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Contracting Riemann tensor with itself

  1. Nov 15, 2015 #1


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    In chapter 8 of Padmanabhan's "Gravitation: Foundations and Frontiers" titiled Black Holes, where he wants to explain that the horizon singularity of the Schwarzchild metric is only a coordinate singularity, he does this by trying to find a scalar built from Riemann tensor and show that its well-behaved at the horizon. He writes (eq. 8.3) ##R_{abcd} R^{abcd}=\frac{48 M^2}{r^6}##. But this can't be correct. The (anti-)symmetries of the Riemann dictate that ##R_{abcd} R^{abcd}## is equal to zero. What is he doing here?
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  3. Nov 15, 2015 #2


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    The anti-symmetries do not dictate that this is zero.
  4. Nov 15, 2015 #3

    Ben Niehoff

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    Also, this quantitiy is called the Kretschmann scalar. You may have seen posts about it recently.
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