Components of the Ricci Tensor

Question

Is there any interpretation of what each of the components of the Ricci tensor corresponds to?

For example, for the stress-energy tensor, $T_{00}$ corresponds to energy density, $T_{0i}$ is the momentum flux in the $i$ direction, etc. Is there something similar for the Ricci tensor?

try http://en.wikipedia.org/wiki/Ricci_tensor#Direct_geometric_meaning — , Jul 27 '13 at 01:41
Possible duplicate: http://physics.stackexchange.com/q/21065/2451 — Qmechanic, Jul 27 '13 at 19:45

score 0 · Accepted Answer · answered Jul 27 '13 at 18:23

It is not totally linked to your question, but there is a interesting point of view .

(Ref : T. Padmanabhan, Gravitation -Fundations and Frontiers, Cambridge, p.259)

If you consider a observer with $4$- velocity $u^i$, then, from Einstein equations, you have :

$$R_{ik}u^iu^k \sim T_{ik}u^iu^k$$

The first term $R_{ik}u^iu^k$ is the scalar curvature of the spatial sections as measured by the observer.

The second term $T_{ik}u^iu^k$, is the energy density as measured by the observer.

So, instead of looking at $T_{ik}$ and $R_{ik}$ alone, we could see them as part of invariants, these invariants being different if they imply different observers with different $4$-velocity.

Not quite what I was looking for, but it helped me understand several other things! So +1 for that! Thanks a lot. — Prahar, Jul 27 '13 at 22:24

Components of the Ricci Tensor

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