Spinning test particles and clock effect in Kerr spacetime

TL;DR

This paper investigates the motion of spinning test particles in Kerr spacetime, analyzing different supplementary conditions and their effects on particle orbits and clock effects, with potential observational implications.

Contribution

It compares various supplementary conditions for spinning particles in Kerr spacetime and explores their impact on orbits and clock effects, revealing conditions that can suppress or magnify these effects.

Findings

The particle's orbit is nearly geodesic in the small spin limit.

Different supplementary conditions significantly affect clock effect magnitudes.

Spin contributions can cancel out, making clock effects indistinguishable from non-spinning particles.

Abstract

We study the motion of spinning test particles in Kerr spacetime using the Mathisson-Papapetrou equations; we impose different supplementary conditions among the well known Corinaldesi-Papapetrou, Pirani and Tulczyjew's and analyze their physical implications in order to decide which is the most natural to use. We find that if the particle's center of mass world line, namely the one chosen for the multipole reduction, is a spatially circular orbit (sustained by the tidal forces due to the spin) then the generalized momentum $P$ of the test particle is also tangent to a spatially circular orbit intersecting the center of mass line at a point. There exists one such orbit for each point of the center of mass line where they intersect; although fictitious, these orbits are essential to define the properties of the spinning particle along its physical motion. In the small spin limit, the particle's orbit is almost a geodesic and the difference of its angular velocity with respect to the geodesic value can be of arbitrary sign, corresponding to the spin-up and spin-down possible alignment along the z-axis. We also find that the choice of the supplementary conditions leads to clock effects of substantially different magnitude. In fact, for co-rotating and counter-rotating particles having the same spin magnitude and orientation, the gravitomagnetic clock effect induced by the background metric can be magnified or inhibited and even suppressed by the contribution of the individual particle's spin. Quite surprisingly this contribution can be itself made vanishing leading to a clock effect undistiguishable from that of non spinning particles. The results of our analysis can be observationally tested.