Questions: Kerr Solution (Rotating Black Holes)

Question 1

The ergosphere of a Kerr black hole is the region where:

• A. All geodesics are ingoing — nothing can escape, including light
• B. No observer can remain at rest relative to distant stars, regardless of thrust, because spacetime itself rotates
• C. The curvature singularity is located
• D. The Kerr metric reduces to the Schwarzschild metric

Answer: (B)

The ergosphere lies between the static limit surface (where g_{tt} = 0) and the outer event horizon. Within it, the frame-dragging effect is so strong that the light cones are tilted in the direction of rotation — no amount of rocket thrust can keep an observer stationary relative to infinity. However, unlike the region inside the event horizon, escape from the ergosphere is possible. This is what enables the Penrose process: a particle entering the ergosphere can split into two pieces, one falling into the black hole with negative energy (as measured at infinity) and the other escaping with more energy than the original particle, extracting rotational energy from the black hole.

Question 2

A maximally rotating Kerr black hole has a = GM/c². At this extremal limit, the inner and outer horizons coincide.

• T. True
• F. False

Answer: True

The horizons are at r± = GM/c² ± √((GM/c²)² - a²). When a = GM/c² (the extremal Kerr limit), the square root vanishes and r₊ = r₋ = GM/c². The event horizon shrinks to half the Schwarzschild radius. For a > GM/c², no horizon exists and the singularity would be 'naked' (visible from the outside), but the cosmic censorship conjecture asserts this cannot form from realistic gravitational collapse. Astrophysical black holes observed via X-ray binary and gravitational wave measurements have spin parameters approaching but not reaching the extremal limit.

Question 3

Explain the Penrose process for extracting energy from a rotating black hole and why it is limited by the irreducible mass.

Model answer: In the Penrose process, a particle enters the ergosphere and splits into two fragments. The ergosphere allows negative-energy orbits (as measured at infinity) because the Killing vector ∂/∂t becomes spacelike there. One fragment falls into the black hole on a negative-energy orbit, decreasing the black hole's mass and angular momentum, while the other escapes with more energy than the original particle — the excess coming from the black hole's rotational energy. The process is limited by the irreducible mass M_irr = (1/2)√(r₊² + a²) c²/G, which is the mass the black hole would have if all its spin were extracted. The area theorem (Hawking) guarantees M_irr can only increase, so at most 29% of a maximally spinning black hole's mass-energy can be extracted.

The Penrose process is the classical mechanism for extracting rotational energy from a black hole. Its quantum analog — superradiance (amplification of waves scattered off a rotating black hole) — is important for black hole stability analysis. The 29% efficiency limit corresponds to reducing the spin from a = GM/c² to a = 0.

Question 4

State the no-hair theorem and explain its physical significance.

Model answer: The no-hair theorem states that a stationary black hole in general relativity (coupled to electromagnetism) is completely characterized by three externally observable parameters: mass M, angular momentum J, and electric charge Q. All other information about the matter that formed the black hole — its composition, shape, multipole moments beyond mass and spin — is radiated away during formation and ringdown, leaving only these three 'hairs.' The physical significance is that black holes are the simplest macroscopic objects in the universe: a black hole of given M, J, Q is identical to every other black hole with the same parameters, regardless of formation history.

The no-hair theorem implies that the Kerr-Newman metric (the charged generalization of Kerr) is the unique stationary black hole solution. For astrophysical black holes, charge is negligible (quickly neutralized by surrounding plasma), so the Kerr metric with parameters M and J suffices. Testing the no-hair theorem — verifying that observed black holes are described by Kerr — is a major goal of gravitational wave astronomy and the Event Horizon Telescope.