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When, if ever, will we see Riemann curvature tensor (RCT) components having 3 or 4 distinct indices?, like for $R_{txxy}$ or $R_{txyz}$ for ex. How this came about was I that I was reading that wonderful exposition "The Meaning of Einstein's Equation" by John C. Baez and Emory F. Bunn. They consider a cluster of free fall test particles on the surface of a sphere which is filled with matter. They say that Einstein's field equations can be derived by considering the fact that the particles will (naturally) converge and the volume of the ball will decrease. However, they caution that it is necessary to consider all frames of reference.

So, to get a better understanding I calculated how the RCT would transform under a Lorentz transformation between 2 locally inertial coord systems: $(t,x,y,z)$ and $(T,X,Y,Z)$ and my calculations showed that there would be non-zero terms like $R_{ytxy}$ and $R_{yxty}$.

(If I knew Mathjax, I would write out the full equation.) These terms having 3 distinct components puzzle me. In all the literature, I have not encountered any RCT non-zero components having 3 or 4 distinct components. Why do they appear here? Does it have to do with torsion?, whether the coordinate system is adapted to the geometry? any ideas?

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  • $\begingroup$ I’ve edited your question to add the MathJax. You should be able to look at what I did so you’ll know how next time. $\endgroup$ – G. Smith Mar 30 at 5:34
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This has nothing to do with torsion.

The Kerr metric has a nonzero $R_{0123}$, or at least it does in rational polynomial coordinates. See equation (84) in this paper. This is probably because $g_{t\phi}$ is a function of $r$ and $\chi$.

So there is nothing too exotic about having nonzero Riemann components with 3 or 4 distinct indices. You just need the metric to be sufficiently generic. For example, for the most generic case, think about a metric where all the components of the metric tensor depend on all the coordinates.

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