The Kretschmann scalar (also Kretschmann invariant or Riemann invariant ; after Erich Kretschmann , who introduced it) designates a scalar invariant in the range of Lorentzian manifolds . It can be interpreted as a measure of the curvature of space-time in general relativity .
definition
Using Einstein's summation convention , the Kretschmann scalar is defined as

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.
Here denotes the Riemann curvature tensor and .


For four-dimensional space-time, the Kretschmann scalar can still be expressed by the Weyl tensor , the Ricci tensor and the Ricci scalar as follows:


example
For the Schwarzschild metric , the Kretschmann scalar with the Schwarzschild radius is given by:


Individual evidence
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^ Richard C. Henry: Kretschmann Scalar for a Kerr-Newman Black Hole . In: The American Astronomical Society (Ed.): The Astrophysical Journal . 535, 2000, pp. 350-353. arxiv : astro-ph / 9912320v1 . bibcode : 2000ApJ ... 535..350H . doi : 10.1086 / 308819 .
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^ Entry on the Kretschmann-Skalar in the Lexicon of Astronomy of the Spektrum Verlag
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↑ Sebastian Boblest, Thomas Müller, Günter Wunner: Special and general relativity theory . Springer, Berlin 2016, p. 225.