Spinning particles in vacuum spacetimes of different curvature types: Natural reference tetrads, and massless particles

Understand

In a previous paper, we considered the motion of massive spinning test particles in the "pole-dipole" approximation, as described by the Mathisson--Papapetrou--Dixon (MPD) equations, and examined its properties in dependence on the spin supplementary condition.

• We decomposed the equations in the orthonormal tetrad based on the time-like vector fixing the spin condition and on the corresponding spin, while representing the curvature in terms of the Weyl scalars obtained in the Newman--Penrose (NP) null tetrad naturally associated with the orthonormal one; the projections thus obtained did not contain the Weyl scalars $\Psi_0$ and $\Psi_4$.
• In the present paper, we choose the interpretation tetrad in a different way, attaching it to the tangent $u^\mu$ of the world-line representing the history of the spinning body.
• Actually {\em two} tetrads are suggested, both given "intrinsically" by the problem and each of them incompatible with one specific spin condition.