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In my class, there is no distinction made between,

$$ C_{ab}{}^{b} $$ and $$ C^{b}{}_{ab}. $$ All I know, and read about so far, is the distinction of covariant and contravariant, form/vector, etc. etc. But what is this slanted business all about?

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what do you mean slanted business? –  nervxxx Apr 7 '13 at 3:21
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@nate tex tip: $C_{ab}{}^b$ is a quick way to get the alignment right automatically. –  Chris White Apr 7 '13 at 3:22
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Well I'm reading about solving a tensor equation by taking traces of it, there are 3, and they are listed as $C^a{}_{ab}$, $C^a{}_{ba}$, $C_{ba}{}^a$. They are distinct and so they are apparently different, but I don't know why... I guess the last 2 in particular. @ChrisWhite Thanks for the tip! –  nate Apr 7 '13 at 3:31

2 Answers 2

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An easy way to see that they are distinct is to consider what happens upon raising (or lowering) all indices.

For example, upon lowering, $$ T_{ab}{}^{cde} $$ becomes $T_{abcde}$, whereas $$ T_{a}{}^{cd}{}_{b}{}^{e} $$ becomes $T_{acdbe}$, and similarly $$ T_{a}{}^{cde}{}_{b} $$ becomes $$ T_{acdeb}. $$

You need to "slant" the indices so as to keep track of the proper order when lowering and raising. For example, if you just write $T_{ab}^{cde}$, and you lower the index $c$, what is the correct tensor: $T_{acb}^{de}$, $T_{cab}^{de}$, or $T_{abc}^{de}$? The notation is ambiguous if you do not "slant" your indices.

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Each of the indices in a tensor have a particular left-right ordering. This ordering cannot be changed unless the tensor has some particular symmetry that permits it (or rather, that equates different components on interchange).

The up-down positions of indices tells us about whether the index is associated with using a basis vector (up) or a basis covector (down) for that index to help extract the component. Let $v$ be a one-index tensor. $v^a$ are the components associated with a set of basis vectors $e_{(a)}$, and $v_a$ are the components associated with a set of basis covectors $e^{(a)}$. In general, $v_a \neq v^a$ for an arbitrary coordinate system. Each index can be associated with basis vectors or basis covectors, and we need not use all the same kinds of basis elements for all the indices of a tensor.

In a space with a metric, we can convert back and forth between using basis vectors and basis covectors to extract components of tensors (we can raise or lower indices more or less at will), so we tend to associate all such combinations of up-down indices with the same inherent object. Still, whether a particular index is up or down for a given situation depends on what's convenient or necessary to use.

Edit: So strictly speaking, writing an indexed object with two indices lined up on top of each other doesn't really have meaning. Still, physicists often do this anyway for, as an example, Christoffel symbols--it's relatively rare that they're used in any other way than as $\Gamma^a_{bc}$. Still, when it comes to the Riemann tensor or other such objects, it's best to think of each index has occupying a whole column--nothing else should come above or below that index, to give room for it to move freely up or down when contracted with the metric.

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