Contracting Riemann tensor with itself

1. Nov 15, 2015

ShayanJ

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?
Thanks

2. Nov 15, 2015

Orodruin

Staff Emeritus
The anti-symmetries do not dictate that this is zero.

3. Nov 15, 2015

Ben Niehoff

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