.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/relativity/kerr/plot_penrose_energy_extraction.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_relativity_kerr_plot_penrose_energy_extraction.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_relativity_kerr_plot_penrose_energy_extraction.py:


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.

.. GENERATED FROM PYTHON SOURCE LINES 25-31

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.relativity.chapters.kerr import KerrBlackHole








.. GENERATED FROM PYTHON SOURCE LINES 32-34

Maximum extractable energy fraction vs. spin
--------------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 34-53

.. code-block:: Python

    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()




.. image-sg:: /api/gallery/relativity/kerr/images/sphx_glr_plot_penrose_energy_extraction_001.png
   :alt: Penrose process efficiency limit
   :srcset: /api/gallery/relativity/kerr/images/sphx_glr_plot_penrose_energy_extraction_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 54-56

A single split: the escaping fragment gains energy
--------------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 56-61

.. code-block:: Python

    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.")




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    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.





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.031 seconds)


.. _sphx_glr_download_api_gallery_relativity_kerr_plot_penrose_energy_extraction.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_penrose_energy_extraction.ipynb <plot_penrose_energy_extraction.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_penrose_energy_extraction.py <plot_penrose_energy_extraction.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_penrose_energy_extraction.zip <plot_penrose_energy_extraction.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_