= Kretschmann scalar
{c}
{title2=$K=R_{abcd}R^{abcd}$}
The Kretschmann scalar is the scalar <tensor contraction> of two copies of the <Riemann curvature tensor>. In <Schwarzschild spacetime>, with metric coefficient $1-2M/r$ and $G=c=1$, it is $48M^2/r^6$. Divergence of this scalar along a curve approaching a proposed regular point proves that there is no twice continuously differentiable, nondegenerate metric extension through that point: such an extension would have continuous finite curvature components and hence a finite scalar contraction. This gives a coordinate-independent test of a <curvature singularity>. Boundedness of this single scalar does not prove regularity.
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