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

Journal article, 2006

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 vectorthat 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 R T, Carini P and Bini D 1992 Ann. Phys. 215 1-50), but the algebraic form of our respective expressions differs. We are also applying the formalism to a novel type of spatial parallel transport introduced in Jonsson (2006 Class. Quantum Grav. 23 1), as well as verifying the validity of the intuitive approach of aforthcoming paper (Jonsson 2006 forthcoming) 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.