Physics Asked on November 23, 2021
The Lorenz-Transformation of the EM-Tensor F is given by the equation
$$ F’^{mu nu} = Lambda^{mu}_{ rho} Lambda^nu_{ sigma} F^{rho sigma}$$
Then it says that this is equivalent to the matrix-equation
$$ F’ = Lambda F Lambda^T $$
How can i see that one of the Lambda-matrices is the transposed of the other, if the indices are at the same spots?
And is it convention that the first index (regardless if upper or lower index) is the row of a matrix and the second index is the row?
$$ Lambda^1_{ 2}$$ would be first row second column of Lambda, and
$$ Lambda_1^{ 2}$$ as well?
One trick that you can use is that summed indices should be near each other (diagonally). In your case this amounts to $$F^{primemunu} = Lambda^mu_{ rho}Lambda^nu_{ sigma} F^{rhosigma} = Lambda^mu_{ color{red}{rho}}F^{color{red}{rho}color{blue}{sigma}}Lambda_{color{blue}{sigma}}^{ nu}$$ Now, by definition $$left(Lambda^Tright)^nu_{ sigma} equiv Lambda_sigma^{ nu}$$ so you get the desired result.
Answered by Davide Morgante on November 23, 2021
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