How can vacuum solutions have curvature if the Ricci tensor vanishes?

[m4r35n357]

When I first started learning about GR, I understood that a vacuum solution is one where the Einstein tensor vanishes, for the simple reason that the stress-energy tensor, T, vanishes. I have since read many times that the Ricci tensor vanishes for a vacuum solution. I am confused because to me this means that the scalar curvature is zero, and if there is no curvature, surely the space is flat?
I would appreciate it if someone could put me straight on this simple (I hope) misunderstanding.

 

[Matterwave]

The scalar curvature is simply one of many measures of curvature. The most general measure is the Reimann curvature tensor which does not vanish for a vacuum solution.

 

[m4r35n357]

The scalar curvature is simply one of many measures of curvature. The most general measure is the Reimann curvature tensor which does not vanish for a vacuum solution.

OK, so are you definitely and unequivocally saying that the Ricci tensor is zero for a vacuum solution?

 

[Matterwave]

Yes. The Ricci tensor, just like the Einstein tensor, is zero for a vacuum solution. With the one potential caveat that this may not hold if there is a cosmological constant.

The Reimann and Weyl curvature tensors are the non-zero curvature tensors in vacuum.

 

[m4r35n357]

Yes. The Ricci tensor, just like the Einstein tensor, is zero for a vacuum solution. With the one potential caveat that this may not hold if there is a cosmological constant.

OK, thanks for replying, sorry to labour the point, but I needed to hear that!
Right, the reason for my question is that I am having problems with Maxima and the Ctensor package, which is my "laboratory" for General Relativity. For some reason Ctensor is giving two non-zero elements (1,2 and 2,2) of the Ricci tensor using its built-in exterior Schwarzschild metric. I also have my own simpler "version" of the metric, but this fares even worse, with five non-zero Ricci elements, and four non-zero Einstein elements. Bearing in mind how simple the Schwarzschild metric is, I am very troubled by these results, and am wondering whether I have a buggy copy of Ctensor.
So, if anyone else here uses Ctensor, I would really appreciate it if you could confirm or refute my findings (using the built-in metric to start with). If you find that Ctensor is reporting a zero Ricci tensor, I can provide snippets that reproduce the problems that I am having.

 

[Mentz114]

OK, thanks for replying, sorry to labour the point, but I needed to hear that!
Right, the reason for my question is that I am having problems with Maxima and the Ctensor package, which is my "laboratory" for General Relativity. For some reason Ctensor is giving two non-zero elements (1,2 and 2,2) of the Ricci tensor using its built-in exterior Schwarzschild metric. I also have my own simpler "version" of the metric, but this fares even worse, with five non-zero Ricci elements, and four non-zero Einstein elements. Bearing in mind how simple the Schwarzschild metric is, I am very troubled by these results, and am wondering whether I have a buggy copy of Ctensor.
So, if anyone else here uses Ctensor, I would really appreciate it if you could confirm or refute my findings (using the built-in metric to start with). If you find that Ctensor is reporting a zero Ricci tensor, I can provide snippets that reproduce the problems that I am having.

I have to correct this slight on ctensor() and Maxima. The problem has been sorted out in PMs between me and the OP. It was finger trouble.
I've used ctensor() for years and with many different spacetimes and never detected an error.

 

[m4r35n357]

I have to correct this slight on ctensor() and Maxima. The problem has been sorted out in PMs between me and the OP. It was finger trouble.
I've used ctensor() for years and with many different spacetimes and never detected an error.

Well it was only a slight slight, I hope ;) I really have found ctensor indispensable and was actually trying hard to put the blame on myself where it ultimately belonged. So yeah, entirely self-inflicted, pardon me everyone, and thanks to Mentz114 for his help!