Kretschmann angelfish

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 . ${\displaystyle K}$

definition

Using Einstein's summation convention , the Kretschmann scalar is defined as ${\displaystyle K}$

${\displaystyle K=R_{\kappa \lambda \mu \nu }\,R^{\kappa \lambda \mu \nu }=\sum _{\kappa =0}^{3}\sum _{\lambda =0}^{3}\sum _{\mu =0}^{3}\sum _{\nu =0}^{3}R_{\kappa \lambda \mu \nu }\,R^{\kappa \lambda \mu \nu }}$.

Here denotes the Riemann curvature tensor and . ${\displaystyle R_{\kappa \lambda \mu \nu }}$ ${\displaystyle \,R^{\kappa \lambda \mu \nu }=\sum _{i=0}^{3}g^{\kappa i}\,\sum _{j=0}^{3}g^{\lambda j}\,\sum _{k=0}^{3}g^{\mu k}\,\sum _{l=0}^{3}g^{\nu l}\,R_{ijkl}\,}$

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: ${\displaystyle C_{\kappa \lambda \mu \nu }}$ ${\displaystyle R_{\kappa \lambda }}$ ${\displaystyle R}$

${\displaystyle K=C_{\kappa \lambda \mu \nu }\,C^{\kappa \lambda \mu \nu }+2R_{\kappa \lambda }\,R^{\kappa \lambda }-{\frac {1}{3}}R^{2}}$

example

For the Schwarzschild metric , the Kretschmann scalar with the Schwarzschild radius is given by: ${\displaystyle r_{s}}$

${\displaystyle K={\frac {12{r_{s}}^{2}}{r^{6}}}={\frac {48G^{2}M^{2}}{c^{4}r^{6}}}}$

Individual evidence

1. ^ 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 .
2. ^ Entry on the Kretschmann-Skalar in the Lexicon of Astronomy of the Spektrum Verlag
3. ↑ Sebastian Boblest, Thomas Müller, Günter Wunner: Special and general relativity theory . Springer, Berlin 2016, p. 225.