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While reading physics textbooks I often come across notation like this; $$J_{\alpha}{}^{\beta},\ \Gamma_{\alpha \beta}{}^{\gamma}, K^\alpha{}_{\beta}.$$ Notice the spacing in indices. I don't understand why they do not write simply $J_{\alpha}^\beta, \Gamma_{\alpha \beta}^\gamma, K^\alpha_{\beta}$.

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It's important to keep track of the ordering if you want to use a metric to raise and lower indices freely (without explicitly writing out $g_{ij}$'s all the time).

For example (using Penrose abstract index notation), if you raise the index $a$ on the tensor $K_{ab}$, then you get $K^a{}_b (=g^{ac} K_{cb})$, whereas if you raise the index $a$ on the tensor $K_{ba}$, you get $K_b{}^a (=g^{ac}K_{bc})$. Since the tensors $K^a{}_b$ and $K_b{}^a$ act differently on $X_a Y^b$ (unless $K$ happens to be symmetric, i.e., $K_{ab}=K_{ba}$), one doesn't want to denote them both by $K^a_b$.

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Tensors can be thought of as multi linear maps from copies of a vector space (and its dual) to a field (usually $\mathbb C$). The placements of the indices tell you which "argument" goes where. E.g. $A_{mn} u^m v^n$ is not the same as $A_{nm}u^m v^v$. Perhaps Penrose's pictorial notation makes this clearest.

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    $\begingroup$ I think the question is actually about the spacing rather than the order in which letters are written, if you compare the displayed examples in the question with the in-line examples at the end of the question. $\endgroup$ – KCd Oct 16 '11 at 22:52
  • $\begingroup$ @KCd: Exactly as you say. I am sorry, my poor English gave rise to a misunderstanding. I'll restate the question to improve clarity. $\endgroup$ – Giuseppe Negro Oct 16 '11 at 22:59
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    $\begingroup$ @Giuseppe: But without the spacing you lose part of the location information... $\endgroup$ – genneth Oct 17 '11 at 10:02

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