# Proving a general formula for the boost transformation of the electromagnetic field

In inertial frame $$\mathcal{O}$$, a region of space-time is filled with constant electric field $$\vec{E}$$ and magnetic field $$\vec{B}$$. Another inertial frame $$\mathcal{O}'$$ has 3-velocity $$\vec{V}$$ relative to $$\mathcal{O}$$. What is the electromagnetic field $$\left(\vec{E}', \vec{B}'\right)$$ measured in $$\mathcal{O}'$$? Express the result in terms of $$\vec{E}$$, $$\vec{B}$$, $$\vec{V}$$, dot product ($$\cdot$$) and cross product ($$\times$$).

This is to get the general formula for the boost transformation of the electromagnetic fields. I know the general form of the Lorentz boost transformation. So, obviously the solution for this problem seems to be applying this boost transformation to the electromagnetic field tensor $$F^{uv}$$. That is, for the boost transformation $$\Lambda^u_v$$, calculate $$F'^{ab}=\Lambda^a_u \Lambda^b_vF^{uv}$$. But this seems like a tremendous amount of work... Is there any more efficient solution than this? Could anyone suggest me?

• Look at how $A^a$ transforms.
– J.G.
Commented Apr 1, 2018 at 17:27
• Then I also must see how $\partial_a$ transforms. Right? Commented Apr 1, 2018 at 17:28
• You're making life too hard for yourself. Forget calculus altogether at the start: just write $\phi',\,\mathbf{A}'$ in terms of their unprimed counterparts.
– J.G.
Commented Apr 1, 2018 at 17:34
• Use w.l.o.g.w.c.a.t. (without loss of generality, we can assume that). For example, assume that the velocity is in the $z$-direction, derive some formulae, then generalize back to dots and crosses based on the patterns. The answer is also in Jackson's electromagnetism textbook. Commented Apr 1, 2018 at 18:06
• In my answer here : Is it a typo in David Tong's derivation of spin-orbit interaction? for the configuration of two inertial systems as in Figure-01 the space-time Lorentz transformation is given by equations (03). The Lorentz transformation of the electromagnetic field is given by equations (04). I would post in an answer the proof of this transformation of the electromagnetic field only if a moderator untags the question as "homework-and-exercises". Commented Jan 28, 2020 at 8:51

• using transformation of the electromagnetic field tensor $$F^{\mu\nu}$$ (which is TOO LONG!!!)
• using the transformation of 4-vector potential $$\mathbf A^\mu$$
• using the Lorentz force $$\mathbf{F}=q \mathbf{E}+q \mathbf{v}\times\mathbf{B}$$