Spinning test body orbiting around a Kerr black hole: Comparing Spin Supplementary Conditions for Circular Equatorial Orbits

Abstract

The worldline of a spinning test body moving in curved spacetime can be provided by the Mathisson-Papapetrou-Dixon (MPD) 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 [ 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 Ohashi-Kyrian-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's I 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.

15 pages, 4 figures, 5 tables, 2 Mathematica notebooks

References in corpus (5)

• Motion of spinning test bodies in Kerr spacetime
• Self-Force Calculations with a Spinning Secondary
• Spinning test-body orbiting around a Kerr black hole: circular dynamics and gravitational-wave fluxes
• Time parameterizations and spin supplementary conditions of the Mathisson-Papapetrou-Dixon equations
• Spinning test body orbiting around a Schwarzschild black hole: Comparing Spin Supplementary Conditions for Circular Equatorial Orbits

Cited by in corpus (6)

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• Charged spinning and magnetized test particles orbiting quantum improved charged black holes
• Prospects for determining the nature of the secondaries of extreme mass-ratio inspirals using the spin-induced quadrupole deformation
• Circular equatorial orbits of extended bodies with spin-induced quadrupole around a Kerr black hole: Comparing spin-supplementary conditions
• Resonant Islands of Effective-One-Body Dynamics
• Extended bodies with spin induced quadrupoles on circular equatorial orbits in Kerr spacetime