Question about 4-vectors/tensors from a complete novice

[AxiomOfChoice]

Is any distinction made between an expression like $A^\nu A_\mu$ and one that looks like $A^\mu A_\mu$? I can justify something like this to myself through some vague hand-waving and mention of a phrase like "just dummy indices," but this isn't convincing even to me...and I'm making the argument!

If the context matters, tell me, and I'll provide some. Thanks!

 

[cristo]

Is any distinction made between an expression like $A^\nu A_\mu$ and one that looks like $A^\mu A_\mu$?

The Einstein summation convention is invoked on repeated indices, so the difference is that A^u A_v is a tensor, but $$A^u A_u=\sum_u A^uA_u$$ which is a scalar.

 

[AxiomOfChoice]

The Einstein summation convention is invoked on repeated indices, so the difference is that A^u A_v is a tensor, but $$A^u A_u=\sum_u A^uA_u$$ which is a scalar.

cristo: Thank you. Of course. I should have known that. I could have saved both of us time and effort if I'd recognized it. Guess that's what happens when you don't really know what you're doing.