# einstein summation notation

• Nov 23rd 2007, 06:57 AM
johnbarkwith
einstein summation notation
Does any body know how to expand HijHij and write with the summation signs??
• Nov 23rd 2007, 07:09 AM
topsquark
Quote:

Originally Posted by johnbarkwith
Does any body know how to expand HijHij and write with the summation signs??

Edit: Sorry for changing the post so many times. I keep forgetting the "KISS" method: "
"Keep It Simple Stupid."

You have two repeated indices.

$H_{ij}H_{ij} = \sum_i \sum_j H_{ij} H_{ij}$

-Dan
• Nov 23rd 2007, 07:10 AM
johnbarkwith
thanks alot, thats what i thought, the order of summation doesnt matter does it?
• Nov 23rd 2007, 07:15 AM
topsquark
Quote:

Originally Posted by johnbarkwith
thanks alot, thats what i thought, the order of summation doesnt matter does it?

(Make sure you double check my last post. I screwed up the original.)

No, the order of summation doesn't matter, since all the components of the tensors are scalars.

-Dan
• Nov 24th 2007, 06:35 AM
topsquark
Okay, I'm going to change my answer one last time. (Doh) (If you feel I need negative rep for changing my answer so many times, go ahead.) This time I took some time to look things up so I can be accurate.

I've been thinking about this and I think what has been confusing me so much is that there really isn't a way to apply the summation convention to $H_{ij}H_{ij}$ in general.

When we apply the summation convention we are using the metric tensor to essentially perform an inner product. (Technically an inner product is either positive or negative definite, which most metrics aren't.) The rule for multiplying two vectors together, for example is
$x_{\mu} \cdot x^{\mu} = g_{\mu \nu}x_{\mu}x_{\nu} \equiv \sum_{\mu} \sum_{\nu} g_{\mu \nu}x_{\mu}x_{\nu}$

Note the position of the indices: The upper index (if I remember right) is a contravariant index and the lower is a covariant index. The only time the summation convention is used is when taking a product of co- and contravariant indices.

So technically we can't do a summation for $H_{ij}H_{ij}$. However there is one kind of space that we can get away with this: in Euclidean space there is no distinction between covariant and contravariant indices. So in an Euclidean space only we can write
$H_{ij}H_{ij} = \sum_i \sum_j H_{ij}H_{ij}$.

-Dan