Bound Kerr Geodesics and Fundamental Frequencies

A bound geodesic in Kerr is generically triperiodic rather than Keplerian. Radial libration, polar libration, and azimuthal circulation have distinct fundamental frequencies, and their differences generate periapsis and orbital-plane precession. For an extreme-mass-ratio inspiral these frequencies form the fast subsystem on which radiation reaction acts slowly.

Helpful background. The Einstein Equation fixes the background field equation, Weak Metric Perturbations and Gauge Freedom supplies perturbative gauge control, and Presymplectic Potentials and Currents clarifies conserved quantities. Black-Hole Perturbation Theory supplies the Kerr perturbations sourced by the orbit, Relativistic Two-Body Dynamics supplies the mass-ratio expansion, and Gravitational Radiation and Waveforms supplies asymptotic flux and waveform conventions.

Separated Kerr motion

Let the large black hole have mass $M_\bullet$, let the secondary have mass $m_\star$, and define $\epsilon=m_\star/M_\bullet\ll1$. Neglect the secondary’s spin and self-force. In Boyer–Lindquist coordinates the specific energy $E=-u_t$, axial angular momentum $L_z=u_\phi$, and Carter constant $Q$ separate the geodesic equations. Here $a=J_\bullet/M_\bullet$ has dimensions of length, $\lvert a\rvert\lt GM_\bullet$, and the stable bound orbit lies outside $r_+=GM_\bullet+\sqrt{G^2M_\bullet^2-a^2}$. Use the Carter convention $Q=K_{\mu\nu}u^\mu u^\nu-(L_z-aE)^2$, so $Q=0$ for equatorial motion; $K_{\mu\nu}u^\mu u^\nu$ is another commonly used constant and must be distinguished from $Q$. Introduce Mino time $\lambda$ by

$$\frac{d\tau}{d\lambda}=\Sigma, \qquad \Sigma=r^2+a^2\cos^2\theta.$$

Then the oscillatory equations decouple:

$$\left(\frac{dr}{d\lambda}\right)^2=R(r), \qquad \left(\frac{d\theta}{d\lambda}\right)^2=\Theta(\theta),$$

where

$$R(r) =\left[E(r^2+a^2)-aL_z\right]^2 -\Delta\left[r^2+(L_z-aE)^2+Q\right],$$ $$\Theta(\theta) =Q-\cos^2\theta \left[ a^2(1-E^2)+\frac{L_z^2}{\sin^2\theta} \right], \qquad \Delta=r^2-2GM_\bullet r+a^2.$$

Factors of the small body’s mass have been divided out. A bound orbit has two radial turning points $r_{\mathrm p}$ and $r_{\mathrm a}$ and a polar turning point $\theta_{\min}$. It may equivalently be labeled by $(p,e,\iota)$, with

$$r_{\mathrm p}=\frac{pGM_\bullet}{1+e}, \qquad r_{\mathrm a}=\frac{pGM_\bullet}{1-e},$$

plus a declared inclination convention. Different definitions of inclination coincide only in special limits, so $Q$, $\theta_{\min}$, and $\iota$ should not be interchanged silently.

From Mino periods to observable frequencies

The Mino-time periods are quadratures,

$$\Lambda_r =2\int_{r_{\mathrm p}}^{r_{\mathrm a}} \frac{dr}{\sqrt{R(r)}}, \qquad \Lambda_\theta =4\int_{\theta_{\min}}^{\pi/2} \frac{d\theta}{\sqrt{\Theta(\theta)}}.$$

Define

$$\Upsilon_r=\frac{2\pi}{\Lambda_r}, \qquad \Upsilon_\theta=\frac{2\pi}{\Lambda_\theta}.$$

The remaining equations split into secular and oscillatory pieces:

$$t(\lambda)=\Gamma\lambda+\Delta t_r+\Delta t_\theta, \qquad \phi(\lambda)=\Upsilon_\phi\lambda+\Delta\phi_r+\Delta\phi_\theta,$$

where each $\Delta$ term is periodic with zero average. Frequencies measured relative to the asymptotically normalized Kerr time are therefore

$$\Omega_r=\frac{\Upsilon_r}{\Gamma}, \qquad \Omega_\theta=\frac{\Upsilon_\theta}{\Gamma}, \qquad \Omega_\phi=\frac{\Upsilon_\phi}{\Gamma}.$$

These coordinate-time frequencies, not the Mino frequencies alone, label the spectral lines seen at infinity. Their invariant use presumes transformations that preserve the asymptotic normalization of time and the axial angle. Calling a proper-time or Mino-time frequency “the orbital frequency” without specifying the clock obscures this condition.

The missing secular coefficients follow from the separated first integrals. Writing $P(r)=E(r^2+a^2)-aL_z$, one has

$$\begin{aligned} \frac{dt}{d\lambda} &=\frac{(r^2+a^2)P(r)}{\Delta} +aL_z-a^2E\sin^2\theta,\\ \frac{d\phi}{d\lambda} &=\frac{aP(r)}{\Delta} +\frac{L_z}{\sin^2\theta}-aE. \end{aligned}$$

Thus $\Gamma=\langle dt/d\lambda\rangle_\lambda$ and $\Upsilon_\phi=\langle d\phi/d\lambda\rangle_\lambda$ are obtained by averaging the radial and polar pieces separately over their Mino periods. Together with the two period integrals, these equations provide a complete frequency calculation. They also check dimensions: $\lambda$ has dimension inverse length, $\Upsilon_i$ has dimension length, and $\Gamma$ has dimension length squared, giving $\Omega_i$ dimension inverse time.

