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Rank and weight of a Tensor

  1. Jan 1, 2009 #1
    Given a Riemann Curvature Tensor. How do you know the weight and rank of each:


    Is the Ricci tensor always a zero tensor for diagonal metric tensors?
  2. jcsd
  3. Jan 1, 2009 #2
    The rank of a tensor can be thought of as the number of distinct indices that the tensor has. Thus [itex]R^i_{jkl}[/itex] is a fourth-rank tensor, while the Ricci tensor [itex]R^k_{ikj}=R_{ij}[/itex] is a second-rank tensor. On the other hand, the Ricci scalar [itex]R=R^i_i[/itex] is a scalar quantity and hence a zero-rank tensor.

    The weight of a tensor is defined to be the power of [itex]\sqrt{-\det g_{ij}}[/itex] that appear in the tensor.
  4. Jan 1, 2009 #3
    What tells the weight?

    [tex](-det(g_{ab}))^{1/2} = r^{2}*sin(\theta)*c*t[/tex]
    Last edited: Jan 1, 2009
  5. Jan 1, 2009 #4


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    No. In fact, it's rarely zero. For instance if you replace 1-2m/r in the Schwarzschild metric with s-2m/r where s is a constant ne to 1, the Ricci tensor gets 2 components.
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