Dono wrote:
How can I find the transformation that ties E,B,E',B'? Any reference
that you can suggest? Thank you.

Look up the Faraday two-form (its dual is called the Maxwell 2-form or
tensor). Its components, when projected onto an inertial frame, consist
of the 3-vector components of E and B:

[ 0 -Ex -Ey -Ez]
F = [ Ex 0 Bz -By]
[ Ey -Bz 0 Bx]
[ Ez By -Bx 0 ]

NOTE: I'm using the sign conventions of MTW, section 3.
This section discusses SR, not GR, but in a general
geometrical framework.
MTW = Misner, Thorne, and Wheeler, _Gravitation_

The two form components, of course, transforms like the components of a
contravariant rank-2 tensor:

F_i'j' = L_i'^i L_j'^j F_ij

Where L_i'^i is the (homogeneous) Lorentz transform between two inertial
frames. NOTE: i' is distinct from i, because I use the modern notation
where the primes go on the indices, not the tensor symbol (here F),
because the tensor itself is unaffected by a coordinate change, only its
components change.

The major thing to notice is that in general E and B get intermixed by a
Lorentz transform; they are NOT independent. That's why we call it the
electromagnetic field.

Exercise for the reader: why is the HOMOGENEOUS Lorentz
transform used, and not the full Poincaré group (aka the
inhomogeneous Lorentz group, which includes translations)?
Can one use the full homogeneous Lorentz group, or only
the component of it which includes the identity?


Tom Roberts