Confused about curvature of vacuum solutions

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Discussion Overview

The discussion revolves around the properties of vacuum solutions in General Relativity (GR), specifically focusing on the Ricci tensor and scalar curvature. Participants explore the implications of these properties and address issues encountered while using the Ctensor package in Maxima for calculations related to the Schwarzschild metric.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Experimental/applied

Main Points Raised

  • Some participants assert that a vacuum solution is characterized by a vanishing Einstein tensor, which implies a vanishing Ricci tensor.
  • Others clarify that while the Ricci tensor is zero for a vacuum solution, the Riemann curvature tensor does not vanish, indicating that curvature can still exist.
  • A participant expresses confusion regarding the relationship between scalar curvature and flatness, questioning if zero scalar curvature implies a flat space.
  • Concerns are raised about the Ctensor package reporting non-zero elements in the Ricci tensor when using the Schwarzschild metric, leading to uncertainty about the software's accuracy.
  • Participants discuss the potential influence of a cosmological constant on the Ricci tensor's behavior in vacuum solutions.
  • One participant mentions that their issues with Ctensor were resolved through private messages, attributing the problem to user error rather than a software bug.
  • Another participant shares their long-term positive experience with Ctensor, indicating confidence in its reliability.

Areas of Agreement / Disagreement

Participants generally agree that the Ricci tensor is zero for vacuum solutions, but there is some uncertainty regarding the implications of this for scalar curvature and the presence of curvature in general. The discussion about the Ctensor package reveals differing experiences and interpretations, leading to some unresolved questions about its functionality.

Contextual Notes

Limitations include the potential influence of a cosmological constant on the Ricci tensor, as well as the specific conditions under which the Ctensor package operates. The discussion does not resolve whether the reported non-zero elements in the Ricci tensor are due to software issues or user error.

Who May Find This Useful

This discussion may be useful for students and practitioners of General Relativity, particularly those using computational tools like Ctensor for their research or studies.

m4r35n357
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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.
 
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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.
 
Matterwave said:
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?
 
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.
 
Matterwave said:
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.
 
m4r35n357 said:
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.
 
Mentz114 said:
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!
 

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