Ricci tensor for Schwarzschild metric

[AleksanderPhy]

Hello I am little bit confused about calculating Ricci tensor for schwarzschild metric:
So we have Ricci flow equation,∂_tg_μν=-2R_μν.
And we have metric tensor for schwarzschild metric:
Diag((1-r_s/r),(1-r_s]/r)^-1,(r²),(sin²Θ) and ∂_tg_μν=0 so 0=-2R_μν and we get that R_μν=0.But R_μν should not equal to zero for schwarzschild metric. I may have some mistakes on symbols beacuse I use them first time.

 

[Orodruin]

The Ricci flow is not the same as the Einstein field equations. The Schwarzschild metric does not undergo Ricci flow.

 

[PeterDonis]

Rμν should not equal to zero for schwarzschild metric.

Yes, it should. The Schwarzschild metric is a vacuum metric, so its Einstein tensor is zero; and it's simple to show that if the Einstein tensor is zero, the Ricci tensor must also be zero.

 

[MattRob]

Yes, it should. The Schwarzschild metric is a vacuum metric, so its Einstein tensor is zero; and it's simple to show that if the Einstein tensor is zero, the Ricci tensor must also be zero.

The Schwarzschild solution is a vacuum solution?

I'm kind of surprised I haven't come across this yet. What does "m" represent in the metric, then?

 

[Orodruin]

The Schwarzschild solution is a vacuum solution?

I'm kind of surprised I haven't come across this yet. What does "m" represent in the metric, then?

If you compute the energy momentum tensor anywhere, you will get zero. So yes, it is a vacuum solution. The mass m is a parameter of a set of possible vacuum solutions.

Note that this is the exterior Schwarzschild solution - a solution outside a spherically symmetric object where there is no matter so you should not be surprised to find that it is a vacuum solition. The interior Schwarzschild solution is not a vacuum solution.

 

[MattRob]

If you compute the energy momentum tensor anywhere, you will get zero. So yes, it is a vacuum solution. The mass m is a parameter of a set of possible vacuum solutions.

Note that this is the exterior Schwarzschild solution - a solution outside a spherically symmetric object where there is no matter so you should not be surprised to find that it is a vacuum solition. The interior Schwarzschild solution is not a vacuum solution.

First time I've heard "exterior" and "interior" Schwarzschild solution. Is this simply referring to regions where r > 2M or r < 2M (in geometric units)?

 

[Orodruin]

First time I've heard "exterior" and "interior" Schwarzschild solution. Is this simply referring to regions where r > 2M or r < 2M (in geometric units)?

No. The interior Schwarzschild solution is a solution inside a spherical mass distribution with zero pressure at the surface. It does not describe a black hole.

Edit: see https://en.wikipedia.org/wiki/Interior_Schwarzschild_metric

Even in Newtonian gravity, the 1/r potential is a vacuum solution. It only describes the potential outside a spherically symmetric distribution.

 

[MattRob]

No. The interior Schwarzschild solution is a solution inside a spherical mass distribution with zero pressure at the surface. It does not describe a black hole.

Edit: see https://en.wikipedia.org/wiki/Interior_Schwarzschild_metric

Even in Newtonian gravity, the 1/r potential is a vacuum solution. It only describes the potential outside a spherically symmetric distribution.

So I decided to look this up. It all fell into place pretty quickly. Saw your edit now - heh, we both immediately turned to the same place.

But yeah, the difference in-between assuming you're outside a spherically symmetric mass or within a sphere of it - I think I get it, now.

This will be something great to look into, though - I'm only an undergrad, now, but I'm working with some faculty at my university and some of the first things I'm interested in really digging into have to do with constant density zero-pressure fluid solutions. Isn't this just a way of describing a uniform energy density in a region of space, though? I thought that's what "dust" was used to describe; a uniform energy density distribution in a region of space (in a static solution)?