Geodesics in Kerr Spacetime
The Carter constant, orbital dynamics, and the black hole shadow
Integrals of Motion
Geodesic motion in Kerr spacetime is remarkably integrable. Despite the metric being far more complex than Schwarzschild, the geodesic equations can be reduced to first-order form using four integrals of motion. Two come from Killing vectors, one from the normalization condition, and the fourth — the Carter constant — comes from a hidden symmetry.
The two Killing vectors ξ = ∂/∂t and ψ = ∂/∂φ provide:
$$E = -g_{\mu\nu}\xi^\mu\frac{dx^\nu}{d\tau} = \left(1 - \frac{2Mr}{\Sigma}\right)\frac{dt}{d\tau} + \frac{2Mar\sin^2\theta}{\Sigma}\frac{d\phi}{d\tau}$$
$$L_z = g_{\mu\nu}\psi^\mu\frac{dx^\nu}{d\tau} = -\frac{2Mar\sin^2\theta}{\Sigma}\frac{dt}{d\tau} + \frac{\mathcal{A}\sin^2\theta}{\Sigma}\frac{d\phi}{d\tau}$$
E is the specific energy and Lz is the z-component of specific angular momentum. Note that in Kerr (unlike Schwarzschild), only the axial component Lz is conserved — there is no spherical symmetry.
The Carter Constant
In 1968, Brandon Carter discovered a fourth integral of motion for Kerr geodesics by separating the Hamilton-Jacobi equation. This was unexpected: with only two Killing vectors, one would expect only two conserved quantities (plus the mass-shell condition), leaving the system non-integrable.
The Carter constant arises from a rank-2 Killing tensor Kμν satisfying:
$$\nabla_{(\alpha} K_{\mu\nu)} = 0$$
The conserved quantity is:
Carter Constant
$$\mathcal{Q} = K_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau} = p_\theta^2 + \cos^2\theta\left(a^2(1 - E^2) + \frac{L_z^2}{\sin^2\theta}\right)$$
The Carter constant Q is related to the total angular momentum of the orbit. For equatorial orbits (θ = π/2 throughout), Q = 0. For orbits inclined to the equatorial plane, Q > 0. Orbits with Q < 0 are called "vortical" and remain confined near the poles.
An equivalent quantity sometimes used is:
$$K = \mathcal{Q} + (L_z - aE)^2 = p_\theta^2 + \cos^2\theta\left(\frac{L_z^2}{\sin^2\theta} + a^2 E^2\right) + (L_z - aE)^2$$
First-Order Geodesic Equations
With all four integrals of motion (E, Lz, Q, and the normalization gμνẋμẋν = -δ where δ = 1 for massive and δ = 0 for massless particles), the geodesic equations reduce to first-order form. Using the Mino parameter λ (defined by dτ = Σ dλ) to decouple r and θ:
Kerr Geodesic Equations (Mino Time)
$$\Sigma\frac{dr}{d\lambda} = \pm\sqrt{R(r)}$$
$$\Sigma\frac{d\theta}{d\lambda} = \pm\sqrt{\Theta(\theta)}$$
$$\Sigma\frac{d\phi}{d\lambda} = -\left(aE - \frac{L_z}{\sin^2\theta}\right) + \frac{a}{\Delta}\left[(r^2 + a^2)E - aL_z\right]$$
$$\Sigma\frac{dt}{d\lambda} = -a(aE\sin^2\theta - L_z) + \frac{r^2 + a^2}{\Delta}\left[(r^2+a^2)E - aL_z\right]$$
where the radial and angular potentials are:
$$R(r) = \left[(r^2+a^2)E - aL_z\right]^2 - \Delta\left[r^2\delta + (L_z - aE)^2 + \mathcal{Q}\right]$$
$$\Theta(\theta) = \mathcal{Q} - \cos^2\theta\left(a^2(\delta - E^2) + \frac{L_z^2}{\sin^2\theta}\right)$$
Equatorial Orbits: ISCO
For equatorial orbits (θ = π/2, Q = 0), the analysis parallels Schwarzschild but with crucial spin-dependent corrections. The ISCO radius depends dramatically on the spin:
ISCO Radius (Equatorial)
$$r_{\text{ISCO}} = M\left(3 + Z_2 \mp \sqrt{(3 - Z_1)(3 + Z_1 + 2Z_2)}\right)$$
Upper sign (-) for prograde, lower sign (+) for retrograde orbits
where the auxiliary quantities are:
$$Z_1 = 1 + (1-a^2/M^2)^{1/3}\left[(1+a/M)^{1/3} + (1-a/M)^{1/3}\right]$$
$$Z_2 = \sqrt{3a^2/M^2 + Z_1^2}$$
The prograde-retrograde asymmetry is one of the most striking consequences of frame dragging:
| Spin (a/M) | rISCO (prograde) | rISCO (retrograde) | ηpro |
|---|---|---|---|
| 0 (Schwarzschild) | 6M | 6M | 5.72% |
| 0.5 | 4.233M | 7.555M | 8.28% |
| 0.9 | 2.321M | 8.717M | 15.58% |
| 0.998 | 1.237M | 8.988M | 32.0% |
| 1.0 (extremal) | M | 9M | 42.3% |
ηpro is the accretion efficiency for prograde orbits
For an extremal Kerr black hole, the prograde ISCO coincides with the horizon at r = M, and the accretion efficiency reaches 42.3% — the most efficient gravitational energy extraction mechanism in nature. This far exceeds nuclear fusion (0.7%).
