Mass in Schwarzschild's Metric: Exploring Vacuum Solutions

[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.

 

[sardar]

The mass M_0 in Schwarzschild's metric corresponds to the mass of the central body, such as a star or a black hole. It is a vacuum solution because it describes the curvature of spacetime in the regions outside of the central body, where there is no matter or energy present. This means that the metric is valid in the vacuum outside of the star or black hole, but it does not account for the mass of the star itself. The vacuum solution is important because it allows us to understand the gravitational effects of the central body on its surroundings, without having to worry about the details of its internal structure. So, while the mass M_0 in the metric may correspond to the mass of the star, the metric itself is describing the vacuum outside of the star.