Spinning particles in general relativity: Momentum-velocity relation for the Mathisson-Pirani spin condition

TL;DR

This paper derives a simple momentum-velocity relation for spinning particles under the Mathisson-Pirani condition in general relativity, clarifying its implications and how to select among multiple possible worldlines in curved spacetime.

Contribution

It provides the first explicit derivation of the momentum-velocity relation for the Mathisson-Pirani spin condition, resolving longstanding ambiguities and explaining the relation's role in describing spinning particles.

Findings

Derived a simple, explicit momentum-velocity relation for the MP condition.

Showed the relation is equivalent to the MP condition and clarifies its degeneracy.

Provided a method to switch between multiple worldlines in curved spacetime.

Abstract

The Mathisson-Papapetrou-Dixon (MPD) equations, providing the "pole-dipole" description of spinning test particles in general relativity, have to be supplemented by a condition specifying the worldline that will represent the history of the studied body. It has long been thought that the Mathisson-Pirani (MP) spin condition -- unlike other major choices made in the literature -- does not yield an explicit momentum-velocity relation. We derive here the desired (and very simple) relation and show that it is in fact equivalent to the MP condition. We clarify the apparent paradox between the existence of such a definite relation and the known fact that the MP condition is degenerate (does not specify a unique worldline), thus shedding light on some conflicting statements made in the literature. We then show how, for a given body, this spin condition yields infinitely many possible representative worldlines, and derive a detailed method how to switch between them in a curved spacetime. The MP condition is a convenient choice in situations when it is easy to recognize its "non-helical" solution, as exemplified here by bodies in circular orbits and in radial fall in the Schwarzschild spacetime.