How is the interior Schwarzschild metric derived?

Question

Where does the interior Schwarzschild metric come from? How is it derived and why does it have NOT a singularity? Would it mean that the singularity is only apparent and for those out of the black hole (who are ruled by the usual, exterior metric)?

Answer 1

Why would it have a singularity if it only applies to homogenous spheres at least 9/8 larger than a black hole, the earth also doesn't have a singularity at its center. The original work with the derivation can be found here (it is in german though, but the equations are self explaining and for the text you can use a translator).

Answer 2

In this context, "interior" does not mean inside the event horizon of a black hole. Rather, one imagines computing the metric for a spacetime which features the presence of a star, modeled as a spherically-symmetric fluid. The exterior solution corresponds to the (vacuum) region outside of the star, and the interior solution corresponds to the (non-vacuum) region within the star itself.

If the entire mass of the star is compressed to a radius less than the Schwarzschild radius $R_s = 2GM/c^2$, then the star undergoes gravitational collapse and a singularity forms. However, this radius is far smaller than the radius of a typical star (the Schwarzschild radius of the sun is $R_s \sim 10$ km while its actual radius is $R_{\odot }\sim 10^5$ km), so no collapse occurs and no singularity exists.

As a non-relativistic analogy, note that if you compute the gravitational field strength of a spherical mass distribution with uniform density, total mass $M$, and radius $R$ in a Newtonian framework, you obtain

$$\mathbf g(r) = \begin{cases} \frac{GM}{R^2} \frac{r}{R} \hat r & r< R \\ \frac{GM}{r^2}\hat r & r\geq R\end{cases}$$

Outside of the star, the gravitational field strength goes like $\sim 1/r^2$ and therefore increases with decreasing $r$. However, once you get inside the star, the gravitational field strength decreases smoothly back to $0$ at the origin and there is no singularity.