- #1

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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]

Please examine the derivation from General Relativity in reference 3.

My question is theoretical, why would the relativistic Equation of State for hydrostatic equilibrium which is the Tolman-Oppenheimer-Volkoff (TOV) equation, 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–Oppenheimer–Volkoff_equation#Derivation_from_General_Relativity"

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