.. 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_kruskal_penrose_diagrams.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_kruskal_penrose_diagrams.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_relativity_spacetime_geometry_plot_kruskal_penrose_diagrams.py:


Kruskal-Szekeres and Penrose-Carter diagrams: the true shape of spacetime
================================================================================

Schwarzschild coordinates :math:`(t, r)` blow up at the horizon
:math:`r=2M` -- a coordinate artifact, since nothing physical happens there
(a free-falling observer notices nothing special crossing it). Kruskal and
Szekeres found coordinates :math:`(X, T)` that are perfectly smooth across
the horizon: in the exterior (:math:`r>2M`),

.. math::

    X = \sqrt{r/2M - 1}\, e^{r/4M} \cosh(t/4M), \qquad
    T = \sqrt{r/2M - 1}\, e^{r/4M} \sinh(t/4M)

(with :math:`\sinh` and :math:`\cosh` swapping roles inside the horizon),
so that light rays always travel at :math:`\pm 45^\circ` and the horizon
itself maps to the lines :math:`X=\pm T`. These coordinates reveal a
startling fact: the maximally extended spacetime contains *two* separate
asymptotically flat exterior regions, joined by a non-traversable
"Einstein-Rosen bridge." Compactifying further -- applying
:math:`\arctan` to the null combinations :math:`u=T-X`, :math:`v=T+X` --
gives the Penrose-Carter diagram, bringing the entire infinite spacetime,
including future and past null infinity, into one finite diagram while
preserving the :math:`\pm 45^\circ` light cones, making the whole causal
structure visible at a glance.

.. GENERATED FROM PYTHON SOURCE LINES 28-33

.. code-block:: Python


    import matplotlib.pyplot as plt

    from physicskit.relativity.visualizers.spacetime_diagrams import plot_kruskal_diagram, plot_penrose_diagram








.. GENERATED FROM PYTHON SOURCE LINES 34-36

Kruskal-Szekeres diagram
----------------------------

.. GENERATED FROM PYTHON SOURCE LINES 36-40

.. code-block:: Python

    fig, ax = plt.subplots(figsize=(6, 6))
    plot_kruskal_diagram(M=1.0, ax=ax)
    plt.tight_layout()




.. image-sg:: /api/gallery/relativity/spacetime_geometry/images/sphx_glr_plot_kruskal_penrose_diagrams_001.png
   :alt: Kruskal-Szekeres diagram
   :srcset: /api/gallery/relativity/spacetime_geometry/images/sphx_glr_plot_kruskal_penrose_diagrams_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 41-43

Penrose-Carter conformal diagram
----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 43-47

.. code-block:: Python

    fig, ax = plt.subplots(figsize=(6, 6))
    plot_penrose_diagram(M=1.0, ax=ax)
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/relativity/spacetime_geometry/images/sphx_glr_plot_kruskal_penrose_diagrams_002.png
   :alt: Penrose-Carter conformal diagram
   :srcset: /api/gallery/relativity/spacetime_geometry/images/sphx_glr_plot_kruskal_penrose_diagrams_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.072 seconds)


.. _sphx_glr_download_api_gallery_relativity_spacetime_geometry_plot_kruskal_penrose_diagrams.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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