Interior metric solution

[Orion1]

Recalling that Birkhoff's theorem guarantees that the exterior spacetime will be the Schwarzschild one, we easily deduce that the metric functions will be given by:

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

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

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"
http://en.wikipedia.org/wiki/Tolman%E2%80%93Oppenheimer%E2%80%93Volkoff_equation"

 

[AndyW]

Thank you for your question. it is important to understand the theoretical basis for any equation or theorem we use in our research. In this case, Birkhoff's theorem states that in a spherically symmetric and static spacetime, the exterior solution must be the Schwarzschild solution. This means that the metric functions for the exterior spacetime will be given by the Schwarzschild solution, which is the second line in the equations you provided.

The reason for this is that the exterior solution is determined by the mass distribution outside of the source, while the interior solution is determined by the mass distribution inside the source. Birkhoff's theorem guarantees that the exterior solution will be the same as the Schwarzschild solution, regardless of the mass distribution inside the source. Therefore, the exterior metric functions will be based on the Schwarzschild solution.

Now, as for the formal definitions of g_{\theta \theta} and g_{\phi \phi} for the interior metric, these can be derived from the Tolman-Oppenheimer-Volkoff (TOV) equation, which describes the hydrostatic equilibrium of a spherical, static source of matter in general relativity. This equation relates the pressure and density of the matter inside the source to the metric functions. The formal equations for g_{\theta \theta} and g_{\phi \phi} can be obtained by solving the TOV equation for these metric functions.

I hope this answers your question and provides some clarity on the theoretical basis for the equations you mentioned.