A Wald's Abstract Index Notation: Explaining T^{acde}_b

[madsmh]

TL;DR Summary

Confusion by the abstract index notation introduced in Wald's General Relativity.

In the second paragraph on page 25 of Wald's General Relativity he rewrites T^{acde}_b as g_{bf}g^{dh} g^{ej}T^{afc}_{hj} . Can anyone explain this? I am confused by the explantion given in the book. Especially puzzling is that the inverse of g seems to be applied twice, which I can't make sese of.

Mads

 

[Orodruin]

The inverse metric is applied twice to raise the last two indices. The metric is used once to lower the second index.

Also note that the index horizontal placement is important.

 

[Ibix]

This isn't particularly to do with abstract index notation - it applies in all tensor index notations. You do need to pay attention to which index is being contracted over and the order of indices is important. But all that's happening here is that the metric is being used to lower two of the indices and the inverse metric is being used to raise one. That's just what the metric does. Just as the metric in $g_{ab}v^b$ lowers the $b$ to give you the one form $v_a$, the metric applied to any tensor lowers the repeated index - so $g_{ad}T^{bcdefg}$ lowers the $d$ to give you $T^{bc}{}_a{}^{efg}$. Note that the repeated index $d$ was replaced with the other index on the metric because we summed over the dummy index.

In the cited section Wald is just randomly lowering a couple of indices and raising one to show you can do it to multiple indices at once.

Quote my post to see a way to do index notation with correct positioning in $\LaTeX$.

 

[Orodruin]

In the cited section Wald is just randomly lowering a couple of indices and raising one to show you can do it to multiple indices at once.

Lowering one and raising two.

 

[Ibix]

Lowering one and raising two.