Inspirals from the Innermost Stable Circular Orbit of Kerr Black Holes: Exact Solutions and Universal Radial Flow

We present exact solutions of test particle orbits spiraling inward from the innermost stable circular orbit (ISCO) of a Kerr black hole. Our results are valid for any allowed value of the angular momentum $a$ parameter of the Kerr metric. These solutions are of considerable physical interest. In particular, the radial four-velocity of these orbits is both remarkably simple and, with the radial coordinate scaled by its ISCO value, universal in form, otherwise completely independent of the black hole spin.

Figure 1

Coefficients $C_0,C_{+}$, and $C_{-}$ appearing in the parametric $\varphi (\psi )$ solution [Eqs. (32) and (33)], as a function of $a/M$. (Spin axis plotted for $-0.998<a/M<0.998$).

Figure 2

Inspiral [Eq. (34)] of a test particle from ISCO ($r\simeq 3.158M$) to event horizon ($r_{+}\simeq 1.661M$) of a Kerr black hole, $a/M=+0.75$.

Figure 3

Inspiral of a test particle from ISCO ($r=9M$) to singularity of an extremal Kerr black hole $a=-M$ [Eq. (37) et seq.]. Upper figure shows the magnitude of the $\varphi$ coordinate over the infall, with the four $r_j(\varphi )$ branches of the solution color coded (see text). The value of the frame-dragging angle ${\varphi }_{\star }$ [Eq. (41)] is shown by the horizontal dashed line. The lower figure shows the inspiral in quasi-Cartesian coordinates with the same color-coding scheme. Note the pronounced effect of counterrotating frame dragging upon the orbit.