Spinning test body orbiting around a Kerr black hole: Comparing spin supplementary conditions for circular equatorial orbits

The worldline of a spinning test body moving in curved spacetime can be provided by the MathissonPapapetrou-Dixon equations when its centroid, i.e., its center of mass, is fixed by a spin supplementary condition (SSC). In the present study, we continue the exploration of shifts between different centroids started in a recently published work [Iason Timogiannis et al., Phys. Rev. D 104, 024042 (2021).], henceforth paper I, for the Schwarzschild spacetime, by examining the frequencies of circular equatorial orbits under a change of the SSC in the Kerr spacetime. In particular, we examine the convergence in the terms of the prograde and retrograde orbital frequencies, when these frequencies are expanded in power series of the spin measure and the centroid of the body is shifted from the Mathisson-Pirani or the OhashiKyrian-Semerak frame to the Tulczyjew-Dixon one. Since in paper I we have seen that the innermost stable circular orbits (ISCOs) hold a special place in this comparison process, we focus on them rigorously in this work. We introduce a novel method of finding ISCOs for any SSC and employ it for the Tulczyjew-Dixon and the Mathisson-Pirani formalisms. We resort to numerical investigation of the convergence between the SSCs for the ISCO case, due to technical difficulties not allowing paper I's analytical treatment. Our conclusion, as in paper I, is that there appears to be a convergence in the power series of the frequencies between the SSCs, which is improved when the proper shifts are taken into account, but there exists a limit in this convergence due to the fact that in the spinning body approximation we consider only the first two lower multipoles of the extended body and ignore all the higher ones.

Automated summary

This study examines the effects of changes in the centroid of a spinning test body on the frequencies of circular equatorial orbits in curved spacetime. By numerically investigating the convergence between different spin supplementary conditions, the study finds that there is a convergence in the power series of the frequencies, but it is limited due to the omission of higher multipoles in the spinning body approximation.