I Tensor Calculus (Einstein notation)

[paperplane]

TL;DR Summary

How to sum over indices when they aren't being contracted?

Hello,

I realize this might sound dumb, but I'm having such a hard time understanding Einstein notation. For something like ∂_uF_v - ∂_vF_u, why is this not necessarily 0 for tensor F_u? Since all these indices are running through the same values 0,1,2,3?

 

[malawi_glenn]

Einstein notation for summation is meant to be done for the same term, here you have two terms. There is no implicit summation in $\partial_\mu F_\nu - \partial_\nu F_\mu$.

Let's call it $G_{\mu \nu}$ i.e. $G_{\mu \nu} = \partial_\mu F_\nu - \partial_\nu F_\mu$. We have
$G_{00} = \partial_0 F_0 - \partial_0 F_0 = 0$
$G_{10} = \partial_1 F_0 - \partial_0 F_1 = - G_{01}$
$G_{11} = \partial_1 F_1 - \partial_1 F_1 = 0 = G_{22} = G_{33}$
$G_{23} = \partial_2 F_3 - \partial_3 F_2 = - G_{32}$
and so on.

Now, lets assume I contract $G_{\mu \nu}$ with $\phi^\mu$, we have due to Einstein summation convention
$\phi^\mu G_{\mu \nu} = \phi^0G_{0 \nu} + \phi^1G_{1\nu} + \phi^2G_{2 \nu} + \phi^3G_{3 \nu}$

Let's define $\psi_\nu = \phi^\mu G_{\mu \nu}$.
We have
$\psi_0 = \phi^\mu G_{\mu 0} = \phi^0G_{0 0} + \phi^1G_{10} + \phi^2G_{20} + \phi^3G_{30} =\phi^1G_{10} + \phi^2G_{20} + \phi^3G_{30}$
$\psi_1 = \phi^\mu G_{\mu 1} = \phi^0G_{0 1} + \phi^1G_{11} + \phi^2G_{21} + \phi^3G_{31} =\phi^0G_{0 1} + \phi^2G_{21} + \phi^3G_{31}$
and so on

 

[paperplane]

Ah I understand now, thank you!

 

[malawi_glenn]

Ah I understand now, thank you!

I updated my reply above with some more examples.

 

[PeroK]

TL;DR Summary: How to sum over indices when they aren't being contracted?

Hello,

I realize this might sound dumb, but I'm having such a hard time understanding Einstein notation. For something like ∂_uF_v - ∂_vF_u, why is this not necessarily 0 for tensor F_u? Since all these indices are running through the same values 0,1,2,3?

Einstein notation omits two standard elements of mathematical notation: the summation symbol ($\sum$) and the universal quantifier ($\forall$). In the above examples we have:
$$G_{\mu \nu} = \partial_\mu F_\nu - \partial_\nu F_\mu$$In full notation this would be:
$$\forall \mu, \nu: G_{\mu \nu} = \partial_\mu F_\nu - \partial_\nu F_\mu$$Note that this is actually $16$ equations! (One for every combination of $\mu = 0, 1,2,3$ and $\nu = 0, 1,2,3$.) And:
$$\psi_\nu = \phi^\mu G_{\mu \nu}$$In full notation this would be:
$$\forall \nu: \psi_\nu = \sum_{\mu = 0}^{3} \phi^\mu G_{\mu \nu}$$And that is four equations.