# Interior metric solution

1. Mar 13, 2010

### Orion1

$$g_{tt} = \left\{ \begin{array}{rcl} \frac{3}{2} \left( 1 - \frac{2GM(r)}{c^2 R} \right)^{\frac{1}{2}} - \frac{1}{2} \left( 1 - \frac{2 G M(r) r^2}{c^2 R^3} \right)^{\frac{1}{2}} \; \; \text{for} \; \; 0 \leq r \leq R \; \text{(interior)} \\ \left( 1 - \frac{2GM(r)}{c^2 R} \right) \; \; \text{for} \; \; r > R \; \text{(Schwarzchild)} \\ \end{array} \right.$$

$$g_{rr} = \left\{ \begin{array}{rcl} \left( 1 - \frac{2G}{c^2 r} \frac{4 \pi r^3}{3} \rho_0 \right)^{-1} \; \; \text{for} \; \; 0 \leq r \leq R \; \text{(interior)} \\ \left( 1 - \frac{2G M(r)}{c^2 r} \right)^{-1} \; \; \text{for} \; \; r > R \; \text{(Schwarzchild)} \\ \end{array} \right.$$

My question is theoretical, why would the relativistic Equation of State for hydrostatic equilibrium be based on the exterior metric as opposed to the interior metric?

What are the formal equation definitions for $$g_{\theta \theta}$$ and $$g_{\phi \phi}$$ for the interior metric?

Reference:
http://www.infn.it/thesis/PDF/getfile.php?filename=3852-Mana-specialistica.pdf"
http://en.wikipedia.org/wiki/Birkhoff%27s_theorem_%28relativity%29" [Broken]
http://en.wikipedia.org/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_equation" [Broken]

Last edited by a moderator: May 4, 2017