Index (Einstein summation) notation question

Question

Question #1:

Lost as to how the second equality in the following equation holds —

$$\frac{\partial}{\partial \tau} (A^2) = \frac{\partial}{\partial \tau} (\eta_{\mu\nu}A^\mu A^\nu) = 2\eta_{\mu\nu}A^\mu \frac{\partial A^\nu}{\partial \tau}.$$

Seems to me like there is a similar trick to like when you use the Euler Lagrange equation to get the Klein Gordan equation,

$$(\partial_\mu \phi)^2=(\partial_\mu \phi)(\partial^\mu \phi)=\eta^{\mu\nu}\partial_\mu \partial_\nu \phi.$$

But I can't put my finger on it.

Answer to question #1 taking eranreches and G. Smith's tips into account:

$$ \begin{eqnarray}\frac{\partial}{\partial \tau} (A^2) &=& \frac{\partial}{\partial \tau} (\eta_{\mu\nu}A^\mu A^\nu) \\ &=& \frac{\partial}{\partial \tau} (\eta_{\mu\nu}A^\mu) A^\nu + \eta_{\mu\nu}A^\mu \frac{\partial}{\partial \tau}(A^\nu)\\ &=& \eta_{\mu\nu}\frac{\partial}{\partial \tau} (A^\mu) A^\nu + \eta_{\mu\nu}A^\mu \frac{\partial}{\partial \tau}(A^\nu) \\ &=& \eta_{\nu\mu}\frac{\partial}{\partial \tau} (A^\nu) A^\mu + \eta_{\mu\nu}A^\mu \frac{\partial}{\partial \tau}(A^\nu) \\ &=& 2\eta_{\mu\nu}A^\mu \frac{\partial}{\partial \tau} (A^\nu) \\ \end{eqnarray}$$

where in the 2nd to last line the index $\mu$ has been renamed to $\nu$, and in the last line, we used the fact that $\eta_{\mu\nu}$ is symmetric.

Question #2:

I have the following "rules" for index-notation "transformations":

• $A'^\mu = \Lambda^\mu_\nu A^\nu$
• $A^\mu = \delta^\mu_\nu A^\nu$
• $\partial^\mu = \eta^{\mu\nu} \partial_\nu$

I am perfectly comfortable with the top equation, but my question is, what is the difference between the 2nd and 3rd? I know that the $\delta^\mu_\nu$ and the $\eta^{\mu\nu}$ are difference (one is the 4x4 identity matrix, and the other the metric tensor with the tt component -1, or vice-versa depending on your metric preference), but I'm not sure in what circumstances to use each when they seem to be used in similar circumstances, i.e. to raise of lower an index.

My only guess is that, one needs to use $\delta^\mu_\nu$ when working with 4-vectors like $A^\mu$ to keep all the components the same. However, I'm confused why then (by implication) we would WANT to change one of the components of the derivative as with using the metric tensor to raise/lower such as with the 3rd equation.

Answer 1

Q1: The second equality is just $\frac{{\rm d}}{{\rm d}\tau}\left(fg\right)=\frac{{\rm d}f}{{\rm d}\tau}g+f\frac{{\rm d}g}{{\rm d}\tau}$ plus re-indexing.

Q2: I would suggest you to post each of your questions separately.

Answer 2

Re #1: Use the fact that $\eta^{\mu\nu}$ is symmetric.

Re #2: The first equation transforms the vector between two different inertial frames. The second and third equations are identities within one reference frame.