# What is the Galilean transformation of the EM field?

Consider a reference frame $$S$$ and which we observe some electric field $$\mathbf{E}$$ and magnetic field $$\mathbf{B}$$.

Let $$S'$$ be a reference frame moving at a constant velocity $$\mathbf{u}$$ with respect to $$S$$. In $$S'$$ we observe an electric field $$\mathbf{E'}$$ and magnetic field $$\mathbf{B'}$$.

The obvious question is how is the EM field in the frame $$S'$$ related to the EM field in the frame $$S$$. We know that special relativity gives us an exact answer, but let's suppose we don't know about SR.

First let's impose that if we observe some charged particle $$q$$ with velocity $$\mathbf{v_0}$$ experiencing the Lorentz force in the frame $$S$$, then it should experience the same force in all other frames.

Hence $$\mathbf{F}=q(\mathbf{E}+\mathbf{v} \times \mathbf{B})=\mathbf{F'}=q(\mathbf{E'}+\mathbf{v'} \times \mathbf{B'}) \tag{*}$$

Assuming the Galilean velocity transformation, we have $$\mathbf{v'}+\mathbf{u}=\mathbf{v}$$. Now, since (*) must hold for all velocities $$\mathbf{v}$$, we deduce that

$$\mathbf{E'}=\mathbf{E}+\mathbf{u}\times\mathbf{B} \tag{I}$$

$$\mathbf{B'}=\mathbf{B} \tag{II}$$

Now, this analysis leads to an obvious paradox. If we choose $$S'$$ to be the frame of reference of the moving charge $$q$$, then in $$S'$$ we have no magnetic field, which would impose by (II) that $$\mathbf{B}=0$$ in all frames $$S$$, a clear contradiction.

Now apparently there is something called the Galilean field transformation which retains (I) but replaces (II) by:

$$\mathbf{B'}=\mathbf{B}-\frac{1}{c^2}\mathbf{u} \times \mathbf{E} \tag{III}$$

Furthermore, apparently the above can be derived without assuming SR.

My question is how can we derive (III) without assuming SR? We obviously have to drop the assumption that the Lorentz force is invariant under a change of reference frame, which is physically unintuitive.

Also, does the Galilean transformation solve the paradox described above?

• Your analysis is totally correct and I don't think that the result at the end of your post can be derived without special relativity. The "c" in that equation is infinity in a truly galilean invariant system. You showed that the assumption of true galilean invariance implies a charged particle moving with constant velocty gives no magnetic field! So you have a theoretical argument that the symmetry obeyed is not galilean. – user12029 Jul 10 '17 at 2:26
• Well there is a reason its called Galilean transformation and not relativistic transformation, and I read that it can be derived without assuming SR. – Joshua Benabou Jul 10 '17 at 2:48
• Would you kindly please provide the link or book reference where you read about this derivation? – user103515 Jul 11 '19 at 4:52

## 1 Answer

In the charge's frame, the Lorentz force law becomes $q({\bf E} + {\bf 0} \times {\bf B}) = q {\bf E}$. The magnetic field doesn't apply a force on the particle, but that doesn't mean it has to be ${\bf 0}$. It just means that you can't use that particular particle to measure the ${\bf B}$ field in that particular frame.

• Are you only assuming that the Lorentz force law holds in all Galilean frame? Or are you also assuming the Biot-Savart law? – tparker Jul 10 '17 at 1:43
• I am assuming in the first part of my question (naive derivation) that Lorentz force is invariant under a change of reference. I then show why this gives a contradiction. I then introduce the full "Galilean field transformtions" and ask where does the formula (III) come from and if this new transformation resolves the paradox. – Joshua Benabou Jul 10 '17 at 1:44
• @JoshuaBenabou What's the contradiction? Your claim that ${\bf B}$ must be zero in every frame is incorrect, so the contradiction goes away. – tparker Jul 10 '17 at 1:45
• Hehe. No I don't claim that B must be zero in every frame. I DEDUCE that B must be zero in every frame as a consequence of equation (II), which was derived assuming that a charged particle $q$ experiencing some force due to an EM field in one frame experiences the same force in every inertial reference frame (in other words, assuming the Lorentz force is invariant under a change of frame). That's the contradiction. – Joshua Benabou Jul 10 '17 at 1:55
• @JoshuaBenabout So you are assuming Ampere's law holds. Are you assuming that all of Maxwell's equations hold? The problem is that Ampere's law is not Galilean invariant, so if it holds in one frame it will not hold in a Galilean boosted frame. You need to be more precise about what you are assuming in your setup. – tparker Jul 10 '17 at 3:56