What does the mass M_0 in Schwarzschild's metric represent in a vacuum solution?

[ChrisVer]

I think this will be a quick question...
If the Schw's metric is a solution of the vacuum, then what does the mass M_0 in the metric correspond to? I thought it was the mass of the star... but if that's true then why is it a vacuum solution?
Or is it vacuum because it describes the regions outside the star of radius R_{0}(r>R_{0},~~ M_0 \equiv M(R_{0})) ?

 

[Markus Hanke]

If the Schw's metric is a solution of the vacuum, then what does the mass M_0 in the metric correspond to?

For the exterior Schwarzschild metric, the parameter M would technically signify the total mass-energy content of the entire space-time.

but if that's true then why is it a vacuum solution?

Because it describes the exterior vacuum region of the mass-energy distribution, i.e. the region where the energy-momentum tensor vanishes everywhere.

 

[ChrisVer]

So would it be wrong to try and describe the interior of the star with a function M(r) instead of M_{0} the total mass?

 

[Nugatory]

So would it be wrong to try and describe the interior of the star with a function M(r) instead of M_{0} the total mass?

Not at all, but the metric won't be the Schwarzschild metric in the interior region. It will, however, have to join smoothly to the Schwarzchilld metric for $M_0$ at the surface.