Spinning test particles in a Kerr field - II

Abstract

The motion of small spinning free test bodies is usually treated within the 'pole-dipole' approximation, which -in general relativity -leads to Mathisson-Papapetrou (MP) equations. These have to be supplemented by three side constraints in order to provide a unique solution. Several different 'spin conditions' have been proposed and used in the literature, each leading to different worldlines. In a previous paper, we integrated the MP equations with the p σ S μσ = 0 condition numerically in Kerr space-time and illustrated the effect of the spincurvature interaction by comparing the trajectories obtained for various spin magnitudes. Here we also consider other spin conditions and clarify their interrelations analytically as well as numerically on particular trajectories. The notion of a 'minimal worldtube' is introduced in order to judge the individual supplementary conditions and to expose the limitations of the pole-dipole approximation.The motion of extended bodies is a difficult problem even if they do not radiate and do not contribute to the field (are treated as 'test'). However, when the body is small in comparison with the characteristic length of the background field (e.g. in comparison with the distance from the field source), its structure can be described by multipoles and its evolution then expressed in terms of their change. In general relativity, the multipoles are given by integrals,of the energy-momentum tensor T μν , calculated around a certain 'representative' worldline X α (τ ) chosen within the body's history; g ≡ det(g μν ) is the metric-tensor determinant, δx α ≡ x α − X α is a deviation from the representative worldline, running through the volume of the body, and τ is the proper time along the representative worldline (dτ 2 = −g μν dX μ dX ν ). Evolution equations are then obtained by integration of the covariant conservation laws T μν ;ν = 0. They have the form of a series of multipole contributions that generally fall off with n as (R/r) n , where R is the characteristic size of the body and r is the characteristic length of the background space-time. The latter is given by the Riemann curvature tensor R μ νκλ which complicates the equations heavily with increasing n, so that only several-term approximations of the evolution equations are usually employed. The approximation involving just the first two terms (the 'pole-dipole particle') leads to the Mathisson-Papapetrou (MP) equations (Papapetrou 1951): 1 We use geometrized units in which c = G = 1 and the metric signature (−+++). Greek indices run from 0 to 3 and the summation convention is followed. Comma/semicolon in the index denotes the partial/covariant derivative. 2 Please note two omissions that crept into appendix A of Paper I: in the first two lines of the Riemann-tensor Boyer-Lindquist components, sin 2 θ should have been added at R φ ttφ and R φ trθ ; namely, these components should have had −[(2Mar sin 2 θ )/( − a 2 sin 2 θ )] in front.

Parameters of innermost stable circular orbits of spinning test particles: Numerical and analytical calculations

“…For the Schwarzschild black hole and non-spinning test particle the efficiency equals to 0.057, and it reaches maximum for the extreme Kerr black hole -0.42 [7], [44]. In the case of spinning test particles we can easily calculate the efficiency with using of expressions for E ISCO presented in (26), (30), see Fig. 12.…”

Motion of a spinning particle in curved space-time

“…There are two complimentary approaches to the subject. Since the gravitating objects possess quasi-rigid rotation along with orbital motion, studies have aimed at keeping track of the centre of mass by using different supplementary conditions with in the Mathisson-Papapetrou model [1][2][3][4][5][6][7][8][9][10][11][12] . In practise, determining the overall motion of the body, by following a detailed microscopic description of a material body is often too complicated.…”

Exotic orbits due to spin–spin coupling around Kerr black holes

“…However, the phenomena and characteristics of extreme spinning particles orbiting near a black hole are very interesting for researchers. [14][15][16][17][18][19][20][21] In this paper, we focus on an exotic orbital configuration whose orbit pattern is asymmetrical about the equatorial plane of the Kerr black hole. We try to study this interesting phenomenon in details and reveal its physical reasons.…”