Symmetric rank-2 tensor, relabelling of indices? (4-vectors)

[rwooduk]

Homework Statement

Homework Equations

Relabelling of indeces, 4-vector notation

The Attempt at a Solution

The forth line where I've circled one of the components in red, I am unsure why you can simply let ν=μ and μ=v for the second part of the line only then relate it to the first part and add them. If you are chosing "dummy indices" as suggested in the image then wouldn't they have to be independent of the first parts indices?

I'm unsure of what is going on here, any ideas would be really helpful, thanks.

 

[Chestermiller]

You are familiar with the summation convention, correct. Just write out the sums both ways, and you will see that they are identical.

Chet

 

[rwooduk]

You are familiar with the summation convention, correct. Just write out the sums both ways, and you will see that they are identical.

Chet

hmm, i think I see what you are getting at, it's because it is symmetric.

when you say write out the sums you are saying if i write out the summation over v=0,1,2,3 and μ=0,1,2,3?

thanks for the reply

 

[Chestermiller]

Yes. With repeated dumny indices, it doesn't matter which letter of the alphabet you use, as long as it's not the same letter as another repeated index.

Chet

 

[rwooduk]

Yes. With repeated dumny indices, it doesn't matter which letter of the alphabet you use, as long as it's not the same letter as another repeated index.

Chet

Yes, I got them to equal, that's very helpful, thank you!