiv-oriented spin or at any radius outside the ergosphere with a specifically given spin orthogonal to the equatorial plane. On the symmetry axis, the equilibrium is possible at any radius in the stationary region and is given by an appropriately tuned spin directed along the axis. At the static radii on the axis the spin of particles in equilibrium must vanish.
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Equilibrium conditions and spin dynamics of spinning test particles are
discussed in the stationary and axially symmetric Kerr de Sitter
black-hole or naked-singularity spacetimes. The general equilibrium
conditions are established, but due to their great complexity, the
detailed discussion of the equilibrium conditions and spin dynamics is
presented only in the simple and most relevant cases of equilibrium
positions in the equatorial plane and on the symmetry axis of the
spacetimes. It is shown that due to the combined effect of the rotation
of the source and the cosmic repulsion the equilibrium is spin dependent
in contrast to the spherically symmetric spacetimes. In the equatorial
plane, it is possible at the so-called static radius, where the
gravitational attraction is balanced by the cosmic repulsion, for the
spinless particles as well as for spinning particles with arbitrarily
large phiv-oriented spin or at any radius outside the ergosphere with a
specifically given spin orthogonal to the equatorial plane. On the
symmetry axis, the equilibrium is possible at any radius in the
stationary region and is given by an appropriately tuned spin directed
along the axis. At the static radii on the axis the spin of particles in
equilibrium must vanish.
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