General Relativity – Comparing Schwarzschild Metric for Stars and Black Holes

Question

Background

The Schwarzschild metric can be used to describe the geometry of the vacuum spacetime outside a spherical massive object. For a star of radius $r$ (which is larger than the corresponding Schwarzschild radius $r_{S}$), we usually use the Schwarzschild metric to describe the spacetime geometry around the star for radial distances larger than $r$. (For reference see the top of page 287 in [1].) On the other hand, when we study the spacetime geometry around a black hole of radius $r_{S}$, we usually describe the spacetime geometry up until the interior singularity.

Question

My question is twofold. First, given a suitable mass density function could we extend the Schwarzschild metric into the interior of a star? Second, why is it that we describe the black hole until the interior singularity and not the star? Moreover, what is the difference in the metric between the center of a black hole and a center of a star?

Due to my lack of knowledge I understand that it is possible that my questions are ill posed. Hence, if that is the case, I would gladly welcome any clarifications.

References

[1] Bernard F. Schutz, A FIRST COURSE IN GENERAL RELATIVITY. Cambridge Univ. Pr., Cambridge, UK, 2009.

Answer 1

Second, why is it that we describe the black hole until the interior singularity and not the star?

Because it can be shown that if the entire mass of an object is inside its Schwarzschild radius then it will collapse to a singularity according to classical general relativity. See Tolman–Oppenheimer–Volkoff limit.

The Schwarzschild metric is only valid in the vacuum around a spherically symmetrical object. For a classical black hole, the Schwarzschild metric will work for any point apart from the singularity which contains the entire mass. Your confusion probably arose because you thought that there is mass everywhere inside the event horizon. But classically everything apart from the singularity is vacuum.

After including quantum gravity effects, things will become complex and are not well understood. Quantum gravity usually won't allow exact singularities. In string theory Fuzzball and in loop quantum gravity Planck star are some alternatives to exact singularities.