Abstract
Suppose we have a coordinate system \({x}^{\mu }\) in a region of an \(n\)-dimensional Riemann (or pseudo-Riemann) manifold [20]. Components of the metric tensor \(g_{\mu \nu }\) are given as functions of \({x}^{\mu }\). We want to calculate the Riemann curvature tensor \({R}^{\mu }\,_{\nu \alpha \beta }\) and related quantities (the Ricci tensor \(R_{\mu \nu }\), the scalar curvature \(R\)).
The metric tensor is symmetric; therefore, it is reasonable to ask the user to provide only the components with \(\mu \geq \nu\).
Catching a lion, the Einstein’s method: Enter the cage and lock it from inside. Then the Universe will be subdivided into two disjoint regions in such a way that you are in one of them and the lion is in the other one. It depends on one’s point of view whom to consider caught; for convenience, let’s say it’s the lion.
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Grozin, A. (2014). Riemann Curvature Tensor. In: Introduction to Mathematica® for Physicists. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-00894-3_21
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