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Riemann tensor: indipendent components

  1. May 22, 2010 #1
    Hi, thanks for the attention and excuse for my bad english.
    I'm studying general relativity and I have a doubt about the number of indipendent component of the riemann curvature tensor.
    We have two kind of riemann tensor:

    type (3,1) Rikml

    type (4,0) Rrkml

    There are also some symmetry rule:

    first skew-symmetry
    Rrkml = -Rkrml
    secondo skew-symmetry
    Rrkml = -Rrklm
    (Rikml = -Riklm)​
    block symmetry
    Rrkml = Rmlrk
    Bianchi's first identity
    Rrkml+Rrlkm+Rrmlk=0​
    (Rikml+Rilkm+Rimlk=0)​

    Now, the (4,0) R-type has
    n2(n2-1)/12
    indipendent components. What about the (3,1) R-type? The indipendent component are also n2(n2-1)/12 or they are n2(n2-1)/3?

    (In the case n=4, (4,1) R-type has 20 indipendent component; (3,1) R-type are 20 or 80?)
     
  2. jcsd
  3. May 22, 2010 #2

    haushofer

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    Science Advisor

    Why would acting with the metric on the Riemann tensor change the number of independent components?
     
  4. May 22, 2010 #3
    I give up. Why would it not, though it does seems reasonable that raising or lowering indices would have no effect?


    Come to think of it, we could take our N independent variables, and transform to Riemann normal coordinates, where the metric is Minkowskian, without loss of generality. With the metric now having constant entries, indices are raised and lowered without changing the number of independent variables.
     
    Last edited: May 22, 2010
  5. May 22, 2010 #4
    I suppose that raising or lowering indices leaves unchanged the number of indipendent components, but i was not able to find a confirm of that. I thought that, given the metric tensor, the Rikml are (linear?) functions of the Rrkml component. So, can 20 indipendent quantities generate more than 20 indipendent quantities?
     
  6. May 26, 2010 #5
    Of course rising and lowering indicies does not change the number of independent
    components of any tensor. However the number of independent components
    should depend of the choice of base. For example in coordinate (holonomic)
    base the number of independent components of metric tensor is 10 but
    in ortonormal base it's of course 0. The number of independent components
    of Riemann tensor in 4 dimension in coordinate base is 20 but what about
    ortonormal base. Does the extra condidtion imposed on base give in this example any
    extra condition which have to be fullfiled by Riemann tensor?
     
  7. May 26, 2010 #6
    We cast around this word 'indendent' so casually at times, it's sometimes hard to tell at times what is independent of what. In the case of a vector, we like to cast it as an object that is independent of coordinate system or metric. The same applies to any tensor object.

    In the case of the components of a tensor, I believe we are looking for the linear independence of tensor element, and perhaps eliminate additional degrees of freedom for other reasons.
     
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