The Kretschmann scalar is a quantity in general relativity that is used to characterize the curvature of spacetime. It is defined as the squared norm of the Riemann curvature tensor, which encodes information about the curvature of a manifold.
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The Kretschmann scalar is the scalar tensor contraction of two copies of the Riemann curvature tensor. In Schwarzschild spacetime, with metric coefficient and , it is . 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.