How do you differentiate between the inner and lower indices on a Lambda tensor?

[QuarkMaster]

TL;DR Summary

tensors relativity

$$ {\Lambda}^{i}_{j} $$

When indices are written on top of one another I am confused wich is the inner index and which is the lower one when we lower the upper index.

 

[strangerep]

Yeah, I hate that sloppiness too (welcome to theoretical physics).

Sometimes you can figure it out from the context. Do you have a specific example where you saw this?

 

[QuarkMaster]

Yeah, I hate that sloppiness too (welcome to theoretical physics).

Sometimes you can figure it out from the context. Do you have a specific example where you saw this?

Just in general I often see this in physics books and get confused. So how am one supposed to know?

 

[strangerep]

You'll have to dig up a specific example. I can't help you in a vacuum.

 

[QuarkMaster]

You'll have to dig up a specific example. I can't help you in a vacuum.

$$ {\eta}^{\mu\nu} = {\Lambda}^{\mu}_{\sigma} {\Lambda}^{\nu}_{\gamma}{\eta}^{\sigma\gamma} $$
This expression. I am trying to write it without the indices, but don't know where they go on the lambda tensor

 

[strangerep]

I'd think of it like this:$$ {\eta}^{\mu\nu} ~=~ {\Lambda}^{\mu}_{~\;\sigma} {\Lambda}^{\nu}_{~\gamma} \, \eta^{\sigma\gamma} ~=~ {\Lambda}^{\mu}_{~\;\sigma} \, \eta^{\sigma\gamma} {(\Lambda^T)}_{\gamma}^{~\;\nu}~.$$If you can re-express the expression in matrix form, in this case ##\,\eta' ~=~ \Lambda \, \eta\, \Lambda^T \,##, it's usually obvious what indices go where.

 

[QuarkMaster]

makes sense. Thanks!

 

[vanhees71]

TL;DR Summary: tensors relativity

$$ {\Lambda}^{i}_{j} $$

When indices are written on top of one another I am confused wich is the inner index and which is the lower one when we lower the upper index.

You are right. It's simply a bad notation. You have to write {\Lambda^{i}}_j, resulting in ##{\Lambda^{i}}_j##. In the Ricci calculus it's crucial to have all the indices placement in both vertical as well as horizontal direction accurate.

The only exception are symmetric tensors, i.e., such that for both indices contravariant or both indices covariant the components don't change under exchange of these arguments, because if
$$T_{ij}=T_{ji}$$
then
$${T^k}_j=\eta^{ki} T_{ij} = \eta^{ki} T_{ji} = {T_j}^k.$$
Then, obviously the horizontal placement is irrelevant.

 

[vanhees71]

$$ {\eta}^{\mu\nu} = {\Lambda}^{\mu}_{\sigma} {\Lambda}^{\nu}_{\gamma}{\eta}^{\sigma\gamma} $$
This expression. I am trying to write it without the indices, but don't know where they go on the lambda tensor

That's exactly the prime example, where physics texts written in such a sloppy way, get useless!

 

[pervect]

$$ {\eta}^{\mu\nu} = {\Lambda}^{\mu}_{\sigma} {\Lambda}^{\nu}_{\gamma}{\eta}^{\sigma\gamma} $$
This expression. I am trying to write it without the indices, but don't know where they go on the lambda tensor

The object in question, ##\Lambda##, are transformations.

There are unfortunately multiple conventions, but I use the conventions in MTW, and re-write expressions that don't follow the convention I use so that the follow the conventions I use. The conventions I use are as follows, from the text "Gravitation" by MTW, sec $2.9, pg 66.

One never need to memorize the index positions in these transformation laws. One need only line the indices up so that (1) free indices on each side of the equation are in the same position; and (2) summed indices appear once up and once down. Then all will be correct! (Note:the indices on ##\Lambda## always run from "northwest" to "southeast").

If I see something that doesn't meet these rules, I re-write it so that it does. A corollary of this is that transformation objects like ##\Lambda## are written in tensor notation, but aren't

 

[vanhees71]

The index position is crucial, or how else would you make sense of an equation like
[EDIT: Corrected in view of #12]
$${(\Lambda^{-1})^{\mu}}_{\nu} ={\Lambda_{\nu}}^{\mu} = \eta^{\mu \rho} \eta_{\nu \sigma} {\Lambda^{\sigma}}_{\rho}?$$
In matrix notation it's saying that
$$\hat{\Lambda}^{-1}=\hat{\eta} \hat{\Lambda}^{\text{T}} \hat{\eta}.$$

 

[weirdoguy]

@vanhees71 something is wrong with the indices: there is only one ##\rho## - the lowest ##\mu## on the right should also be ##\rho##.

 

[vanhees71]

You are right. It's not only the positioning of the indices that's important but also their naming ;-). I've corrected the formula.