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[tex]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.[/tex] ref. 1 - pg. 17 said:Recalling that Birkhoff's theorem guarantees that the exterior spacetime will be the Schwarzchild one, we easily deduce that the metric functions will be given by:

[tex]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.[/tex]

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 [tex]g_{\theta \theta}[/tex] and [tex]g_{\phi \phi}[/tex] 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]

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# Interior metric solution

Can you offer guidance or do you also need help?

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