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I am trying to understand isometries and how to work with tensors.

I know that in special relativity metric transforms as follows $$ g_{\alpha^{\prime} \beta^{\prime}}=g_{\alpha \beta} \Lambda_{\bullet \alpha^{\prime}}^\alpha \Lambda_{\bullet \beta^{\prime}}^\beta $$ or in an analogous form $$ g_{\mu \nu}=g_{\kappa \lambda} \frac{\partial x^\kappa}{\partial x^{\prime \mu}} \frac{\partial x^\lambda}{\partial x^{\prime \nu}} $$ And we define the matrix of Lorents transformations as $$ \Lambda_{\bullet \nu}^\mu=\frac{\partial x^{\prime \mu}}{\partial x^\nu} $$ So it seems that we can get from the first equation $$ \hat{g}=\hat{g} \Lambda^{-1} \Lambda^{-1}, $$ and it is a wrong formula.

Of course, we should get $$ \hat{g}=\Lambda^{\mathrm{T}} \hat{g} \Lambda, $$ and I know how to do it, but what is wrong with my first attempt to get it?

Thank you in advance!

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  • $\begingroup$ Hi Хранитель Рощи. Welcome to Phys.SE. You're almost there. Think about how to multiply matrices, and what a transposed matrix means. $\endgroup$
    – Qmechanic
    Jan 9, 2023 at 16:40
  • $\begingroup$ @Qmechanic Dear Qmechanic, I understand that the proper transformations are $g_{\alpha^{\prime} \beta^{\prime}}=g_{\alpha \beta} \Lambda_{\bullet \alpha^{\prime}}^\alpha \Lambda_{\bullet \beta^{\prime}}^\beta=\left(\Lambda^{\mathrm{T}}\right)_{\alpha^{\prime}}^{\bullet\alpha } g_{\alpha \beta} \Lambda_{\bullet \beta^{\prime}}^\beta$, but I don't fully understand the mistake in the transformations in the question. $\endgroup$ Jan 9, 2023 at 17:33
  • $\begingroup$ @Qmechanic hi, I got the answer, could you please check if I'm thinking correctly? $\endgroup$ Jan 9, 2023 at 19:00
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    $\begingroup$ What the dot before the indices? Is this specific to the material or are you trying to write indices that are spaces out? If so you can just use a space like \; in the subscript to space things out as needed like \Lambda^{\mu}_{\;\nu} to get $\Lambda^{\mu}_{\;\nu}$ $\endgroup$
    – Triatticus
    Jan 9, 2023 at 22:36
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    $\begingroup$ That's why I mentioned using the more standard notation of using spacing instead, in my above comment $\nu$ comes after $\mu$ and they are not in line. The inverse would look like $\Lambda^{\; \nu}_{\mu}$ which displays the needed spacing also. $\endgroup$
    – Triatticus
    Jan 9, 2023 at 23:06

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I think I know the answer: we should better understand how we are writing multiplications of matrices. In indexes, the multiplication of matrices should contain both column and raw indexes. For example, here we don't need to change the position of $\beta$ index, because $\beta$ is in a column and in a raw position, but $\alpha$ is only in a column position, so we are changing it: $$ g_{\alpha^{\prime} \beta^{\prime}}=g_{\alpha \beta} \Lambda_{\bullet \alpha^{\prime}}^\alpha \Lambda_{\bullet \beta^{\prime}}^\beta=\left(\Lambda^{\mathrm{T}}\right)_{\alpha^{\prime}}^{\bullet\alpha } g_{\alpha \beta} \Lambda_{\bullet \beta^{\prime}}^\beta $$ We can't multiply matrices if both indices are in a column position.

Also, it is important that we start from the row, that's why we put $\Lambda^{\mathrm{T}}$ in the beginning.

Please, if someone understands this, can he comment, am I thinking correctly?

Hope it will be useful to someone.

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