Summation convention and index placement

[ramparts]

Hey all,

The way I was taught GR, the summation convention applies on terms where an index is repeated strictly with one covariant, one contravariant. But reading through a translation of Einstein's GR foundations paper just now it looks like the index placement doesn't matter (I've seen it this way on Wikipedia too! :P). I've never actually seen a term like, say, a_\mu b_\mu where you have repeated upper indices or repeated lower indices, so as yet this hasn't been an issue, but I'm curious what the consensus on the convention is, and whether it actually matters (are there terms/can there be terms in GR with repeated upper/lower indices?). Thanks!

 

[dx]

If you have a tensor A^ab_cd, then A^ab_ad is a tensor, but A^aa_cd is not. There's nothing wrong with summing over two upper indices or two lower indices, but you just won't get a very useful object when you do that.

 

[ramparts]

Thanks! That's what I figured, it seemed like the heavens were conspiring to keep summed indices in separate positions. So is there really never a time in GR where something like:

T^aU^a

or

T_aU_a

comes up and needs to be summed?

 

[dx]

I don't think so.

 

[DrGreg]

I think in the early days when the summation convention had just been invented, the "upstairs downstairs" convention for contravariant/covariant hadn't been fully established, so you may see some early documents that have the index in the wrong place according to the modern convention, or where the summation could occur with indexes in the same position. In the case of Wikipedia, it's probably just a mistake.