Confusion with Lorentz indices notation

Question

Apologies in advance if this question has been asked before (if it has, I can't find it).

I am really confused with the right-left index notation of the Lorentz matrix.

In the very first exercise of Voja Radovanovic's problem book on QFT, one has to show that $\Lambda^Tg\Lambda=g$. For this, Radovanovic uses $x'^2 = x^2$ to arrive at the following equation:

$$\Lambda^\mu{}_\rho g_{\mu\nu}\Lambda^\nu{}_\sigma = g_{\rho\sigma}.$$

Now, at this point, I would naively claim that "well you can just sum up over the indices" which means that this is a simple matrix multiplication and thus $\Lambda g \Lambda = g$, which is of course not correct.

Apparently, to get an actual matrix multiplication, one has to write $$(\Lambda^T)_\rho{}^\mu g_{\mu\nu}\Lambda^\nu{}_\sigma = g_{\rho\sigma},$$ or simply $\Lambda^Tg\Lambda = g$.

So, why does $(\Lambda^T)_\rho{}^\mu g_{\mu\nu}\Lambda^\nu{}_\sigma = g_{\rho\sigma} \Longleftrightarrow\Lambda^Tg\Lambda = g$ but not $\Lambda^\mu{}_\rho g_{\mu\nu}\Lambda^\nu{}_\sigma = g_{\rho\sigma} \Longleftrightarrow\Lambda g\Lambda = g$?

Answer 1

You need to bear in mind the following two definitions about matrices and matrix products:

• The matrix elements $a_{ij}$ are defined in such a way that the left index $(i)$ is the row number, and the right index $(j)$ is the column number.
  
  _{(image from Wikipedia - Matrix (mathematics))}
• The matrix multiplication $C=AB$ is defined to be $$c_{ij}=\sum_k a_{ik}b_{kj}$$ i.e. the summing index (here $k$) is the right index of the first matrix $(A)$ and the left index of the second matrix $(B)$. (Mnemonic: the two $k$ are next to each other.)

By applying the rules from above to the two matrix products involved in $$\Lambda^Tg\Lambda = g$$ you find $$(\Lambda^T)_\rho{}^\mu g_{\mu\nu}\Lambda^\nu{}_\sigma = g_{\rho\sigma}.$$ (Summing indices are $\mu$ and $\nu$.)

Answer 2

Having vague memories, but I have an idea what's going on.

Most often I see a metric conversion written as $g_{\alpha \beta}=g_{\mu \nu}\lambda^\mu_\alpha \lambda^\nu_\beta$ and this is not to be confused with the matrix multiplication $g\lambda \lambda$.

They are closely related though. For orthogonal matrix A, $A^T=A^{-1}$ by definition.

So $A^TgA=A^{-1}gA$, something similar to what you have written.

Let's depart from covariant and contravariant indices for now. If we have matrix $A_{mn}$, where m is the row and n is the column and maintain the implication that repeated indices implies summation over the latin index, then $AB_{mp}=A_{mn}B_{np}$

To make that work out with covariant/contravariant indices you probably need $A^m_p=A^m_nB^n_p. $ Note $A^p_m \ne A^m_p$ in general. $(AB)^T=B^TA^T$. I think a combination of these is in play here.

Multiplication isn't generally commutive with matrices and tensors, so you have to be careful the order in which you present your matrices. You can cancel out non-commutative by shuffling the indices.