A spinning (Kerr) black hole drags spacetime around with it. Outside the event horizon there is a region — the ergosphere — where nothing can stay at rest relative to a distant observer, yet particles can still escape. Inside it, a particle's orbital energy as measured at infinity can be negative.
Penrose's 1969 trick: drop a particle of energy E₀ into the ergosphere and split it into two fragments. If fragment 2 is placed on a negative-energy orbit (E₂ < 0) and falls through the horizon, energy conservation forces fragment 3 to escape with more energy than the original particle carried in:
E₀ = E₂ + E₃, E₂ < 0 ⟹ E₃ > E₀
efficiency η = (E₃ − E₀)/E₀ = −E₂/E₀
r₊ = M + √(M² − a²) event horizon
r_ergo(θ) = M + √(M² − a²cos²θ) ergosphere boundary
η_max(a*) = 1 − (1/√2)·√(1 + √(1 − a*²))
η_max grows from 0 at a* = 0 (a non-rotating hole has no ergosphere — no Penrose process is possible) to ≈ 29.3% for an extremal hole (a* → 1), and is largest for particles that enter near the equator, where the ergosphere is widest. The infalling fragment's negative energy is subtracted from the black hole itself: each successful extraction shaves a little mass and, in this simplified model, a little spin off the hole — real rotational energy leaves as escaping kinetic energy, exactly mirroring the physical process (the same reservoir tapped, at far higher efficiency, by the Blandford–Znajek mechanism thought to power relativistic jets from real accreting black holes).
- Spin a* — sets how fast the hole rotates (0 = Schwarzschild, up to 0.998 = near-extremal); the ergosphere and maximum efficiency both grow with it.
- Entry angle — how close to the equatorial plane the particle enters the ergosphere; the effect scales as sin²θ and vanishes at the poles.
- Split efficiency — how close the chosen split trajectory comes to the theoretical maximum for that spin and angle.
- Rest-energy — the mass-energy of the particle you drop in, in solar masses.
This is a physically-grounded but simplified visual model (the spin-down rule is illustrative, not a full geodesic integration) — it is exact for the two textbook results it reproduces: the E₀ = E₂ + E₃ energy budget and the η_max(a*) formula above.