I have to prove that $A_{\mu} B^{\mu}$ is Lorentz invariant, and I'd like to check my understanding with you if you don't mind.
My first question is about the definition of Lorentz invariance. Does it mean the following:
$A'_{\mu} B'^{\mu} = A_{\mu} B^{\mu}$ ?
If so, I think I have an idea how to prove it but I might get confused with the indexes. I would express $A'_{\mu}$ and $B'^{\mu}$ as
$A'_{\mu} = a_{\mu}^{\phantom{\mu} \nu} A_{\nu}$ and
$B'^{\mu} = a^{\mu}_{\phantom{\mu} \nu} B^{\nu}$.
Is that correct? Moreover, if I was to calculate the product above, can I use the same indices for both terms? That is:
$A'_{\mu} B'^{\mu} = a_{\mu}^{\phantom{\mu} \nu} A_{\nu} a^{\mu}_{\phantom{\mu} \nu} B^{\nu}$?
If so, I can rewrite the following as:
$A'_{\mu} B'^{\mu} = a_{\mu}^{\phantom{\mu} \nu} A_{\nu} a^{\mu}_{\phantom{\mu} \nu} B^{\nu} = \underbrace{a_{\mu}^{\phantom{\mu} \nu} a^{\mu}_{\phantom{\mu} \nu}}_{= \delta_{\nu}^{\nu}} A_{\nu} B^{\nu} = 1 \cdot A_{\mu} B^{\mu} = A_{\mu} B^{\mu}$.
Does all of that makes sense? I'm still trying to get a feel of this new notation for me.
Thank you very much in advance for your comments.
Julien.
Lorentz invariance means that the value does not change under Lorentz transformations, so we need to show that $A'_\mu B'^\mu=A_\mu B^\mu$. Let the transformation be given by the matrix $a^\nu_\mu$$~$ ($v'^\nu=a^\nu_\mu v^\mu$ for any vector $v^\mu$).
Then $$A'_\mu \,B'^\mu=a^\nu_\mu \, A_\nu \, a^\mu_\sigma\, B^\sigma =A_\nu \,a^\nu_\mu\, a^\mu_\sigma\, B^\sigma = A_\nu \, \delta^\nu_\sigma B^\sigma = A_\nu \, B^\nu.$$
So in fact, your approach was almost correct, but you need to introduce new indices for each transformation.
Regarding your comment: The objects $a^\mu_\nu$ denote the components of the matrix (they are numbers!), therefore their order can always be exchanged. The order of the tensors is encoded in the summation indices.
For example, let $(A)^i_j=a^i_j$, $(B)^i_j=b^i_j$, $(C)^i_j=c^i_j$, then $(ABC)^i_j= a^i_k b^k_l c^l_j$ whereas $(ACB)^i_j=a^i_k c^k_l b^l_j = a^i_k b^l_j c^k_l$.