Embedding diagrams of the equatorial plane of the Kerr-de Sitter black-hole or naked-singularity spacetimes are constructed for the optical reference geometry. The embedding diagrams do not cover whole stationary parts of the Kerr-de Sitter spacetimes. Hence, limits of embeddability are discussed. The Kerr-de Sitter spacetimes are then classified according to the number of embeddable regions and the number of the turning points of the diagrams.
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Equilibrium configurations that are solutions of spherically symmetric hydrostatic equations of General Relativity for an ideal fluid obeying a polytropic (or adiabatic) equation of state are given in the framework of general relativity. The equilibrium configurations are given in terms of the polytropic index n and the so called relativistic parameter σ (for polytropes) or α (for adiabates). First, simple models of polytropic and adiabatic spheres for non-relativistic and ultra-relativistic case of the equation of state are introduced. Then, the comparison of polytropic and adiabatic spheres is given in some special characteristic cases and the influence of the relativistic parameter on the structure of the spheres and the gravitational and binding energy.
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The integrals of photon motion in the Kerr spacetimes are given in terms of the emission angles related to emitters moving along the circular geodesic orbits. The local frames of the circular geodesic emitters are given in relation to the locally nonrotating frames and local directional angles of the escape cones are given in terms of the motion constants of photons.
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We examine halo orbits of electrically charged particles near a magnetized compact star. We compare the Newtonian and Pseudo-Newtonian approaches to the Störmer problem as a preliminary discussion of the full general relativistic approach. We show the differences in the effective potential that arise due to strong gravity near a gravitating body.
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Perfect fluid tori with uniform distribution of the specific angular momentum, l(r θ)= const, orbiting the Kerr-de Sitter naked singularities are discussed. Closed equipotential surfaces corresponding to stationary thick discs are allowed only in the spacetimes admitting stable circular geodesics. The last closed surface crosses itself in the cusp(s) enabling outflows of matter from the torus due to the violation of hydrostatic equilibrium. The inner cusp enables the accretion onto the ring singularity. Influence of the repulsive cosmological constant, Λ > 0, resides in the existence of the outer cusp enabling the excretion (outflow of matter from the torus into the outer space) and gives rise to completely new type of a disc called the excretion disc. The plus-family accretion and excretion discs can be both the corotating or counterrotating discs, the minus-family ones are always the counterrotating discs, as related to locally non-rotating frames. If the parameters of naked-singularity spacetimes are very close to the parameters of the 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, while the outer one can be the corotating or counterrotating excretion disc, as well as the counterrotating accretion disc.
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Properties of the gravitational perturbation force caused by a small inhomogeneity located on the surface of a neutron star are studied. The oscillating perturbation force in both the accretion disc rotating around the star and in the interior of a differentially rotating star is determined. Both vertical and radial components of the force are given and their relations are discussed. The frequency of the oscillations is given by the difference of the frequency of the rotation of the star surface, and the orbital frequency in the disc (the frequency of rotation of matter in the star interior). Outside the star, in the disc, the vertical and radial forces vary with the same phase. Inside a differentially rotating star, the variations of the forces are in the opposite phase in an internal part of the star, while they are in the same phase in an external layer of the star. In an intermediate part of the star, an additional oscillatory change appears. Is is shown that the anharmonic character of the oscillatory forces is limited to the seventh non-negligible harmonics. For completeness, we present the perturbation force generated by a symmetric accretion column.
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The equation governing small radial oscillations and the related Sturm-Liouville eigenvalue equation for eigenmodes of the oscillations are determined for spherically symmetric configurations of perfect fluid in spacetimes with a nonzero cosmological constant. The Sturm-Liouville equation is then applied in the cases of spherically symmetric configurations with uniform distribution of energy density and polytropic spheres. 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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The Friedman models of the Universe with the superstring dark energy are constructed. According to the spacetime foam approach the stringy dark energy appears to be inversely proportional to the cosmic scale factor. Evolution of the Friedman models is discussed under this assumption and compared with the standard models.
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The Swiss cheese model of the Universe with the superstring dark energy is constructed. The junction conditions are shown to be fulfilled and time evolution of the matching hypersurface of the internal Schwarzschild spacetime and homogeneous external Friedman Universe is studied.
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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 case of equilibrium positions in the equatorial plane 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.
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