The Penrose process: extracting a black hole’s rotational energy

Roger Penrose realized in 1969 that the ergosphere of a rotating (Kerr) black hole – see Frame dragging and the ergosphere of a Kerr black hole – is not just a region where no observer can sit still, but a genuine energy resource. Inside it, the t-Killing vector associated with time-translation symmetry becomes spacelike, so a particle there can have negative energy as measured by an observer at infinity. If a particle entering the ergosphere splits in two, with one negative-energy fragment falling into the horizon, energy conservation forces the escaping fragment to carry away more energy than the original particle had – extracting rotational energy from the black hole itself. The process is capped by Hawking’s area theorem: the hole’s irreducible mass can never decrease, limiting the maximum extractable fraction to

\[\eta_{\max} = 1 - \sqrt{\frac{r_+}{2M}},\]

which rises from 0 at \(a=0\) to \(1 - 1/\sqrt{2} \approx 29.3\%\) for a maximally (extremal) spinning hole.

import matplotlib.pyplot as plt
import numpy as np

from physicskit.relativity.chapters.kerr import KerrBlackHole

Maximum extractable energy fraction vs. spin

a_values = np.linspace(0.0, 0.9999, 100)
efficiency = [KerrBlackHole(M=1.0, a=a).max_penrose_efficiency() for a in a_values]

plt.figure(figsize=(6, 4))
plt.plot(a_values, np.array(efficiency) * 100.0)
plt.axhline(
    (1.0 - 1.0 / np.sqrt(2.0)) * 100.0,
    color="k",
    linestyle="--",
    linewidth=1,
    label="extremal limit, 29.3%",
)
plt.xlabel("spin a/M")
plt.ylabel("max extractable energy (% of M)")
plt.title("Penrose process efficiency limit")
plt.legend()
plt.tight_layout()
plt.show()

A single split: the escaping fragment gains energy

bh = KerrBlackHole(M=1.0, a=0.9)
e_out = bh.penrose_energy_gain(initial_energy=1.0, fragment_energy_infalling=-0.1)
print("Particle falls in with E=1.0, splits inside the ergosphere;")
print("one fragment falls in with E=-0.1 (negative energy, only possible there);")
print(f"the escaping fragment carries away E={e_out:.3f} -- more than it started with.")

Particle falls in with E=1.0, splits inside the ergosphere;
one fragment falls in with E=-0.1 (negative energy, only possible there);
the escaping fragment carries away E=1.100 -- more than it started with.