Trying to learn tensor algebra

[Rick16]

TL;DR

relabeling indices

This is exercise 1.8.3 from Foster & Nightingale:

Show that if $\sigma_{ab} = \sigma_{ba}$ and $\tau^{ab} =-\tau^{ba}$ for all $a$, $b$, then $\sigma_{ab}\tau^{ab}=0$.

I began writing down $\sigma_{ab}\tau^{ab}=\sigma_{ba}(-\tau^{ba})=-\sigma_{ba}\tau^{ba}$. Here I got stuck and looked at the solution in the back of the book. In the solution, the authors use the fact that the suffixes are dummy indices and can be relabeled: $-\sigma_{ba}\tau^{ba}=-\sigma_{ab}\tau^{ab}$. So they end up with $\sigma_{ab}\tau^{ab}=-\sigma_{ab}\tau^{ab}$, which can only be true if $\sigma_{ab}\tau^{ab}=0$.

It seems to me that this relabeling clashes with the definition of the tensors in the problem statement. If I can simply relabel the indices, then what keeps me from relabeling them right away? The skew-symmetric tensor is defined as $\tau^{ab}=-\tau^{ba}$. Since the indices are dummy indices, I relabel them and write $-\tau^{ba}=-\tau^{ab}$ and I end up with $\tau^{ab}=-\tau^{ba}=-\tau^{ab}\Rightarrow \tau^{ab}=0.$ This would imply that all skew-symmetric tensors are zero, which is certainly not true.

So why can I apply the relabeling in the case of $\sigma_{ba}\tau^{ba}$, but not in the case of $\tau^{ba}$? I don’t really see the difference.

 

[Rick16]

Looking at my question after I posted it, I just noticed something. The indices in $\tau^{ba}$ are not dummy incides, because dummy incides are summation indices and there is no summation in this expression. Nevertheless, I fail to see why the relabing is possible in the case of $\sigma_{ba}\tau^{ba}$. It still seems to me that it clashes with the definitions of the tensors.

 

[weirdoguy]

You can relabel if you are summing up. If you don't see it, try do write down the sum explicitly. $\sigma_{ab}\tau^{ab}$ is a sum, $\tau^{ab}$ is not.

 

[Rick16]

You mean something like this:

$\begin{align}\sigma_{ab}\tau^{ab}&=\sigma_{11}\tau^{11}+\sigma_{12}\tau^{12}+\sigma_{21}\tau^{21}+\sigma_{22}\tau^{22} \nonumber \\ &=-\sigma_{ba}\tau^{ba} \nonumber \\ &=-\sigma_{11}\tau^{11}-\sigma_{21}\tau^{21}-\sigma_{12}\tau^{12}-\sigma_{22}\tau^{22} \nonumber \\ &=-(\sigma_{11}\tau^{11}+\sigma_{21}\tau^{21}+\sigma_{12}\tau^{12}+\sigma_{22}\tau^{22}) \nonumber \\ &=-(\sigma_{11}\tau^{11}+\sigma_{12}\tau^{12}+\sigma_{21}\tau^{21}+\sigma_{22}\tau^{22}) \nonumber \\ &=-\sigma_{ab}\tau^{ab} \nonumber \end{align}$

Even if this looks okay, I still have a problem with it. Since per definition $\sigma_{ab}=\sigma_{ba}$ and $\tau^{ab}=-\tau^{ba}$, then per definition $\sigma_{ab}\tau^{ab}=-\sigma_{ba}\tau^{ba}$, and relabeling the indices seems to invalidate this definition.

 

[weirdoguy]

then what keeps me from relabeling them right away?

The fact that you'll get nonsense results

I don’t really see the difference.

The difference is that $\sigma_{ab}\tau^{ab}$ is a shorthand for a double sum: $$\sum_{a,b=1}^{2}\sigma_{ab}\tau^{ab}$$
so each index takes all of the values. Thus you can name them whatever you like:
$$\sum_{a,b=1}^{2}\sigma_{ab}\tau^{ab}=\sum_{k,l=1}^{2}\sigma_{kl}\tau^{kl}=\sum_{G,H=1}^{2}\sigma_{GH}\tau^{GH}=\sum_{\circ,\square =1}^{2}\sigma_{\circ\square}\tau^{\circ\square}=\sum_{b,a=1}^{2}\sigma_{ba}\tau^{ba}$$

Since the indices are dummy indices, I relabel them and write

You can't relabel indices in one expression, you have to have some equality: $\tau^{ab}=-\tau^{ba}$ and then relabelling would give $\tau^{ba}=-\tau^{ab}$.

 

[Ibix]

In 2d, $\sigma^{ab}=-\sigma^{ba}$ is a shorthand way of writing$$\begin{eqnarray*}
\sigma^{00}&=&-\sigma^{00}\\
\sigma^{01}&=&-\sigma^{10}\\
\sigma^{10}&=&-\sigma^{01}\\
\sigma^{11}&=&-\sigma^{11}
\end{eqnarray*}$$If you swap indices on one side you get $\sigma^{ab}=-\sigma^{ab}$, which expands to$$\begin{eqnarray*}
\sigma^{00}&=&-\sigma^{00}\\
\sigma^{01}&=&-\sigma^{01}\\
\sigma^{10}&=&-\sigma^{10}\\
\sigma^{11}&=&-\sigma^{11}
\end{eqnarray*}$$That is a different set of constraints. Switching indices changed something.

However, if you are summing over indices you are free to swap indices. The expression $\tau_{ab}\sigma^{ab}=\tau_{ba}\sigma^{ba}$ is valid because both sides expand to $$\tau_{00}\sigma^{00}
+\tau_{01}\sigma^{01}
+\tau_{10}\sigma^{10}
+\tau_{11}\sigma^{11}$$Switching indices changed nothing.

The general point is that indices you sum over can be freely relabelled because the implied sum contains all combinations. But free indices imply you are matching components between terms, and relabelling them changes the relationship between the terms.

 

[Orodruin]

These may also be somewhat helpful: https://www.physicsforums.com/insights/the-10-commandments-of-index-expressions-and-tensor-calculus/

 

[haushofer]

Compare with integrals. Integrating f(x) over dx is the same as f(y) over dy (with boundaries unchanged). The x and y are dummy variables; they don't appear in the final answer and are merely intermediate. But f(x) depends on x, and f(y) depends on y. So on their own, x and y now are not dummie variables.

 

[Rick16]

Thank you! I will particularly pay attention to the first commandment. I will write it down in large friendly letters.