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Comments to the question (v1): Indices are raised and lowered vertically by the pertinent metric tensor of the theory. The horizontal position of indices is important for a tensor that is not totally symmetric, e.g., the EM field strength $F_{\mu\nu}$ or the Riemann curvature tensor $R_{\mu\nu\lambda\kappa}$, etc, in order to properly identify which ...


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On a two-index tensor, swapping the two indices is equivalent to transposing a matrix. You may not see many authors spending a lot of effort on this issue simply because an awful lot of the tensors we deal with are symmetric. This includes the metric, Ricci tensor, Einstein tensor, and stress-energy tensor. Therefore there is no special interest in ...


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Comments to the question (v1): As usual, be prepared that different authors use different conventions and notations. E.g. what some authors call a vielbein might be what other authors call a transposed vielbein. A curved index (aka. as coordinate index) is raised and lowered vertically with the curved metric tensor, while a flat index (aka. as vielbein ...


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This graphic on Things Made Thinkable uses the $\bar\Delta^-$ notation, which corroborates rob's prediction.



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