The title completely expresses the question. I have come across for several times the mixed tensors where the lower index is written closer to the tensor symbol and the upper index is written a bit farther away, like: $T_a \ ^b$. It also sometimes appeared in the other way around, as in: $T^b \ _a$. What's the difference?

In case of higher rank tensors, what's the difference between: $$T^a\ _{bc},\quad T_b\ ^a\ _c, \quad and \quad T_{bc} \ ^a$$


1 Answer 1


When it comes to nonsymmetric tensors, the order of indices matter, even between covariant and contravariant indices. Let us take the difference between $T^a{}_b$ and $T_b{}^a$, multiplied by the metric $g_{ab}$ to raise and lower indices:


which is zero only if $T_{bc}=T_{cb}$, i.e., if $T$ is a symmetric tensor.

Higher rank tensors are harder to represent explicitly, but the idea is the same: e.g., $g_{ad}(T^a{}_{bc}-T_b{}^a{}_c)=T_{dbc}-T_{bdc}\ne 0$ unless $T$ is symmetric in its first two indices.

This becomes especially relevant in practice when one deals with the indices of the Riemann curvature tensor, for instance.


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