Lense-Thirring Precession
In the weak-field, slow-rotation limit, the Kerr metric reduces to the linearized form and frame dragging manifests as a precession of gyroscope spin axes and orbital planes. This is the Lense-Thirring effect (1918).
For a gyroscope at position r from a rotating body with angular momentum J:
Lense-Thirring Precession Rate
$$\vec{\Omega}_{\text{LT}} = \frac{G}{c^2 r^3}\left[3(\vec{J}\cdot\hat{r})\hat{r} - \vec{J}\right]$$
For Earth (J ≈ 5.86 × 10³³ kg m²/s), the Lense-Thirring precession at satellite altitude is:
Gravity Probe B (2004-2011): Measured the Lense-Thirring precession of gyroscopes in Earth orbit at 39 milliarcseconds/year, agreeing with GR prediction to ~19% accuracy. The geodetic (de Sitter) precession was measured to 0.28% accuracy.
The precession of an orbital plane (nodal precession) follows a similar formula:
$$\dot{\Omega}_{\text{node}} = \frac{2GJ}{c^2 a^3(1-e^2)^{3/2}}$$
where a is the semi-major axis and e the eccentricity
This has been measured for the LAGEOS and LARES satellites, confirming the prediction to a few percent.
Photon Orbits and the Black Hole Shadow
The photon region in Kerr spacetime is far richer than in Schwarzschild. Photons can orbit at various radii depending on their angular momentum and inclination. The critical photon orbits satisfy R(r) = 0 and R'(r) = 0 simultaneously.
For equatorial photon orbits, the photon sphere radii are:
$$r_{\text{ph}} = 2M\left[1 + \cos\left(\frac{2}{3}\cos^{-1}\left(\mp\frac{a}{M}\right)\right)\right]$$
Upper sign for retrograde, lower sign for prograde photon orbits
Photon orbit radii by spin:
- a = 0: rph = 3M (both prograde and retrograde)
- a = 0.5M: rph,pro = 2.347M, rph,ret = 3.532M
- a = M: rph,pro = M (at horizon), rph,ret = 4M
The Black Hole Shadow
The black hole shadow (as observed by a distant observer) is the boundary in the observer's sky between photons that escape to infinity and those captured by the hole. Parameterized by celestial coordinates (α, β) on the observer's sky:
$$\alpha = -\frac{L_z}{E\sin\theta_o}$$
$$\beta = \pm\frac{1}{E}\sqrt{\mathcal{Q} + a^2\cos^2\theta_o - L_z^2\cot^2\theta_o}$$
where θo is the observer's inclination angle. The shadow boundary is traced out by the critical curve parameterized by the photon orbit radius rph.
Key properties of the Kerr shadow:
- For a = 0, the shadow is a perfect circle of radius 3√3 M ≈ 5.196M
- For a > 0, the shadow is displaced toward the approaching side and becomes asymmetric (D-shaped)
- The displacement is due to frame dragging: prograde photons orbit closer, appearing on the approaching side
- The Event Horizon Telescope image of M87* (2019) and Sgr A* (2022) are consistent with Kerr shadows