Kerr Metric: Rotating Black Holes
The Kerr solution, discovered by Roy Kerr in 1963, describes rotating black holes—the most astrophysically relevant black holes in the universe. We derive the Kerr metric in Boyer-Lindquist coordinates, analyze its horizons, ergosphere, and frame-dragging effects.
1. Boyer-Lindquist Coordinates
The Kerr metric in Boyer-Lindquist coordinates (t, r, θ, φ) is:
$ds^2 = -\left(1 - \frac{r_s r}{\rho^2}\right) c^2 dt^2 - \frac{r_s r a \sin^2\theta}{\rho^2} (c\,dt\,d\phi + d\phi\,c\,dt) + \frac{\rho^2}{\Delta} dr^2 + \rho^2 d\theta^2 + \frac{\Sigma}{\rho^2} \sin^2\theta\, d\phi^2$
where the auxiliary functions are defined as:
$\rho^2 = r^2 + a^2 \cos^2\theta$
$\Delta = r^2 - r_s r + a^2$
$\Sigma = (r^2 + a^2)^2 - a^2 \Delta \sin^2\theta$
Parameters:
- $r_s = 2GM/c^2$ : Schwarzschild radius
- $a = J/(Mc)$ : specific angular momentum ($0 \leq a \leq M$ for physical black holes)
- $J$ : total angular momentum of the black hole
Limits
- a → 0: Reduces to Schwarzschild metric
- a = M (extremal Kerr): Maximal rotation, single degenerate horizon
- a > M (naked singularity): Violates cosmic censorship, likely unphysical
2. Event Horizons
The horizons occur where $\Delta = 0$. Solving the quadratic equation:
$r^2 - r_s r + a^2 = 0$
Using the quadratic formula:
$r_{\pm} = \frac{r_s \pm \sqrt{r_s^2 - 4a^2}}{2} = \frac{GM}{c^2} \pm \sqrt{\left(\frac{GM}{c^2}\right)^2 - a^2}$
This gives two horizons:
- Outer (event) horizon: $r_+ = (GM/c^2)[1 + \sqrt{1 - a^2/M^2}]$
- Inner (Cauchy) horizon: $r_- = (GM/c^2)[1 - \sqrt{1 - a^2/M^2}]$
Special Cases
Non-rotating (a = 0):
$$r_+ = r_s = 2GM/c^2, \quad r_- = 0$$
Extremal (a = M):
$$r_+ = r_- = GM/c^2 = r_s/2$$
For typical astrophysical black holes, $a/M \approx 0.6$-$0.9$, so:
$$r_+ \approx 1.0 - 1.4\, GM/c^2$$
The outer horizon is where escape becomes impossible; the inner horizon is unstable and likely replaced by a different structure in realistic collapse scenarios.
3. Ring Singularity
Unlike the point singularity in Schwarzschild spacetime, the Kerr singularity is a ring in the equatorial plane ($\theta = \pi/2$).
The singularity occurs where $\rho^2 \to 0$:
$\rho^2 = r^2 + a^2 \cos^2\theta = 0$
This requires both $r = 0$ and $\theta = \pi/2$ simultaneously, which describes a ring of radius $a$ in the equatorial plane.
The Kretschmann scalar diverges at the ring singularity:
$K = R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma} \sim \frac{M^2}{\rho^6} \to \infty$
Exotic feature: The ring singularity has a "hole" at its center. In principle, geodesics could pass through the ring without hitting the singularity, potentially accessing other regions of spacetime (or even other universes in the maximal extension). However, the inner horizon is unstable, so these exotic features may not persist in realistic black holes.
4. Frame Dragging (Lense-Thirring Effect)
The Kerr metric has an off-diagonal term $g_{t\phi} \neq 0$, which causes frame dragging—spacetime itself rotates around the black hole.
The coefficient is:
$g_{t\phi} = -\frac{r_s r a \sin^2\theta}{\rho^2} c$
Angular Velocity of Dragged Frames
A locally non-rotating observer (LNRF) at radius r and angle θ is dragged with angular velocity:
$\omega_{\text{LNRF}} = -\frac{g_{t\phi}}{g_{\phi\phi}} = \frac{2GMar}{\Sigma c}$
Key observations:
- $\omega_{\text{LNRF}} \to 0$ as $r \to \infty$ (no dragging far from the black hole)
- $\omega_{\text{LNRF}}$ increases as $r$ decreases
- At the outer horizon, $\omega_{\text{LNRF}} = \omega_H$, the horizon angular velocity
- Maximum effect in equatorial plane ($\theta = \pi/2$)
Horizon Angular Velocity
The event horizon rotates with angular velocity:
$\omega_H = \frac{ac}{r_+^2 + a^2} = \frac{c^3 a}{2GMr_+}$
For extremal Kerr ($a = M$), this becomes:
$$\omega_H^{\text{ext}} = \frac{c}{2M}$$
This is the maximum angular velocity possible for a black hole horizon.
