How would you write this in Index notation?


So a bit of an off-topic question, but I think a lot of you here have an interest in math. If you had an expression you wanted to put in index notation given for one value of i as:


Where x is a vector with i = 1 → 3:

Where superscripts a and b refer to particles, for example see this picture:

image - Picture source no need to click.

How would you write this in index notation using Einstein summation convention? I’ve been trying something like this:
Where the nominator seems fine to me, but the denominator I just don’t see how to make it “sum” as I want.

It is primarily the subscripts I want help with, the superscripts are fine to me.

Kind regards

As far as I know, the Einstein summation is primarily used for linear expressions.
Hence, \lambda^i b_i = \sum_i \lambda^i b_i.

But, summation is not automatically performed on addition of terms with indices, i.e. \lambda^i + \gamma^i \neq \sum_i (\lambda^i + \gamma^i).
Instead, this expression makes sense if combined with the basis, namely (\lambda^i + \gamma^i) b_i = \sum_i (\lambda^i + \gamma^i) b_i

In Riemannian geometry, one could use a Riemannian metric (g) to compute the length of a tangent vector x^i b_j as \sqrt{ g_{ij} x^i x^j } = \sqrt{ \sum_{ij} g_{ij} x^i x^j }.

Now, to compute the distance between two points in Riemannian geometry is a bit envolved (geodesics), which is maybe the reason why finding a good expression is a bit unintuitive, but you could write

\sqrt{ \delta_{ij} (x^i - y^i) (x^j - y^j) }

where \delta_{ij} = 1 if i=j and otherwise \delta_{ij} = 0 (which are the coordinates of the Riemannian tensor for Euclidean geometry with respect to the cannonical basis).

For the whole expression, you could write

\frac{x^i - y^i}{\sqrt{ \delta_{\alpha \beta} (x^\alpha - y^\alpha) (x^\beta - y^\beta) }}.

Notes on notation: Everyone uses different notation. I must admit that for me the Einstein summation makes only sense if one differentiates between covariant and contravariant coordinates. That’s why I use upper and lower indices a bit different than you.

(Also, you are most likely aware of it, but is probably a better place where you could get more answers for this type of question.)