Difference between mixed tensor notation

[blankvin]

Hi,

Can someone explain the difference between, say, \Lambda_\nu^\mu, {\Lambda_\nu}^\mu and {\Lambda^\mu}_\nu (i.e. the positioning of the contravariant and covariant indices)?

I have found: http://books.google.ca/books?id=lLP...onepage&q=mixed tensor index position&f=false but maybe someone can shed some more light on this for me.

 

[WWGD]

Hi,

Can someone explain the difference between, say, \Lambda_\nu^\mu, {\Lambda_\nu}^\mu and {\Lambda^\mu}_\nu (i.e. the positioning of the contravariant and covariant indices)?

I have found: http://books.google.ca/books?id=lLP...onepage&q=mixed tensor index position&f=false but maybe someone can shed some more light on this for me.

I assume the issue is that you have more than one covariant/contravariant index in your tensor, and the different $\Lambda$ tell you which index you are making use of, i.e., are you
using the 1st covariant index together with the second contravariant, or the 2nd covariant together with the third contravariant, etc.

 

[Matterwave]

Hi,

Can someone explain the difference between, say, \Lambda_\nu^\mu, {\Lambda_\nu}^\mu and {\Lambda^\mu}_\nu (i.e. the positioning of the contravariant and covariant indices)?

I have found: http://books.google.ca/books?id=lLP...onepage&q=mixed tensor index position&f=false but maybe someone can shed some more light on this for me.

The positions of the indices indicate to you which is the first and second. For a symmetric tensor, it wouldn't matter really, but for a non-symmetric tensor, it might matter which index you raise/lower.

For example: $F^{\mu}_{~~\nu}=g^{\mu\rho}F_{\rho\nu}$ versus $F^{~~\mu}_\nu=g^{\mu\rho}F_{\nu\rho}$