Geodesics in Kerr

We work in a Kerr background with angular momentum and mass parameters \((J, M)\) and use Boyer-Lindquist coordinates \((t, r, \theta, \phi)\).

Bound periodic timelike geodesics in Kerr spacetime are defined in terms of the turning points of the motion:

• \(r_\mathrm{min}\) : minimum Boyer-Lindquist radius

• \(r_\mathrm{max}\) : maximum Boyer-Lindquist radius

• \(\theta_\mathrm{min}\) : minimum Boyer-Lindquist polar angle

• \(\pi-\theta_\mathrm{min}\) : maximum Boyer-Lindquist polar angle

From these we parametrize the geodesic in terms of the generalized Keplerian parameters:

• \(a\) : the dimensionless Kerr spin parameter

• \(p\) : the dimensionless semilatus rectum

• \(e\) : the orbital eccentricty

• \(x\) : cosine of the orbital inclination

where \(a = J/M^2\), \(pM = 2r_\mathrm{max}r_\mathrm{min}/(r_\mathrm{min}+r_\mathrm{max})\) and \(e = (r_\mathrm{max}-r_\mathrm{min})/(r_\mathrm{min}+r_\mathrm{max})\). The motion can also be described in terms of the conserved orbital constants:

• \(E\) : the specific orbital energy

• \(L_z\) : the specific orbital angular momentum

• \(Q\) : the Carter constant

which have units \(\{1, M, M^2\}\), respectively, along with the mass of the small body \(\mu\).

With these conserved quantities, we obtain four first-order ordinary differential equations (ODEs) for \(x_p^\mu\), which decouple when parametrized in terms of the Mino(-Carter) time parameter \(\lambda\),

\[\begin{split}\begin{align} \frac{dt_p}{d\lambda} &= V_{tr}(r_p) + V_{t\theta}(\theta_p), \\ \frac{dr_p}{d\lambda} &= \pm \sqrt{V_r(r_p)}, \\ \frac{d\theta_p}{d\lambda} &= \pm \sqrt{V_\theta(\theta_p)}, \\ \frac{d\phi_p}{d\lambda} &= V_{\phi r}(r_p) + V_{\phi \theta}(\theta_p), \end{align}\end{split}\]

where \(d\lambda = \Sigma^{-1} d\tau\), and the potential functions are given by

\[\begin{split}\begin{align} V_{r}(r) &= { P^2(r) - \Delta\left(r^2 + {K} \right),} & V_{\theta}(\theta) &= {{Q} - {L}_z^2 \cot^2\theta - a^2 (1 -{E}^2)\cos^2\theta,} \\ V_{tr}(r) &= \frac{r^2+a^2}{\Delta}P(r), & V_{t\theta}(\theta) &= a{L}_z - a^2 {E} \sin^2\theta, \\ V_{\phi r}(r) &= \frac{a}{\Delta}P(r), & V_{\phi \theta}(\theta) &= {L}_z \csc^2\theta - a {E}, \end{align}\end{split}\]

with \(P(r) = (r^2+a^2){E} - a {L}_z\).

The resulting bound solutions can be separated into terms that are periodic with respect to the Mino time radial and polar frequencies \(\Upsilon_r\) and \(\Upsilon_\theta\) and terms that grow secularly with the Mino time rates \(\Upsilon_t\) and \(\Upsilon_\phi\). Therefore, the fundamental coordinate time frequencies are given by

\[\begin{align} \Omega_r &= \frac{\Upsilon_r}{\Upsilon_t}, & \Omega_\theta &= \frac{\Upsilon_\theta}{\Upsilon_t}, & \Omega_\phi &= \frac{\Upsilon_\phi}{\Upsilon_t}. \end{align}\]

We then represent the radial and polar motion by

\[\begin{split}\begin{align} r_p(\lambda) &= \Delta r^{(r)}(\Upsilon_r\lambda) = \Delta r^{(r)}(\Upsilon_r\lambda + 2\pi), \\ \theta_p(\lambda) &= \Delta \theta^{(\theta)}(\Upsilon_\theta\lambda) = \Delta \theta^{(\theta)}(\Upsilon_\theta\lambda + 2\pi), \end{align}\end{split}\]

while time and azimuthal angle grow as

\[\begin{split}\begin{align} t_p(\lambda) &= \Upsilon_t \lambda + \Delta t^{(r)}(\Upsilon_r\lambda) + \Delta t^{(\theta)}(\Upsilon_\theta\lambda), \\ \phi_p(\lambda) &= \Upsilon_\phi \lambda + \Delta \phi^{(r)}(\Upsilon_r\lambda) + \Delta \phi^{(\theta)}(\Upsilon_\theta\lambda), \end{align}\end{split}\]

where \(\Delta t^{(r)}\), \(\Delta \phi^{(r)}\), \(\Delta t^{(\theta)}\), and \(\Delta \phi^{(\theta)}\) are \(2\pi\)-periodic odd functions.