Is Λa_b the Same as Λba in Tensor Notation?

[dyn]

Index notation in GR is really confusing ! I'm confused about many things but one thing is the order of index placement , ie. is Λ^a _b the same as Λ_b^a ? And if not what is the difference ? Thanks
If anyone knows of any books or lecture notes that explain index gymnastics step by step that would be great.

 

[PeterDonis]

is Λ^a_b the same as Λ_b^a ?

Strictly speaking, no. See below.

if not what is the difference ?

The best simple explanation of how a tensor works that I've seen is in Misner, Thorne, & Wheeler, the classic GR textbook. Basically, a tensor is a linear machine with some number of slots, that takes geometric objects as input into the slots and outputs numbers; each slot corresponds to an index. If the index is an upper index, the slot takes a vector as input; if the index is a lower index, the slot takes a covector (or 1-form) as input. The order of the slots matters, so Λ^a_b, which takes a vector in the first slot and a 1-form in the second, is not the same as Λ_b^a, which takes a 1-form in the first slot and a vector in the second.

In a manifold with metric (which is all we work with in GR), you can always use the metric to convert vectors to 1-forms or vice versa. So you could take a vector and a 1-form that you inserted into the slots of Λ^a_b, and insert them into the slots of Λ_b^a, by converting the vector to a 1-form (so it will go in the first slot of Λ_b^a) and the 1-form to a vector (so it will go in the second slot of Λ_b^a). If these two operations both give the same number as output, then the two tensors Λ^a_b and Λ_b^a can be considered "the same"; in this case, we say the second is just the first with one index lowered and one index raised, using the metric.

 

[dyn]

Thanks for your answer. Does that mean indices should never be directly in a vertical line as in that case we wouldn't know the order of the "slots" ?

 

[PeterDonis]

Does that mean indices should never be directly in a vertical line as in that case we wouldn't know the order of the "slots" ?

Yes, although some sources are sloppy about this, probably because in some cases it doesn't actually matter. For example, if a two-index tensor is symmetric, its indexes can be exchanged (i.e., slots swapped) without changing its output. Many key tensors that appear in GR are symmetric (e.g., the metric and the stress-energy tensor).

 

[Mentz114]

Thanks for your answer. Does that mean indices should never be directly in a vertical line as in that case we wouldn't know the order of the "slots" ?

Just to add (pedantically) that some tensors are anti-symmetric so $T_{ab}=-T_{ba}$.