Precisely computing bound orbits of spinning bodies around black holes II: Generic orbits

Abstract

In this paper, we continue our study of the motion of spinning test bodies orbiting Kerr black holes. Non-spinning test bodies follow geodesics of the spacetime in which they move. A test body's spin couples to the curvature of that spacetime, introducing a "spin-curvature force" which pushes the body's worldline away from a geodesic trajectory. The spin-curvature force is an important example of a post-geodesic effect which must be modeled carefully in order to accurately characterize the motion of bodies orbiting black holes. One motivation for this work is to understand how to include such effects in models of gravitational waves produced from the inspiral of stellar mass bodies into massive black holes. In this paper's predecessor, we describe a technique for computing bound orbits of spinning bodies around black holes with a frequency-domain description which can be solved very precisely. In that paper, we present an overview of our methods, as well as present results for orbits which are eccentric and nearly equatorial (i.e., the orbit's motion is no more than O(S) out of the equatorial plane). In this paper, we apply this formulation to the fully generic case -orbits which are inclined and eccentric, with the small body's spin arbitrarily oriented. We compute the trajectories which such orbits follow, and compute how the small body's spin affects important quantities such as the observable orbital frequencies Ωr, Ω θ and Ω φ .

Motion of a spinning particle under the conservative piece of the self-force is Hamiltonian to first order in mass and spin

“…Note that in order to apply the result of Sec. III, we must use the form ( 13), ( 22) of the dynamical system for which the symplectic form is constant and so not modified by the substitutions (39), rather than the original form ( 8), ( 9). This is because the result requires that the symplectic form be unperturbed, cf.…”

“…(7) which incorporates O(S ) corrections to the worldline. We next make the replacements (39) in the Hamiltonian (13). This yields the pseudo-Hamiltonian…”

“…[30]. They can also be obtained by making the substitutions (39) in the equations of motion (7). As discussed in the introduction, we keep only terms of order O(m 2 ) , O(S ) and O(mS ) in the self-force, and O(mS ) in the self-torque, which explains why we have replaced h µν R with h µν R (m) [cf.…”

“…However, it may yield useful information for the effective one body framework [20,21] (since our calculation validates the Hamiltonian dynamics assumption) and thereby aid waveform modeling for comparable mass binary systems for LIGO. We also note that our anal- First order conservative spin-induced self-force and self-torque 30,39,40] TABLE I. A summary of different conservative effects in the dynamics of binary systems with spinning components in general relativity, in the small mass ratio limit µ ≪ M, where M is the mass of the primary and µ the mass of the secondary.…”

Particle Motion under the Conservative Piece of the Self-Force is Hamiltonian

“…The explicit Hamiltonian we derive may be useful for computations of waveforms for extreme mass ratio inspirals by LISA. In that context, incorporating the spin of the small body will be necessary to obtain accurate waveforms [30][31][32][33]. Spin effects will be comparable to effects that arise from the subleading point particle self-force, assuming that S /µ 2 ∼ 1 for typical compact object sources.…”