Null geodesics and embedding diagrams of central planes in the ordinary space geometry and the optical reference geometry of the interior Schwarzschild-de Sitter spacetimes with uniform density are studied. For completeness, both positive and negative values of the cosmological constant are considered. The null geodesics are restricted to the central planes of these spacetimes, and their properties can be reflected by an ``effective potential.'' If the interior spacetime is extremely compact, the effective potential has a local maximum corresponding to a stable circular null geodesic around which bound null geodesics are concentrated. The upper limit on the size of the interior spacetimes containing bound null geodesics is R=3M, independently of the value of the cosmological constant. The embedding diagrams of the central planes of the ordinary geometry into three-dimensional Euclidean space are well defined for the complete interior of all spacetimes with a repulsive cosmological constant, but the planes cannot be embedded into the Euclidean space in the case of spacetimes with subcritical values of an attractive cosmological constant. On the other hand, the embedding diagrams of the optical geometry are well defined for all of the spacetimes, and the turning points of these diagrams correspond to the radii of the circular null geodesics. All the embedding diagrams, for both the ordinary and optical geometry, are smoothly matched to the corresponding embedding diagrams of the external vacuum Schwarzschild-de Sitter spacetimes.
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Null geodesics and embedding diagrams of central planes in the ordinary
space geometry and the optical reference geometry of the interior
Schwarzschild-de Sitter spacetimes with uniform density are studied. For
completeness, both positive and negative values of the cosmological
constant are considered. The null geodesics are restricted to the
central planes of these spacetimes, and their properties can be
reflected by an ``effective potential.'' If the interior spacetime is
extremely compact, the effective potential has a local maximum
corresponding to a stable circular null geodesic around which bound null
geodesics are concentrated. The upper limit on the size of the interior
spacetimes containing bound null geodesics is R=3M, independently of the
value of the cosmological constant. The embedding diagrams of the
central planes of the ordinary geometry into three-dimensional Euclidean
space are well defined for the complete interior of all spacetimes with
a repulsive cosmological constant, but the planes cannot be embedded
into the Euclidean space in the case of spacetimes with subcritical
values of an attractive cosmological constant. On the other hand, the
embedding diagrams of the optical geometry are well defined for all of
the spacetimes, and the turning points of these diagrams correspond to
the radii of the circular null geodesics. All the embedding diagrams,
for both the ordinary and optical geometry, are smoothly matched to the
corresponding embedding diagrams of the external vacuum Schwarzschild-de
Sitter spacetimes.
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The hydrodynamical structure of perfect fluid orbiting Schwarzschild-de Sitter black holes is investigated for configurations with uniform distribution of angular momentum density. It is shown that in the black-hole backgrounds admitting the existence of stable circular geodesics, closed equipotential surfaces with a cusp, allowing the existence of toroidal accretion disks, can exist. Two surfaces with a cusp exist for the angular momentum density smaller than the one corresponding to marginally bound circular geodesics; the equipotential surface corresponding to the marginally bound circular orbit has just two cusps. The outer cusp is located nearby the static radius where the gravitational attraction is compensated by the cosmological repulsion. Therefore, due to the presence of a repulsive cosmological constant, the outflow from thick accretion disks can be driven by the same mechanism as the accretion onto the black hole. Moreover, properties of open equipotential surfaces in vicinity of the axis of rotation suggest a strong collimation effects of the repulsive cosmological constant acting on jets produced by the accretion disks.
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Discussion of the equatorial photon motion in Kerr-Newman black-hole and naked-singularity spacetimes with a non-zero cosmological constant is presented. Both repulsive and attractive cosmological constants are considered. An appropriate `effective potential' governing the photon radial motion is defined, circular photon orbits are determined, and their stability with respect to radial perturbations is established. The spacetimes are divided into separated classes according to the properties of the `effective potential'. There is a special class of Kerr-Newman-de Sitter black-hole spacetimes with the restricted repulsive barrier. In such spacetimes, photons with high positive and all negative values of their impact parameter can travel freely between the outer black-hole horizon and the cosmological horizon due to an interplay between the rotation of the source and the cosmological repulsion. It is shown that this type of behaviour of the photon motion is connected to an unusual relation between the values of the impact parameters of the photons and their directional angles relative to the outward radial direction as measured in the locally non-rotating frames. Surprisingly, some photons counterrotating in these frames have a positive impact parameter. Such photons can be both escaping or captured in the black-hole spacetimes with the restricted repulsive barrier. For the black-hole spacetimes with a standard, divergent repulsive barrier of the equatorial photon motion, the counterrotating photons with positive impact parameters must all be captured from the region near the black-hole outer horizon as in the case of Kerr black holes, while they all escape from the region near the cosmological horizon. Further, the azimuthal motion is discussed and photon trajectories are given in typical situations. It is shown that for some photons with negative impact parameter turning points of their azimuthal motion can exist.
