Abstrakt
We consider the motion of spinning test particles with nonzero rest mass in the "pole-dipole" approximation, as described by the Mathisson-Papapetrou-Dixon (MPD) equations, and examine its properties in dependence on the spin supplementary condition added to close the system. In order to better understand the spin-curvature interaction, the MPD equation of motion is decomposed in the orthonormal tetrad whose time vector is given by the four-velocity V-mu chosen to fix the spin condition (the "reference observer") and the first spatial vector by the corresponding spin s(mu); such projections do not contain the Weyl scalars. 0 and. 4 obtained in the associated Newman-Penrose (NP) null tetrad.
One natural option of how to choose the remaining two spatial basis vectors is shown to follow "intrinsically" whenever V-mu has been chosen; it is realizable if the particle's four-velocity and four-momentum are not parallel. In order to see how the problem depends on the algebraic type of curvature, one first identifies the first vector of the NP tetrad k(mu) with the highest-multiplicity principal null direction of the Weyl tensor, and then sets V-mu so that k(mu) belong to the spin-bivector eigenplane.
In spacetimes of any algebraic type but III, it is known to be possible to rotate the tetrads so as to become "transverse," namely so that psi(1) and psi(3) vanish. If the spin-bivector eigenplane could be made to coincide with the real-vector plane of any of such transverse frames, the spinning particle motion would consequently be fully determined by psi(2) and the cosmological constant; however, this can be managed in exceptional cases only.
Besides focusing on specific Petrov types, we derive several sets of useful relations that are valid generally and check whether/how the exercise simplifies for some specific types of motion.