Ergosphere and Frame Dragging
The ergosphere is a unique feature of rotating black holes—a region where spacetime is dragged so violently that nothing can remain stationary. We derive the ergosurface location, analyze the physics of frame dragging, and explore energy extraction mechanisms.
1. The Ergosurface
The ergosurface is the boundary where the timelike Killing vector ∂/∂t becomes null. Inside this surface, ∂/∂t becomes spacelike—time and space swap character.
The Killing vector ξ = ∂/∂t has norm:
$\xi^\mu \xi_\mu = g_{tt} = -\left(1 - \frac{r_s r}{\rho^2}\right)$
The ergosurface occurs where g_tt = 0:
$1 - \frac{r_s r}{\rho^2} = 0 \quad \Rightarrow \quad r_s r = r^2 + a^2 \cos^2\theta$
Solving for r:
$r_E(\theta) = \frac{r_s}{2} + \sqrt{\frac{r_s^2}{4} - a^2 \cos^2\theta} = \frac{GM}{c^2} + \sqrt{\left(\frac{GM}{c^2}\right)^2 - a^2 \cos^2\theta}$
Ergosurface Properties
- Equatorial plane (θ = π/2): r_E = r_s = 2GM/c² (touches Schwarzschild radius)
- Polar axis (θ = 0): r_E = GM/c² + √[(GM/c²)² - a²] = r_+ + a²/(r_+-r_-) (maximum extent)
- Oblate spheroid: Flattened at poles due to rotation
- Outside event horizon: r_E > r_+ everywhere
The ergosphere: The region between the ergosurface (r = r_E) and the outer event horizon (r = r_+) is called the ergosphere. Objects can enter and exit the ergosphere, but cannot remain stationary there.
2. Physics Inside the Ergosphere
Inside the ergosphere, the nature of spacetime changes dramatically. Let's analyze what happens to observers and their possible trajectories.
No Static Observers
A "static" observer has four-velocity proportional to the Killing vector ∂/∂t:
$u^\mu = \frac{1}{\sqrt{-g_{tt}}} (1, 0, 0, 0)$
For this to be timelike (physical), we need g_tt < 0. Inside the ergosphere, g_tt > 0, so ∂/∂t is spacelike. A static observer would need to move faster than light—impossible!
Forced Co-rotation
All observers inside the ergosphere must co-rotate with the black hole. The minimum angular velocity is:
$\Omega_{\min} = -\frac{g_{t\phi}}{g_{\phi\phi}} \quad \text{(same direction as black hole spin)}$
Even if an observer tries to move counter to the rotation, they are dragged forward. The ergosphere is sometimes called the "dragging region."
Light Cones
Outside the ergosphere, light cones are tilted but still allow both prograde and retrograde motion. Inside the ergosphere:
- Light cones tilt further toward the rotation direction
- All future-directed timelike paths have dφ/dt > 0 (forced co-rotation)
- Retrograde motion (dφ/dt < 0) becomes spacelike (forbidden)
3. Frame Dragging Quantification
We can quantify frame dragging by computing the angular velocity of locally non-rotating frames (LNRFs).
LNRF Angular Velocity
A locally non-rotating observer (zero angular momentum in local rest frame) moves with angular velocity:
$\omega = \frac{d\phi}{dt} = \frac{2GMar}{\Sigma c} = \frac{r_s a r c}{2\Sigma}$
where Σ = (r² + a²)² - a² Δ sin²θ. Substituting:
$\omega(r,\theta) = \frac{2GMar c}{[(r^2+a^2)^2 - a^2(r^2 - r_s r + a^2)\sin^2\theta]}$
Limiting Cases
Far from black hole (r → ∞):
$\omega \approx \frac{2GMa}{r^3 c} = \frac{GJ}{r^3 c^2}$
(Lense-Thirring precession, agrees with weak-field limit)
At outer horizon (r = r_+):
$\omega_H = \frac{ac}{r_+^2 + a^2}$
(Horizon angular velocity)
Extremal Kerr (a = M, r_+ = GM/c²):
$\omega_H^{\text{ext}} = \frac{c}{2GM}$
The dragging effect is strongest near the horizon and decreases as 1/r³ at large distances.
4. Negative Energy Orbits
One of the most remarkable features of the ergosphere is the existence of negative energy orbits—trajectories with energy E < 0 as measured by observers at infinity.
Energy at Infinity
The energy per unit mass of a particle is:
$E = -\xi_\mu u^\mu = -g_{tt} \frac{dt}{d\tau} - g_{t\phi} \frac{d\phi}{d\tau}$
Using conserved angular momentum L_z = g_tφ (dt/dτ) + g_φφ (dφ/dτ):
$E = -g_{tt} \frac{dt}{d\tau} - g_{t\phi} \frac{d\phi}{d\tau}$
Condition for Negative Energy
For counter-rotating particles (L_z < 0) in the ergosphere, it's possible to have E < 0. The condition is approximately:
$L_z < -mc \frac{g_{t\phi}}{g_{\phi\phi}} r_E$
This is possible because:
- The particle orbits counter to the black hole's rotation
- Frame dragging forces it to move forward (in the rotating frame)
- The particle "swims against the current" so hard that its energy becomes negative
Physical interpretation: Negative energy doesn't mean the particle has negative mass. From the perspective of an observer at infinity, the particle's motion is so opposed to the rotation that dropping it into the black hole would decrease the black hole's mass-energy. This is the basis of the Penrose process.
5. Astrophysical Manifestations
Accretion Disk Warping
Frame dragging causes accretion disks around rotating black holes to precess and warp. The Bardeen-Petterson effect: the inner disk aligns with the black hole's equatorial plane due to frame dragging, even if the outer disk is misaligned.
Quasi-Periodic Oscillations (QPOs)
X-ray binaries show QPOs with frequencies matching the ISCO frequency and frame-dragging precession rates, providing strong evidence for the Kerr metric and allowing spin measurements.
Relativistic Jets
The Blandford-Znajek mechanism extracts rotational energy from spinning black holes via magnetic fields threading the ergosphere, potentially powering relativistic jets seen in quasars and active galactic nuclei.
6. Experimental Evidence
Gravity Probe B
Frame dragging around Earth (a much weaker effect than near black holes):
$\text{Predicted: } 39 \pm 1 \text{ milliarcsec/year}$
$\text{Measured: } 37.2 \pm 7.2 \text{ milliarcsec/year}$
LIGO Binary Mergers
Gravitational wave signals encode black hole spins. Precession of orbital planes due to spin-orbit coupling matches Kerr predictions. Typical measured spins: a/M ≈ 0.3-0.9.
EHT Imaging
The M87* image shows asymmetry consistent with a rotating black hole with spin a/M ≈ 0.5-0.9. The bright southern side matches frame dragging predictions for gas orbiting a spinning black hole.
Key Takeaways
- Ergosurface at r_E = (GM/c²) + √[(GM/c²)² - a²cos²θ] where g_tt = 0
- Ergosphere: region between ergosurface and event horizon
- No static observers possible inside ergosphere (forced co-rotation)
- LNRF angular velocity ω = 2GMar/(Σc)
- Negative energy orbits exist in the ergosphere
- Frame dragging ~ 1/r³ at large distances (Lense-Thirring)
- Astrophysical effects: disk warping, QPOs, jet launching
- Confirmed by Gravity Probe B, LIGO, and EHT observations