To review my special relativity I tried to work out the inverse lorentz transformation explicitly. This led to a lot of confusion; I would like to ask what the issue was with the assumptions I made in the last steps & what the actual expression should be, in index notation. In other SE posts it seems index notation is avoided, maybe precisely because of this confusion.
The derivation follows. We start with
$$\Lambda^\alpha_{\,\,\mu} \,g_{\alpha \beta} \Lambda^\beta _{\,\,\nu} = g_{\mu \nu}$$
Contract with/multiply by $g^{\nu \sigma}$
$$\Lambda ^\alpha_{\,\,\mu} \, g_{\alpha \beta} \Lambda^\beta _{\,\,\nu} g^{\nu \sigma}= g_{\mu \nu}g^{\nu \sigma}=\delta_\mu ^{\,\,\sigma}$$
Write $g_{\alpha \beta} \Lambda^\beta _{\,\,\nu} g^{\nu \sigma}=\Lambda_\alpha^{\,\,\sigma}$. Then we have
$$\Lambda^\alpha _{\,\,\mu}\Lambda_\alpha^{\,\,\,\sigma}=\delta_\mu ^{\,\,\sigma} \tag 1$$
We would like to identify this as matrix multiplication, and thus identify the rightmost $\Lambda$ in this equation as the inverse. To do this, the inner indices need to be the same. Therefore we first take the transpose:
$$\Lambda_\alpha^{\,\,\sigma}=(\Lambda^T)_\sigma^{\,\,\alpha} \tag 2$$
Note 1: The answer to this question is that $(2)$ was incorrect and we actually have $\Lambda_\alpha^{\,\,\sigma}=(\Lambda^T)^\sigma_{\,\,\alpha}$. The correct derivation continues with Equation $(3)$.
Plugging into $(1)$ gives
$$\Lambda^\alpha _{\,\,\mu}(\Lambda^T)_\sigma^{\,\,\alpha}=(\Lambda^T)_\sigma^{\,\,\alpha}\Lambda^\alpha _{\,\,\mu}=\delta_\mu ^{\,\,\sigma}$$
…which should be the answer. However, ever since taking the transpose, this equation is just plain weird: the indices do not match on the RHS and LHS so it is no longer lorentz covariant.
This suggests that taking the transpose actually implicitly inverts the transformation properties of each index, changing upper <-> lower in addition to just permuting them:
$$\Lambda_\alpha^{\,\,\sigma}=(\Lambda^T)^\sigma_{\,\,\alpha}$$
In this case, we have
$$\Lambda^\alpha _{\,\,\mu}(\Lambda^T)^\sigma_{\,\,\alpha}=(\Lambda^T)^\sigma_{\,\,\alpha}\Lambda^\alpha _{\,\,\mu}=\delta_\mu ^{\,\,\sigma} \tag 3$$
However, now there is a new problem: It doesn't work! As noted in this question, if you write the matrix representation of a Lorentz boost, the transpose is clearly not the inverse.
So, how can I write the components of the inverse? And what went wrong here? Also greatly appreciated would be a source where all of this is very explicitly done. I did not find MTW or Griffiths especially clear.
Note 2: I submitted an answer which explains that Equation $(3)$ is in fact consistent.
Best Answer
In addition to the suggestion of walber97, I'd also propose that the transpose should not lower or raise indices. It just interchanges the order of the indices. So your 4th equation should perhaps be $$\Lambda^\alpha_{\,\,\mu}=(\Lambda^T)_\mu^{\,\,\alpha} . $$ The matrix multiplication then remains as a contraction between an upper and a lower pair of indices $$(\Lambda^T)_\mu^{\,\,\alpha} \Lambda_\alpha^{\,\,\,\sigma} = \delta_\mu^{\,\,\sigma}$$ Then the Lorentz transformation properties remain in tact.