# The Lorentz Transformation of the electric field of a moving charge

If I have a moving charge observed in frame $$S$$, may a Lorentz boost from $$S$$ in a direction not parallel to the charge's velocity in $$S$$ result in an electric field that has a different magnitude of the component in the direction of that boost? In other words, is a single Lorentz Boost w.r.t. a frame where a charge is moving really not a pure Lorentz boost?

And what if in frame $$S$$ there was another charge that is not moving? Does it mean that whether or not a Lorentz boost is pure is not really an intrinsic property of that boost, but rather dependent also on the frames in which the electric field of one or more charges are not "transformed" (i.e. the source charge's/charges' rest frame(s))?

• Pure boost - no rotation of coordinates – Kevin Marinas Jul 14 '19 at 16:21
• Also, the electric field projected in the direction of a pure boost shouldn't change as a result of that boost. – Kevin Marinas Jul 14 '19 at 16:23
• If it is not a pure boost, but a boost + rotation, I am wondering if that projected E-field will change. – Kevin Marinas Jul 14 '19 at 16:25
• Please make it more clear, is your particle moving in x direction w.r.t. S, while S' is moving in y axis w.r.t S? Also fields are measured in particle's frame or S? Which you want to transform to S' frame? – Paradoxy Jul 14 '19 at 17:41
• Let's have the charged particle move in the $x$ direction in $S$. The observer is either stationary in $S$ or moving at some arbitrary velocity in $S$ (stationary in $S'$). Both positions are arbitrary. All velocities are steady. Fields are measured by an observer at constant velocity. Perhaps think of two such observers making an observation at the moment that they cross paths at the same point in space time. Do those observers see the same E-field projected along their difference of velocity? They should, but does that follow from applying non-collinear boost w.r.t. the charge frame $S'''$? – Kevin Marinas Jul 14 '19 at 20:31