.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/relativity/spacetime_geometry/plot_flamm_paraboloid.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_spacetime_geometry_plot_flamm_paraboloid.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_relativity_spacetime_geometry_plot_flamm_paraboloid.py:


Flamm's paraboloid: gravity as curved geometry, not a force
==================================================================

General Relativity's central insight is that gravity is not a force
propagating through space, but the curvature of spacetime itself -- massive
objects don't pull on other objects, they bend the geometry that other
objects move through in a straight line (a geodesic). Restricting the
Schwarzschild metric to a constant-time, equatorial (:math:`\theta=\pi/2`)
slice leaves a curved 2D spatial geometry with proper length element

.. math::

    d\ell^2 = \frac{dr^2}{1 - 2M/r} + r^2 d\phi^2

Flamm's paraboloid makes this curvature literal: embedding this 2-surface
as a surface of revolution :math:`z(r)` in ordinary flat 3D Euclidean space,

.. math::

    z(r) = 2\sqrt{2M(r - 2M)}, \qquad r \ge 2M

reproduces exactly the same proper distances as the curved metric above, so
that walking along the resulting funnel-shaped surface covers the same
proper distance as walking through the real curved space around the black
hole -- for masses :math:`M` of increasing size (and correspondingly larger
horizons :math:`r=2M`).

.. GENERATED FROM PYTHON SOURCE LINES 29-35

.. code-block:: Python


    import matplotlib.pyplot as plt

    from physicskit.relativity.chapters.schwarzschild import SchwarzschildBlackHole
    from physicskit.relativity.visualizers.spacetime_3d import plot_flamm_paraboloid








.. GENERATED FROM PYTHON SOURCE LINES 36-38

The embedding surface for black holes of increasing mass
------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 38-46

.. code-block:: Python

    fig = plt.figure(figsize=(13, 4.5))
    for i, M in enumerate([0.5, 1.0, 2.0]):
        ax = fig.add_subplot(1, 3, i + 1, projection="3d")
        plot_flamm_paraboloid(M, ax=ax, r_max=15.0)
        bh = SchwarzschildBlackHole(M=M)
        ax.set_title(f"M={M} (horizon at r={bh.horizon_radius}M)")
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/relativity/spacetime_geometry/images/sphx_glr_plot_flamm_paraboloid_001.png
   :alt: M=0.5 (horizon at r=1.0M), M=1.0 (horizon at r=2.0M), M=2.0 (horizon at r=4.0M)
   :srcset: /api/gallery/relativity/spacetime_geometry/images/sphx_glr_plot_flamm_paraboloid_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.168 seconds)


.. _sphx_glr_download_api_gallery_relativity_spacetime_geometry_plot_flamm_paraboloid.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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