# Does Schwarzschild Solution work inside a massive body?

1. Sep 13, 2009

### nickyrtr

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. Sep 13, 2009

### atyy

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.

3. Sep 13, 2009

### nickyrtr

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?

4. Sep 13, 2009

### A.T.

For constant density, there is also the interior Schwarzschild Solution:

5. Sep 13, 2009

### atyy

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