The Mathisson-Papapetrou-Dixon (MPD) equations describe the motion of an extended test body in general relativity. This system of equations, though, is underdetermined and has to be accompanied by constraining supplementary conditions, even in its simplest version, which is the pole-dipole approximation corresponding to a spinning test body. In particular, imposing a spin supplementary condition (SSC) fixes the center of the mass of the spinning body, i.e. the centroid of the body. In the present study, we examine whether characteristic features of the centroid of a spinning test body, moving in a circular equatorial orbit around a massive black hole, are preserved under the transition to another centroid of the same physical body, governed by a different SSC. For this purpose, we establish an analytical algorithm for deriving the orbital frequency of a spinning body, moving in the background of an arbitrary, stationary, axisymmetric spacetime with reflection symmetry, for the Tulzcyjew-Dixon, the Mathisson-Pirani and the Ohashi-Kyrian-Semer\'{a}k SSCs. Then, we focus on the Schwarzschild black hole background and a power series expansion method is developed, in order to investigate the discrepancies in the orbital frequencies expanded in power series of the spin among the different SSCs. Lastly, by employing the fact that the position of the centroid and the measure of the spin alters under the centroid's transition, we impose proper corrections to the power expansion of the orbital frequencies, which allows to improve the convergence between the SSCs. Our concluding argument is that when we shift from one circular equatorial orbit to another in the Schwarzschild background, under the change of a SSC, the convergence between the SSCs holds only up to certain powers in the spin expansion, and it cannot be achieved for the whole power series.

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 | 2 n , n > 2 , under the conditions of the asymptotic flatness and behavior of SF φ ∼ 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 ≤ 40 , | Q | ≤ 60 , M ≤ 60 . 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.

We study static spherically symmetric configurations in the presence of linear massive scalar fields within General Relativity. Static solutions of the Einstein equations are considered under conditions of asymptotic flatness. Each solution is fixed by the configuration mass and the field strength parameter, which are defined at spatial infinity. The metric coefficients and the scalar field for a specific configuration are obtained numerically. Then we study the time-like geodesics describing the test particle motion. The focus is on the distribution of stable circular orbits (SCOs) of the test particles around a configuration. We found that, for the continuum of configuration parameters, there exist two unlinked regions of SCOs that are separated by some annular region, where SCOs do not exist.

Location of the innermost stable circular orbit of binary neutron stars in the post Newtonian approximations of general relativity In this paper, we present results obtained from our recent studies on the location of the innermost stable circular orbit (ISCO) for binary neutron stars (BNSs) in several levels of post Newtonian (PN) approximations. We reach the following conclusion at present: (1) even in the Newtonian case, there exists the ISCO for binary of sufficiently stiff equation of state (EOS). If the mass and the radius of each star are fixed, the angular velocity at the ISCO $\Omega_{ISCO}$ is larger for softer EOS: (2) when we include the first PN correction, there appear roughly two kinds of effects. One is the effect to the self-gravity of each star of binary and the other is to the gravity acting between two stars. Due to the former one, each star of binary becomes compact and the tidal effect is less effective. As a result, $\Omega_{ISCO}$ tends to be increased. On the other hand, the latter one has the property to destabilize the binary orbit, and $\Omega_{ISCO}$ tends to be decreased.

We study static spherically symmetric configurations in the presence of linear massive scalar fields within General Relativity. Static solutions of the Einstein equations are considered under conditions of asymptotic flatness. Each solution is fixed by the configuration mass and the field strength parameter, which are defined at spatial infinity. The metric coefficients and the scalar field for a specific configuration are obtained numerically. Then we study the time-like geodesics describing the test particle motion. The focus is on the distribution of stable circular orbits (SCOs) of the test particles around a configuration. We found that, for the continuum of configuration parameters, there exist two unlinked regions of SCOs that are separated by some annular region, where SCOs do not exist.

We provide constraints on possible configurations and interactions of two coplanar tori orbiting a central Kerr black hole (BH), in dependence on its dimensionless spin. The two-tori configurations can be directly linked to the current models featuring the obscuration of galactic BH X-ray emission. The emergence of each torus instability phases is discussed and tori collision has been also investigated. The first simple evaluation of the center-of-mass energy proves that collision-energy-efficiency increases with the dimensionless BH spin. We explore the phenomenological aspects of the corotating and counterrotating tori by analyzing properties of the orbiting toroidal configurations related to the fluid enthalpy flux, the mass-flux, the mass-accretion-rates, and the cusp luminosity in the two cases of corotating and counterrotating fluids in dependence on the SMBH spin. The analysis resulted ultimately in a comparative investigation of the properties of corotating versus counterrotating tori, demonstrating that two accretion tori can orbit around the central Kerr attractor only under very specific conditions. Our results also demonstrate that the dynamics of the unstable phases of these double tori systems is significant for the high energy phenomena which could be observable in the X-ray emission and extremely energetic phenomena in active galactic nuclei and quasar.