Highly relativistic spin-gravity-$\mathrm{\Lambda }$ coupling

The effects of highly relativistic spin-gravity coupling in the Schwarzschild–de Sitter background that follow from the Mathisson-Papapetrou equations are investigated. The dependence of gravitoelectric and gravitomagnetic components of gravitational field on the velocity of an observer that is moving in Schwarzschild–de Sitter’s background is estimated. The action of gravitomagnetic components on a fast moving spinning particle is considered. Different cases of the highly relativistic circular orbits of a spinning particle that essentially differ from the corresponding geodesic orbits are described.