Comparing spin supplementary conditions for particle motion around traversable wormholes

The Mathisson-Papapetrou-Dixon equations describe the motion of spinning test particles in the pole-dipole approximation. It is well known that these equations, which couple the Riemann curvature tensor with the antisymmetric spin tensor ${S}^{\ensuremath{\alpha}\ensuremath{\beta}}$, together with the normalization condition for the four-velocity, is a system of 11 equations relating 14 unknowns. To ``close'' the system, it is necessary to introduce a constraint of the form ${V}_{\ensuremath{\mu}}{S}^{\ensuremath{\mu}\ensuremath{\nu}}=0$, usually known as the spin supplementary condition (SSC), where ${V}_{\ensuremath{\mu}}$ is a future-oriented reference vector satisfying the normalization condition ${V}_{\ensuremath{\alpha}}{V}^{\ensuremath{\alpha}}=\ensuremath{-}1$. There are several SSCs in the literature. In particular, the Tulczyjew-Dixon, Mathisson-Pirani, and Ohashi-Kyrian-Semer\'ak are the most used by the community. From the physical point of view, choosing a different SSC (a different reference vector ${V}^{\ensuremath{\mu}}$) is equivalent to fixing the centroid of the test particle. In this manuscript, we compare different SSCs for spinning test particles moving around a Morris-Thorne traversable wormhole. To do so, we first obtain the orbital frequency and expand it up to third order in the particle's spin; as expected, the zero order coincides with the Keplerian frequency, the same in all SSCs; nevertheless, we found that differences appear in the second order of the expansion, similar to the Schwarzschild and Kerr black holes. We also compare the behavior of the innermost stable circular orbit (ISCO). Since each SSC is associated with a different centroid of the test particle, we analyze (separately) the radial and spin corrections for each SSC. We found that the radial corrections improve the convergence, especially between Tulczyjew-Dixon and Mathisson-Pirani SSCs. In the case of Ohashi-Kyrian-Semer\'ak, we found that the spin corrections remove the divergence for the ISCO and extend its existence for higher values of the particle's spin.

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