Intuitive description of what the Ricci tensor & scalar represent?

[andrewkirk]

Is there a simple intuitive description of what the Ricci tensor and scalar represent?

I have what seems to me a straightforward understanding of what the Riemann tensor R^a_bcd represents, as follows. If you parallel transport a vector b around a tiny rectangle, the sides of which are determined by two other vectors c and d, the change in the transported vector when it arrives back at the start will have component in direction a given by the application of the Riemann tensor to vectors b, c, d and one-form a.

The Ricci tensor is a contraction of the Riemann tensor, and the Ricci scalar is a contraction of the Ricci tensor. However I can't think of a physical interpretation of these items that is similarly intuitive to the one above for the Riemann tensor.

Does anybody have such an interpretation?

 

[WannabeNewton]

This is something that is explained in pretty much every textbook on Riemannian geometry. It has also been asked on the forum numerous times in the past e.g. https://www.physicsforums.com/showthread.php?t=698693&highlight=ricci+tensor

 

[pervect]

Is there a simple intuitive description of what the Ricci tensor and scalar represent?

I have what seems to me a straightforward understanding of what the Riemann tensor R^a_bcd represents, as follows. If you parallel transport a vector b around a tiny rectangle, the sides of which are determined by two other vectors c and d, the change in the transported vector when it arrives back at the start will have component in direction a given by the application of the Riemann tensor to vectors b, c, d and one-form a.

The Ricci tensor is a contraction of the Riemann tensor, and the Ricci scalar is a contraction of the Ricci tensor. However I can't think of a physical interpretation of these items that is similarly intuitive to the one above for the Riemann tensor.

Does anybody have such an interpretation?

I'd suggest Baez's http://math.ucr.edu/home/baez/gr/outline2.html, the "long course". A specific line of interest is:

Well, first of all, it's nice because we have a simple geometrical way of understanding the Ricci tensor Rab in terms of convergence of geodesics. Remember, if v is the velocity vector of the particle in the middle of a little ball of initially comoving test particles in free fall, and the ball starts out having volume V, the second time derivative of the volume of the ball is

$-R_{ab} v^a v^b$ times V

but this won't make much sense until you read the backround.

"The Meaning of Einstein's equation" http://math.ucr.edu/home/baez/einstein/, by the same author, is a simpler look at some of the same material, but it doesn't mention as much details about the tensor names.

[add]
It might not be quite as intuitive, but articles on Raychaudhuri's equation, such as http://arxiv.org/abs/gr-qc/0611123, might be helpful too.

 

[nikols]

I'd suggest Baez's "The Meaning of Einstein's equation" [url]http://math.ucr.edu/home/baez/einstein/, by the same author, is a simpler look at some of the same material, but it doesn't mention as much details about the tensor names.

..

http://settheory.net/cosmology,
http://settheory.net/general-relativity

It's better :
- It is directly applied to an important example (universal expansion)
- The expression is simpler (relating 1 component of the energy tensor
to 3 components of the Riemann tensor)
- The relation between energy and curvature is not only expressed but
also justified
- Both (diagonal) space and time components of the relation are
expressed and justified, resulting in showing their similarity "like a
coincidence".

 

[haael]

Is there a simple intuitive description of what the Ricci tensor and scalar represent?

How much "intuitive" do you mean?

Imagine a thin rubber membrane. Now you select some point on it and you take 4 needles. You pin the needles in the neighbourhood of that point, so the entry points form a cross and your original point is in the center.

Now you use the needles to squeeze the membrane. Notice that in the original position the needles form a circle and the cross is aligned horizontally-vertically. You can do 3 things:
1. You can move the needles closer or further so the size of the circle is smaller or larger.
2. You can move the horizontal needles closer (further) and the vertical further (closer). The circle is deformed to an ellipse.
3. You can turn all needles (counter)clockwise around the central point. The cross formed by the needles is no longer aligned.

Now you can fetch the following information from your deformations:
A. If you consider the ellipse eccentricity (deviation from the circular shape) you get the Ricci tensor.
B. If you consider only the ellipse area (actually the ratio to the area of the undeformed circle) you get the Ricci scalar.
C. If you consider the angle of the cross relative to the horizontal-vertical grid you get the Weyl tensor.

Our membrane was 2-dimensional, so we needed 4 needles. In 3 dimensions we would need 6, in 4 we would need 8.

I couldn't make it any more "intuitive". I hope it helps.

 

[robphy]

In my opinion, interpretations in Riemannian geometry books don't necessarily provide satisfying physical interpretations (say, in terms of an experiment) in our Lorentz-signature spacetime.
Here's an old post of mine
https://www.physicsforums.com/showthread.php?t=648969
with links to some attempts at physical interpretation.

(I really want to understand the Conformal http://en.wikipedia.org/wiki/Weyl_tensor [and the http://en.wikipedia.org/wiki/Schouten_tensor ].)