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

TL;DR

This paper investigates how different Spin Supplementary Conditions affect the orbital frequencies of spinning test bodies in Kerr spacetime, focusing on the innermost stable circular orbits and demonstrating convergence in frequency expansions.

Contribution

It introduces a new method for finding ISCOs under various SSCs and analyzes the convergence of orbital frequencies between different SSCs in Kerr spacetime.

Findings

Convergence in frequency expansions improves with proper centroid shifts.

ISCO frequencies show convergence between SSCs, especially when shifts are considered.

Limitations arise due to neglecting higher multipole moments of the spinning body.

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.