r"""
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 :doc:`plot_ergosphere_and_penrose` -- 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

.. math::

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

which rises from 0 at :math:`a=0` to :math:`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.")