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Lovelock gravity

  1. Sep 12, 2011 #1
    I read about lovelock gravity:

    I dont undestand how generate a Riemman tensor [tex]R_{ijkl}[/tex] in the expression:

    [tex]\delta^{abcd}_{ABCD}R^{AB}_{ab}R^{CD}_{cd} = R_{ijkl}R^{ijkl}-4R_{ij}R^{ij}+R^2[/tex]
    [tex]\delta^{abcd}_{ABCD}[/tex] is the generalized kronecker delta,

    i understan that a term

    [tex]\delta^a_A \delta^b_B \delta^c_C \delta^d_D R^{AB}_{ab}R^{CD}_{cd} = R^2[/tex]

    but , for example
    [tex]\delta^a_A \delta^b_B \delta^c_D \delta^d_C R^{AB}_{ab}R^{CD}_{cd} ??[/tex]

    for example , how i find the term [tex]R_{ijkl}[/tex] ?
    Last edited: Sep 12, 2011
  2. jcsd
  3. Feb 14, 2012 #2
    use the defination of the generalized kronecker delta as determinant
  4. Feb 14, 2012 #3


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    In your second example all you did was switch C and D. The Riemann tensor is antisymmetric on C and D so this term is just -R2. To get a term like RabcdRabcd you would need to use for example RCDabRABcd.
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