Spinning particles in vacuum spacetimes of different curvature types

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. The MPD equation of motion is decomposed in the orthonormal tetrad whose time vector is given by the four-velocity $V^μ$ chosen to fix the spin condition (the "reference observer") and the first spatial vector by the corresponding spin; such projections do not contain the Weyl scalars $Ψ_0$ and $Ψ_4$ obtained in the associated Newman-Penrose (NP) null tetrad. One natural choice of the remaining two spatial basis vectors is shown to follow "intrinsically"; it is realizable if the particle's four-velocity and four-momentum are not parallel. To see how the problem depends on the curvature type, one first identifies the first vector of the NP tetrad $k^μ$ with the highest-multiplicity principal null direction of the Weyl tensor, and then sets $V^μ$ so that $k^μ$ belong to the spin-bivector eigenplane. In spacetimes of any algebraic type but III, it is possible to rotate the tetrads so as to become "transverse", namely so that $Ψ_1$ and $Ψ_3$ vanish. If the spin-bivector eigenplane could be made coincide with the real-vector plane of any of such transverse frames, the motion would consequently be fully determined by $Ψ_2$ 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 valid generally and check whether/how the exercise simplifies for some specific types of motion. The particular option of having four-velocity parallel to four-momentum is advocated and a natural resolution of nonuniqueness of the corresponding reference observer $V^μ$ is suggested.

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