  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.
