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The Mathisson-Papapetrou-Dixon (MPD) equations describe the motion ofspinning test particles. It is well-known that these equations, which couplethe Riemann curvature tensor with the antisymmetric spin tensor S, togetherwith the normalization condition for the four-velocity, is a system of elevenequations relating fourteen unknowns. To ``close'' the system, it is necessaryto introduce a constraint of the form V_\mu S^{\mu \nu} = 0, usually known asthe spin supplementary condition (SSC), where V_\mu is a future-orientedreference vector satisfying the normalization condition V_\alpha V^\alpha = -1.There are several SSCs in the literature. In particular, the Tulzcyjew-Dixon,Mathisson-Pirani, and Ohashi-Kyrian-Semer\'ak are the most used by thecommunity. From the physical point of view, choosing a different SSC (adifferent reference vector $V^\mu$) is equivalent to fixing the centroid of thetest particle. In this manuscript, we compare different SSCs for spinning testparticles moving around a Morris-Thorne traversable wormhole. To do so, wefirst obtain the orbital frequency and expand it up to third-order in theparticle's spin; as expected, the zero-order coincides with the Keplerianfrequency, the same in all SSCs; nevertheless, we found that differences appearin the second order of the expansion, similar to the Schwarzschild and Kerrblack holes. We also compare the behavior of the innermost stable circularorbit (ISCO). Since each SSC is associated with a different centroid of thetest particle, we analyze (separately) the radial and spin corrections for eachSSC. We found that the radial corrections improve the convergence, especiallybetween Tulzcyjew-Dixon and Mathisson-Pirani SSCs. In the case ofOhashi-Kyrian-Semer\'ak, we found that the spin corrections remove thedivergence for the ISCO and extend its existence for higher values of theparticle's spin.

Comparing spin supplementary conditions for particle motion around traversable wormholes