A new representation, which does not contain the third-order derivatives of the coordinates, of the exact Mathisson-Papapetrou-Dixon equations, describing the motion of a spinning test particle, is obtained under the assumption of the Mathisson-Pirani condition in a Kerr background. For this purpose the integrals of energy and angular momentum of the spinning particle as well as a differential relationship following from the Mathisson-Papapetrou-Dixon equations are used. The form of these equations is adapted for their computer integration with the aim to investigate the influence of the spin-curvature interaction on the particle's behavior in the gravitational field without restrictions on its velocity and spin orientation. Some numerical examples for a Schwarzschild background are presented.PACS numbers: 95.30.Sf
Using the Mathisson-Papapetrou-Dixon (MPD) equations, we investigate the trajectories of a spinning particle starting near r (−) ph in a Kerr field and moving with the velocity close to the velocity of light (r (−) ph is the Boyer-Lindquist radial coordinate of the counter-rotation circular photon orbits). First, as a partial case of these trajectories, we consider the equatorial circular orbit with r = r (−) ph . This orbit is described by the solution that is common for the rigorous MPD equations and their linear spin approximation. Then different cases of the nonequatorial motions are computed and illustrated by the typical figures. All these orbits exhibit the effects of the significant gravitational repulsion that are caused by the spin-gravity interaction. Possible applications in astrophysics are discussed. PACS numbers: 04.20.-q, 95.30.Sf I.
Different types of essentially nongeodesic motions of highly relativistic spinning particles in Schwarzschild's and Kerr's background which follows from the Mathisson-Papapetrou (MP) equations are considered. It is shown that dependently on the correlation of signs of the spin and the particle's orbital velocity the spin-gravity coupling acts as a significant repulsive or attractive force. Numerical estimates for electrons, protons, and neutrinos in the gravitational field of black holes are presented. The correspondence between the general relativistic Dirac equation and MP equations is discussed. It is stressed that for the highly relativistic motions the adequate supplementary condition for the MP equations is the Mathisson-Pirani condition. In the following it is important to study the possible role of the highly relativistic spin-gravity coupling in astrophysics, cosmology, and high energy physics.
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