.. _sphx_glr_api_gallery_relativity_schwarzschild:

Schwarzschild black holes
--------------------------

Orbits, perihelion precession, light bending, the photon sphere, and a
radial infall's finite proper time to the horizon.


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


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

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    <div class="sphx-glr-thumbcontainer" tooltip="Every GPS satellite carries an atomic clock that runs measurably faster than an identical clock on the ground -- about 38 microseconds per day, a combination of gravitational blueshift (from sitting higher in Earth&#x27;s potential well) partially offset by special-relativistic time dilation (from orbital speed). A clock at radius r moving with angular velocity \Omega around a mass M (Earth, here) has a proper-to-coordinate time rate that combines both effects in one formula:">

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  .. image:: /api/gallery/relativity/schwarzschild/images/thumb/sphx_glr_plot_gps_relativistic_correction_thumb.png
    :alt:

  :doc:`/api/gallery/relativity/schwarzschild/plot_gps_relativistic_correction`

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      <div class="sphx-glr-thumbnail-title">GPS: General Relativity in your pocket</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Starlight grazing the Sun is deflected by \delta\phi \approx 4M/b -- twice the (wrong) Newtonian prediction -- confirmed by Eddington&#x27;s 1919 solar eclipse expedition and the observation that made Einstein a household name overnight. Push the impact parameter down toward the critical value b_c = 3\sqrt{3}M and deflection diverges: photons can orbit (unstably) forever at the photon sphere r=3M, and anything with b &lt; b_c is captured. This example traces individual light rays at a range of impact parameters, then renders the full 2D shadow silhouette by ray-tracing an entire camera image -- the calculation behind the 2019 Event Horizon Telescope image of M87*.">

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  .. image:: /api/gallery/relativity/schwarzschild/images/thumb/sphx_glr_plot_light_bending_and_shadow_thumb.png
    :alt:

  :doc:`/api/gallery/relativity/schwarzschild/plot_light_bending_and_shadow`

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      <div class="sphx-glr-thumbnail-title">Light bending, the photon sphere, and the black hole shadow</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Mercury&#x27;s orbit precesses by about 43 arcseconds per century more than Newtonian gravity (including the perturbations of the other planets) can account for -- a puzzle known since Le Verrier&#x27;s 1859 analysis. Einstein&#x27;s 1915 calculation of exactly this excess, from General Relativity alone with no free parameters, was his first observational triumph and a major reason the theory was taken seriously. Unlike a Newtonian ellipse, a bound Schwarzschild orbit of semi-major axis a and eccentricity e advances its periapsis by, at leading post-Newtonian order,">

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  .. image:: /api/gallery/relativity/schwarzschild/images/thumb/sphx_glr_plot_orbit_precession_thumb.png
    :alt:

  :doc:`/api/gallery/relativity/schwarzschild/plot_orbit_precession`

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      <div class="sphx-glr-thumbnail-title">Perihelion precession: the anomaly that first confirmed General Relativity</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Long before Penrose&#x27;s 1965 proof that trapped surfaces force geodesic incompleteness regardless of symmetry, a simpler, coordinate-dependent puzzle already hinted that something was wrong with treating Schwarzschild coordinate time t as a universal clock: a particle dropped from rest at radius R reaches the true singularity in finite proper time,">

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  .. image:: /api/gallery/relativity/schwarzschild/images/thumb/sphx_glr_plot_radial_infall_and_singularity_thumb.png
    :alt:

  :doc:`/api/gallery/relativity/schwarzschild/plot_radial_infall_and_singularity`

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      <div class="sphx-glr-thumbnail-title">Penrose's singularity theorems: finite proper time, divergent coordinate time</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="An object falling toward a black hole isn&#x27;t crushed from all sides -- it is stretched along the radial direction (the near side is pulled harder than the far side) and squeezed along the two transverse directions. In the local orthonormal frame of a freely-falling observer, two points separated by a small proper length \ell experience a relative (geodesic deviation) acceleration">

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  .. image:: /api/gallery/relativity/schwarzschild/images/thumb/sphx_glr_plot_tidal_forces_thumb.png
    :alt:

  :doc:`/api/gallery/relativity/schwarzschild/plot_tidal_forces`

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      <div class="sphx-glr-thumbnail-title">Spaghettification: tidal forces near a black hole</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/relativity/schwarzschild/plot_gps_relativistic_correction
   /api/gallery/relativity/schwarzschild/plot_light_bending_and_shadow
   /api/gallery/relativity/schwarzschild/plot_orbit_precession
   /api/gallery/relativity/schwarzschild/plot_radial_infall_and_singularity
   /api/gallery/relativity/schwarzschild/plot_tidal_forces