Homoclinic orbits around spinning black holes. II. The phase space portrait

In paper I in this series, we found exact expressions for the equatorial homoclinic orbits: the separatrix between bound and plunging, whirling and not whirling motion. As a companion to that physical space study, in this paper we paint a phase space portrait of the homoclinic orbits that includes exact expressions for the actions and fundamental frequencies. Additionally, we develop a reduced Hamiltonian description of Kerr motion that allows us to track groups of trajectories with a single global clock. This facilitates a variational analysis, whose stability exponents and eigenvectors could potentially be useful for future studies of families of black hole orbits and their associated gravitational waveforms.

Figure 1

The dimensionless real-valued stability exponent ${\lambda }_r$ (measured in units of $M^{-1}$) for unstable circular orbits with $E<1$ for various spins $a$. Left: Prograde orbits. Right: Retrograde orbits.

Figure 2

Projections of the eigenvector ${\mathbf{u}}_r^{(u)}$, to which the linearized separation $\delta {\mathbf{X}}^{(u)}$ is proportional, overlayed with the actual coordinate differences ${\mathbf{X}}^{\mathrm{hc}}-{\mathbf{X}}^{\mathrm{circ}}$ in the phase space. In the $\Delta \phi$ plot, we have identified $-\pi$ at the bottom of the plot and $\pi$ at the top. The plots, intended to be schematic, are around an unstable circular orbit at $r_u=2.2$ for $a=0.8$.

Figure 3

Schematic of a homoclinic tangle. The curve above represents the repeated intersection of a homoclinic orbit of the perturbed system with the $r$, $p_r$ phase plane. The large dot represents the intersection of a circular orbit in the perturbed system along with the eigenvectors denoting its local stable and unstable manifolds.