Relativity and electromagnetic field I have a question regarding Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973), Gravitation ISBN 978-0-7167-0344-0. It is a book about Einstein's theory of gravitation.
At the page 79, in the Chapter 3.4. about the Maxwell's equation, we assume that we are in a Rocket with velocity $\beta^j$ with $j$ being x, y and z. But we change to a system of coordinates so that the rocket is moving in the z direction.
The first result introduced is the calculation of the electric field along the z after the Lorentz tranformation:
$\bar E_z = F_\bar {30} =\Lambda^\alpha_{\bar 3}\Lambda^\beta_{\bar 0} F_{\alpha \beta} = (1-\beta^2) F_{30} = E_z$
I have no problem with that. However, in the calculation along the x axis:
$\bar E_x = F_\bar {10} =\Lambda^\alpha_{\bar 1}\Lambda^\beta_{\bar 0} F_{\alpha \beta} = \gamma F_{10} + \beta \gamma F_{13} = \gamma(E_x-\beta B_y)$
However, I don't understand how come $F_{13}$ appears, I would have said that $F_{01}$  appears. So what am I missing?
 A: Actually the last of eqs 3.4 on page 79 on my "Gravitation" reads
$\bar E_x = F_\bar {10} =\Lambda^\alpha_{\bar 1}\Lambda^\beta_{\bar 0} F_{\alpha \beta} = \gamma F_{10} + \beta \gamma F_{13} = \gamma(E_x-\beta B_y)$
which is perfectly correct.
Please note that the $ \Lambda^\alpha_{\bar \beta}$ that they are using is given by eq 2.45 on page 69 (boost in the z direction).
Sure you got the original MTW ? :-)
PS After the OP edited the question...
In the equation
$\bar E_x = F_\bar {10} =\Lambda^\alpha_{\bar 1}\Lambda^\beta_{\bar 0} F_{\alpha \beta} $ the $\alpha$ and $\beta$ indices are summed over, giving in general 4*4 terms. However most of these terms are 0 because $\Lambda$ has mostly empty components (see  eq 2.45 right, on page 69). For example if $\alpha=0$ and $\beta=1$ the term $\Lambda^0_{\bar 1}\Lambda^1_{\bar 0} F_{0 1} $ is null  since $\Lambda^0_{\bar 1}$ and $\Lambda^1_{\bar 0} $ are null
A: I simply misread the tensor.
The first term is the product of $\Lambda^1_{\bar 1}=1$ and $\Lambda^0_{\bar 0}=\gamma$, while the second is the product of $\Lambda^1_{\bar 1} = 1$ and $\Lambda^3_{\bar 0}=\beta \gamma$.
