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

摘要

We derive an effectively three-dimensional relativistic spin precessionformalism. The formalism is applicable to any spacetime where an arbitrarytimelike reference congruence of worldlines is specified. We employ what wecall a stopped spin vector which is the spin vector that we would get if wemomentarily make a pure boost of the spin vector to stop it relative to thecongruence. Starting from the Fermi transport equation for the standard spinvector we derive a corresponding transport equation for the stopped spinvector. Employing a spacetime transport equation for a vector along aworldline, corresponding to spatial parallel transport with respect to thecongruence, we can write down a precession formula for a gyroscope relative tothe local spatial geometry defined by the congruence. This general approach hasalready 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 expressionsdiffer. We are also applying the formalism to a novel type of spatial paralleltransport introduced in Jonsson (Class. Quantum Grav. 23 (2006) 1), as well asverifying the validity of the intuitive approach of a forthcoming paper(Jonsson, Am. Journ. Phys. 75 (2007) 463) where gyroscope precession isexplained entirely as a double Thomas type of effect. We also present theresulting formalism in explicit three-dimensional form (using the boldfacevector notation), and give examples of applications.