Part II: Rotating & Charged Black Holes
Astrophysical black holes rotate—often rapidly. The Kerr solution describes rotating black holes and exhibits phenomena absent in Schwarzschild: frame dragging, the ergosphere, and extractable rotational energy. Charged black holes, though rare astrophysically, reveal deep theoretical structure.
Part Overview
The Kerr metric, discovered by Roy Kerr in 1963, is one of the most important exact solutions in GR. It describes a rotating black hole characterized by mass and angular momentum . The ergosphere is a region where spacetime is dragged so strongly that nothing can remain at rest. The Penrose process allows extraction of rotational energy. The no-hair theorem states that stationary black holes are uniquely determined by mass, charge, and angular momentum.
Key Topics
- • Kerr metric in Boyer-Lindquist coordinates
- • Ergosphere: region of inevitable frame dragging
- • Penrose process: extracting rotational energy from Kerr BHs
- • Reissner-Nordström solution: electrically charged black holes
- • Kerr-Newman solution: rotating and charged
- • No-hair theorem: uniqueness of stationary black holes
6 chapters | Realistic black holes | Rotation and charge
Chapters
Chapter 1: Kerr Metric
The Kerr solution in Boyer-Lindquist coordinates. Rotation parameter . Horizons: outer and inner . Ring singularity at , . Dragging of inertial frames: the Lense-Thirring effect. Extremal Kerr: . Geodesics are significantly more complex than Schwarzschild.
Chapter 2: Ergosphere and Frame Dragging
The ergosurface is at . Between the ergosurface and outer horizon is the ergosphere. Inside, the Killing vector becomes spacelike—no observer can remain stationary. All particles and photons are dragged in the direction of rotation. Experimentally verified via Gravity Probe B satellite.
Chapter 3: Penrose Process
A particle entering the ergosphere can split into two, with one falling into the horizon carrying negative energy (in the rotating frame), while the other escapes with more energy than the original. This extracts rotational energy from the black hole, spinning it down. Maximum extractable energy: 29% of for extremal Kerr. The Blandford-Znajek mechanism may power astrophysical jets via a variant of this process.
Chapter 4: Reissner-Nordström Black Holes
Electrically charged, non-rotating black holes. The metric includes electromagnetic energy: . Two horizons: . Extremal case yields a single degenerate horizon. Naked singularities violate cosmic censorship (if they exist).
Chapter 5: Kerr-Newman Solution
The most general stationary black hole: rotating AND charged. Characterized by mass , angular momentum , and charge . The metric generalizes both Kerr (Q=0) and Reissner-Nordström (a=0). Ring singularity, inner and outer horizons, ergosphere. Astrophysically, black holes are nearly neutral () due to plasma accretion neutralizing charge.
Chapter 6: No-Hair Theorem
"Black holes have no hair"—all information about the matter that formed them is lost except mass, charge, and angular momentum. Stationary black hole solutions are uniquely determined by . Multipole moments are determined by these three parameters. All other "hairs" (magnetic fields, scalar fields, etc.) either radiate away or fall in during collapse. Implications for information loss and quantum gravity.