How does Ricci curvature represent volume deficit ?

[AuraCrystal]

How does Ricci curvature represent "volume deficit"?

Hi all,

I've been reading some general relativity in my spare time (using Hartle). I'm a bit confused about something. I understand that Riemann curvature is defined in terms of geodesic deviation; the equation of geodesic deviation is
$\left ( \mathbf{\nabla}_{\mathbf{u}} \mathbf{\nabla}_{\mathbf{u}} \mathbf{\chi} \right )^{\alpha} = - R^{\alpha}_{\beta \gamma \delta} u^{\beta} \chi^{\gamma} u^{\delta}$
Where, of course, $R^{\alpha}_{\beta \gamma \delta}$ is the Riemann (curvature) tensor, $\mathbf{\nabla}_{\mathbf{u}}$ represents the covariant derivative w.r.t. the four-velocity $\mathbf{u}$ i.e. the unit tangent to the geodesic, $dx/d\tau$, and $\chi$ is the separation vector between two geodesics. This definition makes geometric sense: the Riemann tensor tells you how fast the separation vectors are changing. Of course, the Ricci tensor is defined as the contraction of the Riemann tensor: $R_{\alpha \beta} \equiv R^{\gamma}_{\alpha \gamma \beta}$,
where one sums over $\gamma$, of course.

However, I read somewhere that the Ricci tensor has a geometric meaning: in terms of deviation of the volume of a ball from that in normal, Euclidean space. I've never seen this and I've looked in Schutz's A First Course in General Relativity, Hartle, of course, Schaum's Outline of Tensor Calculus, and Lovelock and Rund's Tensors, Differential Forms, and Variational Principles and have found nothing about this geometric meaning in there. How do you get that geometric meaning?

I also read something about Christoffel symbols being "connections." What is the geometric meaning of this and how do I get it? (Sorry I didn't put this in the title; it wouldn't fit lol.)

As well, does anyone know a good book explaining these geometric meanings? (Either in a book on GR or on tensors/differential geometry? I was thinking of going through either Carrol or Wald after I finish Hartle and Lovelock/Rund.)

 

[WannabeNewton]

Where did you hear that about the Ricci tensor (not denying I just want to read up on it too)? Anyways, a connection, on the tangent bundle (a specific type of a fiber bundle so the connection is more general than this) to a manifold, allows you to relate neighboring elements of the tangent bundle through parallel transport. Christoffel symbols are the components of the levi - civita connection, $\bigtriangledown$, which is a torsion - free connection that, when acting on the metric, gives $\bigtriangledown g = 0$. That is very simplified; I personally found the definition of the connection in O'Neill's Semi - Riemannian Geometry text to be very clear as well as doCarmo's text. Also, I would personally go with Carroll then Wald or at least finishing the appendices in Carroll and then using Wald.

 

[AuraCrystal]

I read it in Wikipedia's article on it.

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space. As such, it provides one way of measuring the degree to which the geometry determined by a given Riemannian metric might differ from that of ordinary Euclidean n-space.

 

[robphy]

Possibly useful reading:
http://math.ucr.edu/home/baez/gr/outline2.html (Ricci)

 

[jfy4]

Possibly useful reading:
http://math.ucr.edu/home/baez/gr/outline2.html (Ricci)

Thanks for that link, it's pretty cool.

Carroll's book takes a geometric approach.

 

[AuraCrystal]

Carrol's book takes a geometric approach.

OK thanks; I'll have to check that out! ^_^

 

[waht]

You might be interested in this explanation (page 7)

http://arxiv.org/abs/gr-qc/0401099v1.

 

[jfy4]

You might be interested in this explanation (page 7)

http://arxiv.org/abs/gr-qc/0401099v1.

Thanks for the paper, I'm going to give it a skimming.