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The hydrodynamical structure of perfect fluid orbiting Schwarzschild-de
Sitter black holes is investigated for configurations with uniform
distribution of angular momentum density. It is shown that in the
black-hole backgrounds admitting the existence of stable circular
geodesics, closed equipotential surfaces with a cusp, allowing the
existence of toroidal accretion disks, can exist. Two surfaces with a
cusp exist for the angular momentum density smaller than the one
corresponding to marginally bound circular geodesics; the equipotential
surface corresponding to the marginally bound circular orbit has just
two cusps. The outer cusp is located nearby the static radius where the
gravitational attraction is compensated by the cosmological repulsion.
Therefore, due to the presence of a repulsive cosmological constant, the
outflow from thick accretion disks can be driven by the same mechanism
as the accretion onto the black hole. Moreover, properties of open
equipotential surfaces in vicinity of the axis of rotation suggest a
strong collimation effects of the repulsive cosmological constant acting
on jets produced by the accretion disks.
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Discussion of the equatorial photon motion in Kerr-Newman black-hole and
naked-singularity spacetimes with a non-zero cosmological constant is
presented. Both repulsive and attractive cosmological constants are
considered. An appropriate `effective potential' governing the photon
radial motion is defined, circular photon orbits are determined, and
their stability with respect to radial perturbations is established. The
spacetimes are divided into separated classes according to the
properties of the `effective potential'. There is a special class of
Kerr-Newman-de Sitter black-hole spacetimes with the restricted
repulsive barrier. In such spacetimes, photons with high positive and
all negative values of their impact parameter can travel freely between
the outer black-hole horizon and the cosmological horizon due to an
interplay between the rotation of the source and the cosmological
repulsion. It is shown that this type of behaviour of the photon motion
is connected to an unusual relation between the values of the impact
parameters of the photons and their directional angles relative to the
outward radial direction as measured in the locally non-rotating frames.
Surprisingly, some photons counterrotating in these frames have a
positive impact parameter. Such photons can be both escaping or captured
in the black-hole spacetimes with the restricted repulsive barrier. For
the black-hole spacetimes with a standard, divergent repulsive barrier
of the equatorial photon motion, the counterrotating photons with
positive impact parameters must all be captured from the region near the
black-hole outer horizon as in the case of Kerr black holes, while they
all escape from the region near the cosmological horizon. Further, the
azimuthal motion is discussed and photon trajectories are given in
typical situations. It is shown that for some photons with negative
impact parameter turning points of their azimuthal motion can exist.
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Properties of the optical reference geometry related to Kerr-Newman black-hole and naked-singularity spacetimes are illustrated using embedding diagrams of their equatorial plane. It is shown that among all inertial forces defined within the framework of the optical geometry, just the centrifugal force plays a fundamental role in connection with the embedding diagrams because it changes sign at the turning points of the diagrams. The embedding diagrams do not cover the stationary part of the Kerr-Newman spacetimes completely. Hence, the limits of embeddability are given, and it is established which of the photon circular orbits hosted by the Kerr-Newman spacetimes appear in the embeddable regions. Some typical embedding diagrams are constructed, and the Kerr-Newman backgrounds are classified according to the number of embeddable regions of the optical geometry as well as the number of their turning points. It is shown that embedding diagrams are closely related to the notion of the radius of gyration which is useful for analysing a fluid rotating in strong gravitational fields.
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Properties of the optical reference geometry related to Kerr-Newman
black-hole and naked-singularity spacetimes are illustrated using
embedding diagrams of their equatorial plane. It is shown that among all
inertial forces defined within the framework of the optical geometry,
just the centrifugal force plays a fundamental role in connection with
the embedding diagrams because it changes sign at the turning points of
the diagrams. The embedding diagrams do not cover the stationary part of
the Kerr-Newman spacetimes completely. Hence, the limits of
embeddability are given, and it is established which of the photon
circular orbits hosted by the Kerr-Newman spacetimes appear in the
embeddable regions. Some typical embedding diagrams are constructed, and
the Kerr-Newman backgrounds are classified according to the number of
embeddable regions of the optical geometry as well as the number of
their turning points. It is shown that embedding diagrams are closely
related to the notion of the radius of gyration which is useful for
analysing a fluid rotating in strong gravitational fields.
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