Tensor algabra, dummy indices manipulation

[roberto85]

Homework Statement

Show by manipulating the dummy indices, that

(Z\underline{abc} + Z\underline{cab} + Z\underline{bca})X\overline{a}X\overline{b}X\overline{c} = 3Z\underline{abc}X\overline{a}X\overline{b}X\overline{c}

Homework Equations

The Attempt at a Solution

This question comes from Ray D'Inverno's book - Introducing Einstein's Relativity and is on page 67 ex. 5.3

 

[AEM]

Homework Statement

Show by manipulating the dummy indices, that

(Z\underline{abc} + Z\underline{cab} + Z\underline{bca})X\overline{a}X\overline{b}X\overline{c} = 3Z\underline{abc}X\overline{a}X\overline{b}X\overline{c}

Homework Equations

The Attempt at a Solution

This question comes from Ray D'Inverno's book - Introducing Einstein's Relativity and is on page 67 ex. 5.3

The thing to realize about dummy indices is that they have no intrinsic meaning except that
they are to be summed over. Thus, in the product

z_{abc}x^a x^b x^c

you can replace a with d and the meaning is the same. You might want to systematically replace each of your given letters with another letter and shuffle the x's around so that you can change back to a,b,c in the order you want.

 

[Gunthi]

The thing to realize about dummy indices is that they have no intrinsic meaning except that
they are to be summed over. Thus, in the product

z_{abc}x^a x^b x^c

you can replace a with d and the meaning is the same. You might want to systematically replace each of your given letters with another letter and shuffle the x's around so that you can change back to a,b,c in the order you want.

Aren't tensors non-commutative? If so you couldn't 'shuffle' the x's as you say right?
I'm trying to solve this exercise but I get a little confused in what one is allowed or not to do with dummy indices...
I can't seem to get the indices in the right order because if I change an 'a' with a 'b' in the Z tensor, then I must change the letters in the x's. Or am I missing something?

Thanks in advance...

 

[TerryW]

Hi Roberto85,

I see you are having problems with the LateX editor on the forum too. I've given up and now have a LateX editor which produces nice PDFs like the one attached. Hope you find it useful.

Regards


TerryW

 

[TerryW]

Hi Roberto,

A few additional thoughts.


Regards


TerryW

 

[roberto85]

Hi Roberto,

A few additional thoughts.


Regards


TerryW

Hi Terry, thankyou so much for such a helpful and detailed post. I've saved the pdf's for future reference because i gave up on tensors for my relativity course but i still have the book and fully intend to learn about relativity in the future. :)

 

[TerryW]

Hi Roberto,

Best of luck. I never did GR in my undergrad years but always wanted to understand it. I tried a couple of times during my working life to get to grips with it but never really had the time. I'm retired now and at last have had the chance to devote some time to it. I'm now up to Chapter 18 and have managed to crack all the problems except for about 4. I t was 8 at one time but I keep going back and having another look at them. As you go on, you just get a better feel for what is going on and then the answers pop up! So it might be many years before you get around to it but I'm sure you'll find it's worth it in the end.RegardsTerryW