A geodesic source has harmonics

$$\omega_{mkn} =m\Omega_\phi+k\Omega_\theta+n\Omega_r.$$

This formula is the bridge to frequency-domain black-hole perturbation theory. The construction and its use for generic Kerr orbits are given in Drasco and Hughes 2004, §§II–IV and reviewed in Pound and Wardell 2021, §6.1, pp. 54–65.

Precession and special limits

For positive azimuthal circulation, two differences have direct physical interpretations:

$$\Omega_{\mathrm{peri}}=\Omega_\phi-\Omega_r, \qquad \Omega_{\mathrm{nodal}}=\Omega_\phi-\Omega_\theta.$$

Here $\Omega_\phi$ is signed while libration frequencies are positive. For retrograde circulation, periapsis and nodal rates must include the chosen orientation; the displayed positive-circulation formulas cannot be used unchanged. In the weak-field limit the frequency magnitudes approach the Kepler value and both relativistic precessions become small. In Schwarzschild, spherical symmetry makes the orbital plane fixed, so after rotating that plane to the equator one has $\Omega_\theta=\Omega_\phi$. Radial and azimuthal frequencies remain distinct and encode periapsis advance.

For a circular equatorial orbit, the absent radial and polar librations are understood by taking epicyclic limits. With $u=GM_\bullet/r$ and signed $\chi=a/(GM_\bullet)$, choose $\Omega_\phi\gt0$, so positive $\chi$ describes prograde motion and negative $\chi$ retrograde motion relative to the black-hole spin. Then

$$GM_\bullet\Omega_\phi =\frac{1}{u^{-3/2}+\chi},$$ $$\frac{\Omega_r^2}{\Omega_\phi^2} =1-6u+8\chi u^{3/2}-3\chi^2u^2, \qquad \frac{\Omega_\theta^2}{\Omega_\phi^2} =1-4\chi u^{3/2}+3\chi^2u^2.$$

At $\chi=0$, $\Omega_\theta=\Omega_\phi$ and $\Omega_r=\Omega_\phi\sqrt{1-6u}$. The radial frequency vanishes at the Schwarzschild ISCO, $r=6GM_\bullet$, while the azimuthal frequency remains finite. This limiting check catches both sign errors and confusion between a separatrix and a vanishing orbital frequency.

Resonances and the slow inspiral

An $r$–$\theta$ orbital resonance occurs when

$$n_r\Omega_r+n_\theta\Omega_\theta=0$$

for nonzero integers $n_r,n_\theta$. Since both frequencies contain the same factor $1/\Gamma$, the same rational condition may be written using $\Upsilon_r$ and $\Upsilon_\theta$. It must not be tested by mixing a coordinate-time frequency with a proper-time frequency.

Away from a resonance, phases cover the two-torus ergodically and long-time averages lose their dependence on the initial radial–polar phase. At a commensurability, one phase combination becomes slowly varying and normally oscillatory self-force terms can accumulate. This is why the geodesic frequency map is not only a kinematic classification: it identifies where ordinary adiabatic averaging fails.

The geodesic approximation itself holds over orbital times. Radiation reaction changes $(E,L_z,Q)$ over a time of order $GM_\bullet/\epsilon$ at fixed dimensionless orbital parameters. Near the separatrix, $\Omega_r$ tends toward zero and the radial period grows, degrading the separation between orbital and inspiral times even before the formal plunge.

Exercises

Recover the Schwarzschild circular limits and identify the ISCO using the epicyclic formulas.

Solution

Setting $\chi=0$ gives

$$\Omega_\theta=\Omega_\phi, \qquad \Omega_r^2=\Omega_\phi^2(1-6GM_\bullet/r).$$

The equality of polar and azimuthal frequencies expresses spherical symmetry. Radial stability requires $\Omega_r^2\gt 0$, and marginal stability occurs at $r=6GM_\bullet$. In the weak-field limit $GM_\bullet/r\to0$, all three frequencies coincide.

Show that an $r$–$\theta$ resonance condition is unchanged when written in Mino time.

Solution

Substituting $\Omega_i=\Upsilon_i/\Gamma$ gives

$$n_r\Omega_r+n_\theta\Omega_\theta =\frac{n_r\Upsilon_r+n_\theta\Upsilon_\theta}{\Gamma}.$$

For a regular bound orbit $\Gamma$ is finite and nonzero, so either numerator vanishes exactly when the coordinate-time combination does. This equivalence works because both terms use the same clock conversion.

References

• Drasco, Steve, and Scott A. Hughes. “Rotating Black Hole Orbit Functionals in the Frequency Domain.” Physical Review D 69 (2004): 044015. DOI.
• Pound, Adam, and Barry Wardell. “Black Hole Perturbation Theory and Gravitational Self-Force.” In Handbook of Gravitational Wave Astronomy, edited by Cosimo Bambi, Stavros Katsanevas, and Konstantinos D. Kokkotas, 1–119. Singapore: Springer (2021). DOI.