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Calculate the elements of the Riemann tensor

  1. Nov 10, 2012 #1
    1. The problem statement, all variables and given/known data
    Compute 21 elements of the Riemann curvature tensor in for dimensions. (All other elements should be able to produce through symmetries)


    2. Relevant equations
    [itex]R_{abcd}=R_{cdab}[/itex]
    [itex]R_{abcd}=-R_{abdc}[/itex]
    [itex]R_{abcd}=-R_{bacd}[/itex]



    3. The attempt at a solution
    I don't see how 21 elements can be enough,
    for example
    abcd
    0000
    1111
    2222
    3333
    1110
    1100
    1000
    2220
    2200
    2000
    3330
    3300
    3000
    0012
    0120
    0023
    0230
    1112
    1122
    1212
    2223
    2233
    2323

    I am already up to 24 elements and I am not even sure if this is enough to derive all the other ones. Am I missunderstanding something?
    According to http://mathworld.wolfram.com/RiemannTensor.html it should be possible, but I really don't see it right now.
    Any hints would be appreciated.
     
  2. jcsd
  3. Nov 11, 2012 #2
    There are 20 components of the tensor which can be non-zero. For example R0000 is certainly 0 (why?)
     
  4. Nov 11, 2012 #3
    see 8th heading in the following,it will guide you how there are only 20 independent components and not 256 as it might be think(4×4×4×4)
    http://www.mathpages.com/rr/appendix/appendix.htm
     
  5. Nov 11, 2012 #4
    Ah of course, there are some elements that are zero. I don't know why I didn't checked that first. Thanks for quick replies, I think I got it right now :)
     
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