.. _sphx_glr_api_gallery_relativity_spacetime_geometry:

Spacetime geometry
-------------------

Curvature, embedding diagrams, and the global causal structure of
Schwarzschild spacetime.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Einstein&#x27;s field equations, G_{\mu\nu} = 8\pi T_{\mu\nu}, reduce to G_{\mu\nu} = 0 in vacuum (no matter or energy present). Schwarzschild and Kerr describe the vacuum spacetime outside a mass, so their Einstein tensors must vanish identically everywhere outside the horizon -- despite the metrics themselves being far from flat. This example uses physicskit.relativity&#x27;s numerical (finite-difference) curvature engine to verify this directly for both metrics at several radii, and shows how the same engine reveals genuine matter content (a nonzero Einstein tensor) for a charged Reissner-Nordstrom black hole, whose electromagnetic field sources spacetime curvature even in the &quot;vacuum&quot; region outside the horizon.">

.. only:: html

  .. image:: /api/gallery/relativity/spacetime_geometry/images/thumb/sphx_glr_plot_curvature_engine_validation_thumb.png
    :alt:

  :doc:`/api/gallery/relativity/spacetime_geometry/plot_curvature_engine_validation`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Validating the curvature engine: Einstein's vacuum field equations</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="General Relativity&#x27;s central insight is that gravity is not a force propagating through space, but the curvature of spacetime itself -- massive objects don&#x27;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 (\theta=\pi/2) slice leaves a curved 2D spatial geometry with proper length element">

.. only:: html

  .. image:: /api/gallery/relativity/spacetime_geometry/images/thumb/sphx_glr_plot_flamm_paraboloid_thumb.png
    :alt:

  :doc:`/api/gallery/relativity/spacetime_geometry/plot_flamm_paraboloid`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Flamm's paraboloid: gravity as curved geometry, not a force</div>
    </div>


.. raw:: html

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

.. only:: html

  .. image:: /api/gallery/relativity/spacetime_geometry/images/thumb/sphx_glr_plot_kruskal_penrose_diagrams_thumb.png
    :alt:

  :doc:`/api/gallery/relativity/spacetime_geometry/plot_kruskal_penrose_diagrams`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Kruskal-Szekeres and Penrose-Carter diagrams: the true shape of spacetime</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/relativity/spacetime_geometry/plot_curvature_engine_validation
   /api/gallery/relativity/spacetime_geometry/plot_flamm_paraboloid
   /api/gallery/relativity/spacetime_geometry/plot_kruskal_penrose_diagrams