# Ricci tensor of the orthogonal space

1. May 27, 2013

### PLuz

While reading this article I got stuck with Eq.$(54)$. 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, $u^a$. 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

$Note:$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.

2. May 28, 2013

### 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.

3. May 28, 2013

### George Jones

Staff Emeritus
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,