5. Killing Vectors and Conserved Quantities
The Kerr spacetime has two Killing vectors due to symmetries:
Timelike Killing Vector
$\xi^{\mu} = (1, 0, 0, 0)$
This gives conserved energy per unit mass:
$$E = -\xi_\mu u^\mu = -g_{tt} \frac{dt}{d\tau} - g_{t\phi} \frac{d\phi}{d\tau}$$
Axial Killing Vector
$\psi^{\mu} = (0, 0, 0, 1)$
This gives conserved angular momentum per unit mass:
$$L_z = \psi_\mu u^\mu = g_{t\phi} \frac{dt}{d\tau} + g_{\phi\phi} \frac{d\phi}{d\tau}$$
Carter Constant
In addition to $E$ and $L_z$, the Kerr spacetime has a hidden symmetry (Killing tensor) giving a third constant of motion, the Carter constant $Q$:
$$Q = p_\theta^2 + \cos^2\theta \left[a^2(m^2 - E^2) + \frac{L_z^2}{\sin^2\theta}\right]$$
These three constants ($E, L_z, Q$) make the geodesic equations completely integrable via the Hamilton-Jacobi method.
6. Innermost Stable Circular Orbit (ISCO)
The ISCO radius in the Kerr metric depends on the spin parameter $a$ and whether the orbit is prograde (co-rotating) or retrograde (counter-rotating).
Prograde Equatorial ISCO
For particles orbiting in the same direction as the black hole spin (equatorial plane, $\theta = \pi/2$):
$r_{\text{ISCO}}^{\text{pro}} = GM/c^2 \left\{3 + Z_2 - \sqrt{(3-Z_1)(3 + Z_1 + 2Z_2)}\right\}$
where:
$Z_1 = 1 + (1-a^2/M^2)^{1/3}[(1+a/M)^{1/3} + (1-a/M)^{1/3}]$
$Z_2 = \sqrt{3a^2/M^2 + Z_1^2}$
Special Cases
Schwarzschild (a = 0):
$$r_{\text{ISCO}} = 6GM/c^2$$
Extremal prograde (a = M):
$$r_{\text{ISCO}}^{\text{pro}} = GM/c^2$$
Extremal retrograde (a = M):
$$r_{\text{ISCO}}^{\text{retro}} = 9GM/c^2$$
Efficiency of Energy Extraction
For prograde extremal Kerr, the ISCO is at $r = GM/c^2$, very close to the horizon at $r_+ = GM/c^2$. The binding energy is:
$$\eta = 1 - E_{\text{ISCO}}/mc^2 \approx 42\%$$
This is vastly more efficient than nuclear fusion (~0.7%) or Schwarzschild accretion (~6%), explaining the enormous luminosities of quasars and active galactic nuclei.
7. Experimental Verification
Gravity Probe B
The Gravity Probe B satellite measured frame dragging around Earth (2004-2005). The gyroscope precession rate matches the prediction:
$\Omega_{\text{LT}} = \frac{GJ}{c^2 r^3}$
Measured: 37.2 ± 7.2 milliarcseconds/year (agreement with GR within error bars).
X-ray Spectroscopy
Iron K$\alpha$ line profiles from accretion disks show relativistic broadening and asymmetry consistent with Kerr metric predictions. Measurements suggest many supermassive black holes have spins$a/M \approx 0.6$-$0.99$.
Gravitational Waves
LIGO/Virgo detections measure black hole spins from binary mergers. The final black hole spin can exceed the initial spins due to orbital angular momentum, with typical values $a/M \approx 0.7$.
Key Takeaways
- Kerr metric describes rotating black holes with spin parameter $a = J/(Mc)$
- Two horizons: outer $r_+ = (GM/c^2)[1 + \sqrt{1-a^2/M^2}]$, inner $r_-$
- Ring singularity at $r = 0, \theta = \pi/2$ (instead of point singularity)
- Frame dragging: spacetime rotates with angular velocity $\omega_{\text{LNRF}}$
- Horizon rotates with $\omega_H = c^3a/(2GMr_+)$
- Three constants of motion: $E, L_z, Q$ (Carter constant)
- ISCO ranges from $GM/c^2$ (extremal prograde) to $9GM/c^2$ (extremal retrograde)
- Up to 42% energy extraction efficiency for extremal Kerr accretion