Some partial solutions of Mathisson-Papapetrou equations in a Schwarzschild field

TL;DR

This paper investigates analytical and numerical solutions of Mathisson-Papapetrou equations for ultrarelativistic spinning particles in Schwarzschild spacetime, revealing how spin influences particle trajectories over short times.

Contribution

It provides explicit expressions for non-equatorial circular orbits and studies the impact of spin on ultrarelativistic particle trajectories in Schwarzschild fields.

Findings

Spin affects the shape of ultrarelativistic trajectories.

Explicit formulas for non-equatorial circular orbits are derived.

Spin influence is significant over short times, less than two revolutions.

Abstract

The analytical and numerical solutions of the Mathisson-Papapetrou equations under the Mathisson-Pirani supplementary condition describing highly relativistic (ultrarelativistic) motions of a spinning particle in a Schwarzschild field are investigated. The known condition S/mr<<1, which is necessary for a test particle, holds on all these solutions. The explicit expressions for the non-equatorial circular orbits, in particular for the space boundaries of the region of existence of these orbits, are obtained. The dynamics of the deviation of a spinning particle from the equatorial ultrarelativistic circular orbit with r=3M caused by the non-zero initial value of the radial particle's velocity is studied. It is shown in the concrete cases that spin can considerable influence the shape of an ultrarelativistic trajectory, as compared to the corresponding geodesic trajectory, for the short time, less than the time of one or two revolutions of a spinning particle around a Schwarzschild mass.