Tensors of Relativity: Inner vs Outer Indices

[praharmitra]

I am a bit confused with tensors here.

now i know that $$\Lambda$$, the transformation matrix has a different meaning when I write

$$\Lambda^\mu\ _{\nu}$$ and when I write $$\Lambda_{\nu}\ ^\mu$$

One is the mu-nu th element of $$\Lambda$$ and the other is the mu-nu th element of $$\Lambda^{-1}$$.

Is it the same for tensors. I mean is $$F^\mu\ _{\nu}$$ different from $$F_{\nu}\ ^\mu$$ ?

If there is a difference of writing inner and outer indices what is it?

 

[Peeter]

From what I've seen, inner and outer indexes is only a convention used to distinguish the two cases, and there are at least two other conventions:

For example, in:

http://www.teorfys.uu.se/people/minahan/Courses/SR/tensors.pdf

both the rotation and the inverse rotation are written lower index out (but primes are used to distinguish if forward or inverse rotation is implied).

An alternate convention appears to be to just use a different symbol for the forward and inverse transformations. In:

http://qmplus.qmul.ac.uk/pluginfile.php/301050/mod_resource/content/2/EMT7new.pdf

That author uses

$${\Lambda^\mu}_\nu$$

and:

$${(\Lambda^{-1})^\alpha}_\beta$$

... everything I know about tensors is self taught, so I'm no authority, but I'm pretty sure that I've seen all three variations of index conventions used in various papers I've attempted to read.

 

[praharmitra]

I understand its just a notation. I guess its you did not understand my question. I am asking whether the same applies for tensors. Remember, $$\Lambda$$ is NOT a tensor. It is only the transformation matrix.

However for tensors, like the electromagnetic field tensor T, does the outer and inner indices have any meaning?

 

[Peeter]

I would give those tensors the following meaning:

$$\begin{align*}<br /> F^{\mu\nu} &= \partial^{\mu} A^{\nu} - \partial^{\nu} A^{\mu} \\<br /> F_{\mu\nu} &= \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu} \\<br /> {F^{\mu}}_{\nu} &= \partial^{\mu} A_{\nu} - \partial_{\nu} A^{\mu} \\<br /> {F_{\mu}}^{\nu} &= \partial_{\mu} A^{\nu} - \partial^{\nu} A_{\mu}<br /> \end{align}$$

But I don't have any texts that cover maxwell's equations in tensor form to confirm. The statement above is from personal notes where I was "translating" between the bivector and tensor forms of maxwell's equations:

http://sites.google.com/site/peeterjoot/geometric-algebra/maxwell_to_tensor.pdf

No guarantee of correctness, so you will have to use your own judgement to verify if this seems right.

 

[Meir Achuz]

I understand its just a notation. I guess its you did not understand my question. I am asking whether the same applies for tensors. Remember, $$\Lambda$$ is NOT a tensor. It is only the transformation matrix.

However for tensors, like the electromagnetic field tensor T, does the outer and inner indices have any meaning?

1. $$\Lambda$$ IS a tensor, as well as defining the LT.
2. Raising and lowereing indices are performed by the metric tensor $$g_{\mu\nu}$$,
and gives different components. It does not always produce an inverse.
This just happens to happen for \Lambda.