7
$\begingroup$

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?

$\endgroup$
2

1 Answer 1

0
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ Not quite what I was looking for, but it helped me understand several other things! So +1 for that! Thanks a lot. $\endgroup$
    – Prahar
    Jul 27, 2013 at 22:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.