Publikace ÚTF

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 Vμ chosen to fix the spin condition (the “reference
observer”) and the first spatial vector by the corresponding spin sμ; such projections do not contain the
Weyl scalars Ψ0 and Ψ4 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 Vμ 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 kμ with the highest-multiplicity principal null direction of theWeyl tensor, and then sets Vμ so that kμ
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 Ψ1 and Ψ3 vanish. If the spinbivector
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 Ψ2 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. The particular option of having four-velocity parallel to four-momentum
is advocated, and a natural resolution of nonuniqueness of the corresponding reference observer Vμ is
suggested.