The contraction of the EM field tensor is Lorentz invariant. Using the standard formulae,(adsbygoogle = window.adsbygoogle || []).push({});

the fields [tex]\vec{E} = ( E_x, 0, 0 )[/tex] and [tex]\vec{B} = (0, B_y, 0)[/tex] when bosted in the x direction go to

[tex]\vec{E'} = ( E_x, 0, \gamma\beta B_y )[/tex]

[tex]\vec{B'} = (0, \gamma B_y, 0)[/tex]

and it is clear that [tex] E_x^2-B_y^2 = E_z'^2 + E_x'^2-B_y'^2 [/tex].

This potential [tex]A^{\mu} = (\phi(x), A_x(z), 0, 0 )[/tex] gives

[tex]\vec{E} = ( -\partial_x\phi(x), 0, 0 )[/tex]

[tex]\vec{B} = (0, -\partial_zA_x(z), 0)[/tex].

I thought that if I boosted the potential as a 4-vector, then calculated the fields again, I

would get the same result. The boosted potential is

[tex]A'^{\mu} = (\gamma\phi(x)-\gamma\beta A_x(z) , \gamma A_x(z)-\gamma\beta\phi(x), 0, 0 )[/tex]

On recalculating the fields, Ex is multiplied by [tex]\gamma[/tex], while the other fields are

correct. So boosting the potential seems to be not Lorentz invariant, or just wrong maybe?

Any references where I might find out more ?

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# Boosting the potential

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