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Does Schwarzschild Solution work inside a massive body?

  1. Sep 13, 2009 #1
    The Schwarzschild Solution to Einstein's Field Equations for gravity are said to be exact when outside a spherically symmetric massive body. My question is, can the Schwarzschild Solution also be used inside the massive body, such as a neutron star.

    In Newtonian gravity we can find the gravity potential at some position inside the body; just find the distance to the center of mass, and count the total mass enclosed by a sphere whose radius equals that distance. It is similar to Gauss' Law in electrostatics.

    Does a similar procedure work in general relativity? To find the gravity field at some position inside a neutron star, can I just calculate an effective Schwarzschild radius by finding the mass enclosed by a sphere passing through that position? If not, is there a correction to the Schwarzschild solution for the interior of a spherically symmetric massive body.
  2. jcsd
  3. Sep 13, 2009 #2


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    No, it doesn't work inside the body. (Huh, does it work inside in Newton, I'd forgotten!) Try the Tolman-Oppenheimer-Volkov (TOV) equations to start.
  4. Sep 13, 2009 #3
    Thank you for the reference to the TOV equation, it is most helpful. Is there a similar equation if the body has some radial flow, for example a dense star that undergoes a rapid expansion such as during a supernova?
  5. Sep 13, 2009 #4


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    For constant density, there is also the interior Schwarzschild Solution:
  6. Sep 13, 2009 #5


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    I don't know exactly, but maybe you can find a useful reference here:

    Quasi-Normal Modes of Stars and Black Holes
    Kostas D. Kokkotas, Bernd G. Schmidt
    http://relativity.livingreviews.org/Articles/lrr-1999-2/ [Broken]

    Rotating Stars in Relativity
    Nikolaos Stergioulas
    http://relativity.livingreviews.org/Articles/lrr-2003-3/ [Broken]
    Last edited by a moderator: May 4, 2017
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