On common solutions of Mathisson equations under different conditions

TL;DR

This paper investigates highly relativistic spinning particles in Schwarzschild spacetime, revealing common solutions of Mathisson equations under different conditions that differ from geodesic orbits, especially in energy characteristics.

Contribution

It demonstrates that Mathisson equations under Frenkel-Mathisson and Tulczyjew-Dixon conditions share solutions for relativistic circular orbits near the photon sphere.

Findings

Common solutions exist for different supplementary conditions.

Relativistic orbits differ significantly from geodesic orbits.

Particle energy on these orbits is notably different from geodesic energy.

Abstract

In the context of investigations of possible highly relativistic motions of a spinning particle in the gravitational field, which can be described by the Mathisson equations under different supplementary condition, we analyze the circular orbits in a Schwarzschild field. The very orbits most clearly demonstrate the effect of the gravitational spin-orbit interaction on the particle's motion. It is shown that the Mathisson equations under the Frenkel-Mathisson and Tulczyjew-Dixon conditions have the common solutions describing the highly relativistic circular orbits in the region $r=3M(1+\delta), |\delta|\ll 1$. These orbits essentially differ from the geodesic circular orbits in the same region, particularly by the value of the particle's energy on the corresponding orbits.