Ricci tensor of the orthogonal space

[PLuz]

While reading this article I got stuck with Eq.[itex](54)[/itex]. I've been trying to derive it but I can't get their result. I believe my problem is in understanding their hints. They say that they get the result from the Gauss embedding equation and the Ricci identities for the 4-velocity, [itex]u^a[/itex]. Is the Gauss equation they refer the one in the wiki article?

Looking at the terms that appear in their equation it looks like the Raychaudhuri equation is to be used in the derivation in order to get the density and the cosmological constant, but even though I realize this I can't really get their result.

Can anyone point me in the right direction?

Thank you very much

[itex]Note:[/itex]The reason why I'm trying so hard to prove their result is because I wanted to know if it would still be valid if the orthogonal space were 2 dimensional (aside some constants). It appears to be the case but to be sure I needed to be able to prove it.

 

[Bill_K]

Yes, the Gauss equation that they're referring to is the same Gauss equation mentioned in the Wikipedia article, relating the Riemann tensor of a surface to its second fundamental form. The second fundamental form, in turn, describes the embedding of the surface and can be expressed in terms of the kinematics of the normal congruence.

If you haven't already, I suggest you look up the cited articles, refs 5 and 6 by Ehlers and Ellis, where this relationship is proved.

 

[George Jones]

If you haven't already, I suggest you look up the cited articles, refs 5 and 6 by Ehlers and Ellis, where this relationship is proved.

I agree with Bill. This kind of "legwork" should be almost second nature. Another place to look is section 6.3 "The other Einstein field equations" in the new book "Relativistic Cosmology" by Ellis, Maartens, and MacCallum,

http://www.physicstoday.org/resource/1/phtoad/v66/i4/p54_s1

 

[PLuz]

I just saw my error. I was forgetting some terms and in the end the expressions were obviously different. Thank you for the references, I had taken a quick look at them but, clearly, I had to read them with more attention...

 

[sardar]

First of all, it's great that you are trying to understand and derive the result in the article. As a scientist, it is important to have a deep understanding of the concepts and equations we use in our work.

To answer your question, the Gauss embedding equation mentioned in the article is most likely the one from the Wikipedia article on Ricci curvature (https://en.wikipedia.org/wiki/Ricci_curvature). This equation relates the Ricci tensor of a manifold to its curvature tensor. It is a fundamental equation in differential geometry and plays a key role in the study of curved spaces.

In order to derive the result in Eq.(54), you will need to use the Gauss embedding equation and the Ricci identities for the 4-velocity, u^a. These identities relate the curvature tensors to the derivatives of the 4-velocity. The Raychaudhuri equation is also important in this derivation as it relates the expansion and shear of a congruence of geodesics to the Ricci tensor.

If you are having trouble understanding the hints in the article, it might be helpful to consult some textbooks on differential geometry or general relativity. They will provide a more detailed explanation of the concepts and equations involved in the derivation.

It is also worth mentioning that the result in Eq.(54) is valid for any dimension, not just in 4 dimensions. However, the specific values of the constants may differ depending on the dimension of the orthogonal space.

In conclusion, keep working on understanding and deriving the result in Eq.(54). It may take some time and effort, but once you have a solid understanding of the concepts and equations involved, you will be able to confidently use and apply them in your work. Good luck!