Orbits in a Kerr Spacetime

The Kerr solution in general relativity describes spacetime geometry about a spinning point-mass of mass $M$. The angular momentum of the black hole is given by the dimensionless angular momentum parameter $a$, which has values in the range $0 \le a \le 1$.

The Kerr metric in Boyer-Lindquist coordinates is

\[ds^2 = -\left(1-\frac{2r}{\Sigma}\right)\, dt^2 - \frac{4ar\sin^2\theta}{\Sigma}\,dt\,d\phi + \sin^2\theta\left(r^2 + a^2 + \frac{2a^2 r \sin^2\theta}{\Sigma}\right)\,d\phi^2 + \frac{\Sigma}{\Delta}\,dr^2 + \Sigma\,d\theta^2,\]

where

\[\begin{aligned} \Sigma &\equiv r^2 + a^2\cos^2\theta,\\ \Delta &= r^2 - 2r + a^2. \end{aligned}\]

Geodesics in the Kerr spacetime have three integrals of motion: the orbital energy $E$, the $z$-component of the orbital angular momentum $L_z$, and the Carter constant $Q$. The geodesic equations can then be written in terms of the particle's proper time $\tau$ as

\[\begin{aligned} \frac{dr}{d\tau} &= \frac{1}{\Sigma} \pm \sqrt{R}\\ \frac{d\theta}{d\tau} &= \frac{1}{\Sigma} \pm \sqrt{\Theta}\\ \frac{d\phi}{d\tau} &= \frac{1}{\Sigma}\left[\frac{a}{\Delta}(2rE - aL_z) + \frac{L_z}{\sin^2\theta}\right]\\ \frac{dt}{d\tau} &= \frac{1}{\Sigma}\left[\frac{(r^2+a^2)^2E - 2arL_z}{\Delta} - a^2 E \sin^2\theta \right]. \end{aligned}\]

Here $R(r)$ and $\Theta(\theta)$ are "quasi-potentials"

\[\begin{aligned} R(r) &= -(1-E^2)r^4 + 2r^3 - \left[a^2(1-E^2) + L_z^2\right]r^2 + 2(aE-L_z)^2 r - Q\Delta ,\\ \Theta(\theta) &= Q - \cos^2\theta\left\{a^2(1-E^2) + \frac{L_z^2}{\sin^2\theta}\right\}. \end{aligned}\]

The equations can be simplified by defining the Mino time $\lambda$ as

\[\frac{d\tau}{d\lambda} \equiv \Sigma \rightarrow \frac{d}{d\lambda} = \Sigma\frac{d}{d\tau}.\]

The geodesic equations of motion in a Kerr spacetime can be written in Mino time as

\[\begin{aligned} \dot r &= \Delta p_r\\ \dot p_r &= \frac{1}{2}\left\{ -\Delta' p_r^2 + \left(\frac{\Delta' R - \Delta R'}{\Delta^2}\right)\right\}\\ \dot \theta &= p_\theta\\ \dot p_\theta &= \frac{1}{2} \Theta^\theta\\ \dot \phi &= -\frac{\partial}{\partial L} \left(\frac{R}{\Delta} + \Theta\right)\\ \dot p_\phi &= 0\\ \dot t &= \frac{1}{2}\frac{\partial}{\partial E} \left(\frac{R}{\Delta} + \Theta\right)\\ \dot p_t &= 0. \end{aligned}\]

Here the prime ($'$) and $\theta$ superscripts indicate differentiation with respect to $r$ and $\theta$, respectively.

Reference: Gabe Perez-Giz, From Measure Zero to Measure Hero: Periodic Kerr Orbits and Gravitational Wave Physics, (PhD Dissertation, Columbia University, 2011).