Spinning test body orbiting around a Schwarzschild 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-Semerák 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 an analytical method of finding ISCOs for any SSC and employ it for the Tulczyjew-Dixon and the Mathisson-Pirani formalisms. However, at the end we resort to numerical investigation of the convergence between the SSCs for the ISCO case, due to technical difficulties with the Ohashi-Kyrian-Semerák SSC. 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.

Self-Force Calculations with a Spinning Secondary

“…A thorough study of motion governed by the MPD equations was produced in [30,31]. The consistency of different physical results for solutions of the MPD equations with several common SSCs was investigated in [32,33]. The results in [33] showed, for example, that varying the SSC can change the expression for the orbital frequency of a spinning body in a circular orbit at quadratic and higher order in spin.…”

“…The consistency of different physical results for solutions of the MPD equations with several common SSCs was investigated in [32,33]. The results in [33] showed, for example, that varying the SSC can change the expression for the orbital frequency of a spinning body in a circular orbit at quadratic and higher order in spin. As the secondary's spin is proportional to the square of its mass, the leading linear spin terms enter the metric perturbation at second order in the mass ratio, and linearizing in spin is necessary to consistently satisfy the equations of motion at each order in the mass ratio.…”

Spin Dynamics of Moving Bodies in Rotating Black Hole Spacetimes

“…This is in agreement with result of ref. [128], where the evolution of test bodies moving on circular equatorial orbits around a Schwarzschild black hole were investigated.…”

Self-force calculations with a spinning secondary

“…A thorough study of motion governed by the MPD equations was produced in [31,32]. The consistency of different physical results for solutions of the MPD equations with several common SSCs was investigated in [33,34]. The results in [34] showed, for example, that varying the SSC can change the expression for the orbital frequency of a spinning body in a circular orbit at cubic and higher order in spin.…”