Spinning test particles and clock effect in Schwarzschild spacetime

TL;DR

This paper investigates the motion of spinning test particles in Schwarzschild spacetime, revealing that spin influences orbital dynamics and causes a measurable clock effect similar to Kerr spacetime, with potential experimental verification.

Contribution

It demonstrates that spinning particles can have circular orbits influenced by spin-induced tidal forces, and identifies a gravitomagnetic clock effect in Schwarzschild spacetime, analogous to Kerr.

Findings

Spinning particles can maintain circular orbits due to spin-induced tidal forces.

A nonzero clock effect occurs for oppositely spinning particles in Schwarzschild spacetime.

The results suggest possible experimental verification of spin effects in gravitational fields.

Abstract

We study the behaviour of spinning test particles in the Schwarzschild spacetime. Using Mathisson-Papapetrou equations of motion we confine our attention to spatially circular orbits and search for observable effects which could eventually discriminate among the standard supplementary conditions namely the Corinaldesi-Papapetrou, Pirani and Tulczyjew. We find that if the world line chosen for the multipole reduction and whose unit tangent we denote as $U$ is a circular orbit then also the generalized momentum $P$ of the spinning test particle is tangent to a circular orbit even though $P$ and $U$ are not parallel four-vectors. These orbits are shown to exist because the spin induced tidal forces provide the required acceleration no matter what supplementary condition we select. Of course, in the limit of a small spin the particle's orbit is close of being a circular geodesic and the (small) deviation of the angular velocities from the geodesic values can be of an arbitrary sign, corresponding to the possible spin-up and spin-down alignment to the z-axis. When two spinning particles orbit around a gravitating source in opposite directions, they make one loop with respect to a given static observer with different arrival times. This difference is termed clock effect. We find that a nonzero gravitomagnetic clock effect appears for oppositely orbiting both spin-up or spin-down particles even in the Schwarzschild spacetime. This allows us to establish a formal analogy with the case of (spin-less) geodesics on the equatorial plane of the Kerr spacetime. This result can be verified experimentally.