The Schwarzschild solution is meant to be a solution of the vacuum Einstein equations. That is


So, the Ricci tensor must be null for $r>0$.

Now, if the scalar curvature is nothing but the Ricci tensor contracted, and the Ricci tensor is null, the cuvature should be zero.

Nonetheless, I have been told that the curvature of the Schwarzschild solution (in the usual coordinates) is


which is obviously non zero.

What am I making wrong?


You're correct that $R=0$. $R_{abcd} R^{abcd} = \frac{12 r_s^2}{r^6}$ is the Kretschmann scalar for the Schwarzschild metric, an invariant used to find the true singularities of a spacetime. In this case, only the singularity at $r=0$ is a spacetime singularity, not a coordinate-system one.

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