Hamilton-Jacobi equation for spinning particles near black holes

A compact stellar-mass object inspiraling onto a massive black hole deviates from geodesic motion due to radiation-reaction forces as well as finite-size effects. Such postgeodesic deviations need to be included with sufficient precision into wave-form models for the upcoming space-based gravitational-wave detector LISA. I present the formulation and solution of the Hamilton-Jacobi equation of geodesics near Kerr black holes perturbed by the so-called spin-curvature coupling, the leading-order finite-size effect. In return, this solution allows one to compute a number of observables such as the turning points of the orbits as well as the fundamental frequencies of motion. This result provides one of the necessary ingredients for waveform models for LISA and an important contribution useful for the relativistic two-body problem in general.

Figure 1

The turning points of particles near a Kerr black hole with $a=0.9M$ and with various choices of particle spin. The fiducial geodesic is always with constants $K=2.9M^2$, $E=0.87$, $L=2.0M$ (or semilatus rectum $p=3M$, eccentricity $e=0.1$, and inclination $\sqrt{z_{-\mathrm{g}}}=0.5$). On the left we plot the turning points of the fiducial geodesic (dotted black) and the turning points of a spin-perturbed orbit with completely aligned spin, $s_{\parallel }=s={10}^{-3}M$ (full dark red). On the right we show the turning points of an orbit with a completely oscillating spin $s_{\parallel }=0$, $s=5\times {10}^{-2}M$ for the values of spin angle $\varphi =0,\pi /2,3\pi /2$.

Figure 2

The relative corrections to fundamental frequencies given in units of the aligned component of spin $s_{\parallel }$ as a function of semilatus rectum $p$ for fiducial geodesics with $e=0.1$, $z_{-\mathrm{g}}=0.1$, and $a=0.9M$. The corrections to the radial frequency become large at small $p$ because the motion is becoming unstable.