GR - Trying to grasp index notation (Levi Civita)

[Clever-Name]

Homework Statement

I'm trying to grasp how the indices are listed when writing out multiple vector products or divergences or gradients, etc. I'm working with 'An Introduction to General Relativity' by Hughston and Tod.

Homework Equations

[tex]
A\wedge B = \varepsilon_{ijk}A_{j}B_{k}
[/tex]

[tex]
[A,B,C] = \varepsilon_{ijk}A_{i}B_{j}C_{k}
[/tex]

The Attempt at a Solution

To use a problem as an indicator of my struggles, Problem 2.1 in the text I mentioned above states:

Using index notation, show that

[tex]
(A\wedge B)\wedge (P\wedge Q) = -A[B,P,Q] + B[A,P,Q] \\
(A\wedge B)\wedge (P\wedge Q) = (\varepsilon_{ijk}A_{j}B_{k})\wedge (\varepsilon_{kpq}P_{p}Q_{q}) = ?
[/tex]

This is where I'm confused, where am I allowed to repeat the indices? How would you go about first writing out the expression?

The text writes:
[tex]
A\wedge (B\wedge C) = \varepsilon_{ijk}A_{j}(\varepsilon_{kpq}B_{p}C_{q})
[/tex]

So I see that the k index is repeated, but why were p and q used? I regurgitated that above in the problem but how do I then take care of the middle [itex] \wedge [/itex], do I use 2 unique indices but repeat j??

Help!

 

[Reptillian]

Theres a sum over repeated indices right? You're allowed to change indices which are summed over since they're dummy indices, but not ones that aren't...right? So,

(A ∧ B)∧(P∧Q)=ε_ijk(ε_mjkA_jB_k)_j(ε_njkP_jQ_k)_k

could be a good starting point...Maybe...it has been a long time since I've worked through a problem like this. Sometimes it helps to write out a few terms in the expression to get an idea of how to proceed.

 

[Muphrid]

What's important is that you begin to understand the difference between free indices and repeated indices. For instance,

[tex]A \wedge B = \epsilon_{ijk} A_j B_k[/tex]

Really means that

[tex]A \wedge B = C \quad C_i = \epsilon_{ijk} A_j B_k[/tex]

An index should never be repeated more than twice. It doesn't matter what letters are used to represent the indices. I could use [itex]\epsilon_{lmn} A_m B_n[/itex] instead. What's important is that the structure is the same: the middle index of the Levi-Civita matches with the index of A, and so on.

I take from the form of Levi-Civita that this must be 3D; that's good because otherwise, these guys' definitions of wedge products make no sense.

 

[Clever-Name]

Yes, Muphrid, it's 3D

Ok, I think I've got it...

[tex]
\begin{align*}
[(A\wedge B)\wedge(P\wedge Q)]_{i} &= \varepsilon_{ijk}(A\wedge B)_{j}(P\wedge Q)_{k} \\
&= \varepsilon_{ijk} \varepsilon_{jlm} A_{l}B_{m} \varepsilon_{knp} P_{n}Q_{p} \\
&= -\varepsilon_{jik} \varepsilon_{jlm} A_{l}B_{m} \varepsilon_{knp} P_{n}Q_{p} \\
&= -(\delta_{il}\delta_{km} - \delta_{im}\delta_{kl})A_{l}B_{m} \varepsilon_{knp} P_{n}Q_{p}\\
&= (-A_{i}B_{k} + A_{k}B_{i}) \varepsilon_{knp} P_{n}Q_{p}\\
&= -A_{i} \varepsilon_{knp} B_{k}P_{n}Q_{p} + B_{i} \varepsilon_{knp} A_{k}P_{n}Q_{p} \\
&= (-A[B,P,Q] + B[A,P,Q])_{i}
\end{align*}
[/tex]

Does that look right? All these indices are confusing me.

 

[Reptillian]

My advice was terrible, the work looks good to me though for what it's worth. :)

 

[Muphrid]

That looks good, Clever-Name. Seeing as this is a GR book, though, I hope they quickly get away from the idea of [itex]A \wedge B[/itex] being a vector, because when you get to 3+1D it absolutely will not be. It's going to be a two-index antisymmetric tensor representing the plane that the two vectors span. For now, however, working in 3D, you seem to have the correct grasp of what you're being taught.

Indices are a pain, which is why I try never to deal with them. (It's actually a good exercise to work this problem without indices at all.)

 

[Clever-Name]

Great! Thanks for the input! The book isn't very good at explaining the process, I'll have my work cut out for me this summer I guess :/

 

[Muphrid]

It may or may not be helpful to consider that index notation is often just a shorthand for components where the basis vectors are left out, but if you put the basis vectors back in, you can reconnect a lot of index notation back to traditional vector algebra that you already know. For instance,

[tex]A \cdot B = A_i B_j (\vec e_i \cdot \vec e_j) = A_i B_j \delta_{ij} = A_i B_i[/tex]

That's a trivial example, of course, but sometimes it's helpful for realizing where Kronecker deltas are Levi-Civitas may end up popping into give you the freedom to cut down on some indices.