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How does Ricci curvature represent volume deficit ?

  1. Sep 30, 2011 #1
    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
    [itex]\left ( \mathbf{\nabla}_{\mathbf{u}} \mathbf{\nabla}_{\mathbf{u}} \mathbf{\chi} \right )^{\alpha} = - R^{\alpha}_{\beta \gamma \delta} u^{\beta} \chi^{\gamma} u^{\delta} [/itex]
    Where, of course, [itex]R^{\alpha}_{\beta \gamma \delta}[/itex] is the Riemann (curvature) tensor, [itex]\mathbf{\nabla}_{\mathbf{u}}[/itex] represents the covariant derivative w.r.t. the four-velocity [itex]\mathbf{u}[/itex] i.e. the unit tangent to the geodesic, [itex]dx/d\tau[/itex], and [itex]\chi[/itex] is the seperation vector between two geodesics. This definition makes geometric sense: the Riemann tensor tells you how fast the seperation vectors are changing. Of course, the Ricci tensor is defined as the contraction of the Riemann tensor: [itex]R_{\alpha \beta} \equiv R^{\gamma}_{\alpha \gamma \beta}[/itex],
    where one sums over [itex]\gamma[/itex], 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.)
     
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  3. Sep 30, 2011 #2

    WannabeNewton

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    Re: How does Ricci curvature represent "volume deficit"?

    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, [itex]\bigtriangledown[/itex], which is a torsion - free connection that, when acting on the metric, gives [itex]\bigtriangledown g = 0[/itex]. 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.
     
  4. Oct 1, 2011 #3
    Re: How does Ricci curvature represent "volume deficit"?

    I read it in Wikipedia's article on it.

     
    Last edited: Oct 1, 2011
  5. Oct 1, 2011 #4

    robphy

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  6. Oct 1, 2011 #5
    Re: How does Ricci curvature represent "volume deficit"?

    Thanks for that link, it's pretty cool.

    Carroll's book takes a geometric approach.
     
    Last edited: Oct 1, 2011
  7. Oct 1, 2011 #6
    Re: How does Ricci curvature represent "volume deficit"?

    OK thanks; I'll have to check that out! ^_^
     
  8. Oct 2, 2011 #7
    Last edited: Oct 2, 2011
  9. Oct 2, 2011 #8
    Re: How does Ricci curvature represent "volume deficit"?

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