A covariant formalism of spin precession with respect to a reference congruence

TL;DR

This paper develops a covariant, three-dimensional formalism for relativistic spin precession relative to an arbitrary reference congruence, introducing a stopped spin vector and applying it to novel spatial transport and gyroscope precession analysis.

Contribution

It introduces a new covariant formalism for spin precession using a stopped spin vector, applicable to any spacetime with a specified timelike congruence, and applies it to novel spatial transport concepts.

Findings

Derived a transport equation for the stopped spin vector.

Formulated a precession formula relative to a spatial geometry.

Validated the formalism through applications and comparisons.

Abstract

We derive an effectively three-dimensional relativistic spin precession formalism. The formalism is applicable to any spacetime where an arbitrary timelike reference congruence of worldlines is specified. We employ what we call a stopped spin vector which is the spin vector that we would get if we momentarily make a pure boost of the spin vector to stop it relative to the congruence. Starting from the Fermi transport equation for the standard spin vector we derive a corresponding transport equation for the stopped spin vector. Employing a spacetime transport equation for a vector along a worldline, corresponding to spatial parallel transport with respect to the congruence, we can write down a precession formula for a gyroscope relative to the local spatial geometry defined by the congruence. This general approach has already been pursued by Jantzen et. al. (see e.g. Jantzen, Carini and Bini, Ann. Phys. 215 (1997) 1), but the algebraic form of our respective expressions differ. We are also applying the formalism to a novel type of spatial parallel transport introduced in Jonsson (Class. Quantum Grav. 23 (2006) 1), as well as verifying the validity of the intuitive approach of a forthcoming paper (Jonsson, Am. Journ. Phys. 75 (2007) 463) where gyroscope precession is explained entirely as a double Thomas type of effect. We also present the resulting formalism in explicit three-dimensional form (using the boldface vector notation), and give examples of applications.