# Mass in Schwarzschild's metric

1. Aug 16, 2014

### 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 $R_{0}$($r>R_{0},~~ M_0 \equiv M(R_{0})$) ?

Last edited: Aug 16, 2014
2. Aug 16, 2014

### Markus Hanke

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

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

3. Aug 16, 2014

### 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?

4. Aug 16, 2014

### Staff: Mentor

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