No. of Independent Components of Riemann Tensor in Schwartzchild Metric

  1. In general 4d space time, the Riemann tensor has 20 independent components.

    However, in a more symmetric metric, does the number of independent components reduce? Specifically, for the Schwartzchild metric, how many IC does the corresponding Riemann tensor have?

    (I think it is 4, but I cannot find a source to confirm this)
     
  2. jcsd
  3. atyy

    atyy 10,616
    Science Advisor

    Once the metric is specified, there are no independent components of the Riemann tensor, since the metric completely determines the Riemann tensor.

    Also, the counting of 20 independent components of the Riemann tensor doesn't take into account the requirement that one should be able to "integrate" the Riemann tensor to get a metric. These additional constraints are listed in Stephani's Relativity: an introduction to special and general relativity, on p143 in the section beginning with "The determination of the metric from a specified curvature tensor amounts ... to the solution of a system of twenty second-order differential equations for the ten metric compoenents ... In general such a system will posess no solutions ..."
     
    Last edited: Jan 14, 2012
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