Index Gymnastics: 2 More Questions

[dyn]

Hi. Got 2 more questions on index placements.

1 - I found the following on an exam paper and it seems wrong to me. It concerns a weak perturbation to the Minkowski metric g_ik = η_ik + h_ik. It then states that to first order in h_ik the contravariant metric tensor is g^ik = η^ik - ηⁱⁿη^kmh_nk. This seems wrong to me as the RHS has a free m index but the LHS does not. I think 2nd term on the RHS should be -η^aiη^bkh_ab which I think is equal to -h^ik. Am I right ?

2 - In some notes I found n_uvT^uv = T_u^v. This doesn't seem right either but I'm not sure exactly what it should be ? If I contract the u I get T_v^v but if I contract the v first I get T^u_u. Are these different ? Do they both equal scalar T ?

 

[Orodruin]

You are correct on both accounts. The two contractions at the end of 2 are equivalent (they must be as they come from the same expression). This is called the trace of T, which is a scalar.

 

[dyn]

Thanks. Just to confirm for me ; η_uvT^uv does not equal T_u^v ? It actually equals T_v^v which equals T^u_u and these are both equal to T ?
For a general tensor does A^u_u always equal A_v^v ?

 

[Orodruin]

Two index expressions with different free indices will in general not be equal.

For a general tensor does Auu always equal Avv ?

Correct.

 

[bcrowell]

Two index expressions with different free indices will in general not be equal.

I would say there are several different cases to consider if you want to say this as a general rule.

If the number or placement of the free indices differ (as in #3), then they are different types of mathematical objects, e.g., a vector and a scalar.

If the number and placement of free indices are the same, and it's concrete index notation, but the indices are different letters, e.g., v^\mu and v^\nu, then there is some ambiguity. We could be talking about components in different coordinate systems, which could be equal but probably wouldn't be. Or comparing them could be a notational mistake.

For the same case in abstract index notation, v^a and v^b, these represent exactly the same mathematical object, and they are guaranteed to be equal to one another. However, we have a notational rule that when we mix them in the same equation, we should not use different letters, e.g., we write v^a=v^a, not v^a=v^b, and v^a+v^a, not v^a+v^b. This is because the whole purpose of using different letters in abstract index notation is to keep straight how we're hooking up the "plumbing."