E AND B UNDER LORENTZ

In 1895, ten years before Einstein gave the geometric meaning, the Dutch physicist Hendrik Lorentz wrote down how the electric and magnetic fields change when you describe them from a frame moving uniformly past the lab. He had a different motivation — he was trying to save the ether — but the formulas survived the new geometry intact. For a boost along the x-axis at velocity $v = \beta c$, with $\gamma = 1/\sqrt{1-\beta^2}$, the transformation reads

The structure is cleaner than the components suggest. Components parallel to the boost direction are unchanged; components perpendicular to it pick up a factor of $\gamma$ and mix linearly with the perpendicular components of the other field. In coordinate-free form,

Two features to note. First, the cross-product structure is exactly the magnetic part of the Lorentz force $q(\mathbf{E} + \mathbf{v}\times\mathbf{B})$ — moving charges feel an effective electric field $\mathbf{v}\times\mathbf{B}$, and the boosted observer sees that as a literal electric field in their own frame. Second, the $\mathbf{v}\times\mathbf{E}/c^{2}$ term is suppressed by $1/c^{2}$ relative to the $\mathbf{v}\times\mathbf{B}$ term, which is why everyday lab life — wires, magnets, batteries — is dominated by the electric mixing and not the magnetic one.

There are two paths to these expressions. The brute-force one applies the boost to the four-potential $A^\mu = (\phi/c, \mathbf{A})$ and recomputes $\mathbf{E} = -\nabla\phi - \partial_t \mathbf{A}$ and $\mathbf{B} = \nabla\times\mathbf{A}$ in the new coordinates. It works, but you spend a page on chain-rule bookkeeping.

The clean path goes through the electromagnetic field tensor $F^{\mu\nu}$ — the antisymmetric rank-2 object that packages both fields into one. Under a Lorentz boost $\Lambda^\mu{}_\nu$ along $+x$, the tensor transforms by the rule every rank-2 tensor obeys,

The components of $F^{\mu\nu}$ are arranged so that $F^{0i} = E_i/c$ and $F^{ij} = -\varepsilon_{ijk}B_k$. Plug those into the tensor transform, multiply out the matrix products, and the six closed-form expressions of EQ.01 and EQ.02 fall out at once — no chain rule, no integrations by parts. §11.3 builds $F^{\mu\nu}$ from scratch and shows the same calculation done in index notation; this section just notes that the strange "perpendicular components mix" pattern is the geometric fingerprint of a tensor's two indices being boosted independently.

A simple check: take a parallel-plate capacitor, charged so that in the lab frame there is only a uniform $\mathbf{E} = E_y\,\hat{\mathbf{y}}$ between the plates and $\mathbf{B} = 0$. Boost along $+x$. EQ.02 gives $B'_z = -\gamma v E_y / c^{2}$ — non-zero. The boosted observer sees the same plates, the same charges, but a magnetic field they swear is being produced by something. That something is just the moving charge sheet, which now constitutes a surface current.

The lab in this scene contains a uniform magnetic field pointing out of the page, no electric field at all. To you, sitting in the lab, that is the entire story — a magnet, period. To the observer gliding past at $\beta$, the magnetic field is still there (slightly weaker, since the perpendicular component of $\mathbf{B}$ picks up a $\gamma$ but the parallel one does not, and the closed-form rearrangement leaves $B'_z = \gamma B_z$ in this configuration), but they also measure an electric field perpendicular to both the boost and the original magnetic field, with magnitude $E' = \gamma v B$.

That is the practical consequence: it is meaningless to insist either field is "the real one." A wire carrying a steady current produces, in its rest frame, only a magnetic field. Step into the rest frame of the drifting electrons inside that wire and you find a non-zero electric field — the same wire now looks slightly negatively charged because Lorentz contraction has shifted the apparent ion-electron density balance. The classic two-parallel-wires experiment of §11.4 is the place this argument is made airtight; for now, the scene above is enough to fix the picture: turning your boost knob exchanges electric for magnetic, smoothly.

The numerical scaling is worth noting because it is precisely what makes magnetism feel like a small relativistic correction at human velocities and a peer of electricity at relativistic ones. A drift speed of $1\,\text{mm/s}$ in copper gives $\beta \sim 10^{-12}$, and the induced $E'$ from a $1\,\text{T}$ magnetic field is $vB \sim 10^{-3}\,\text{V/m}$ — small but measurable, and the source of every motional EMF. At $\beta = 0.9$ the same $1\,\text{T}$ produces an $E'$ of order $2.7\times10^{8}\,\text{V/m}$, comparable to a strong DC capacitor. Same field, two regimes, one transformation.

Although $\mathbf{E}$ and $\mathbf{B}$ are observer-dependent, there are exactly two scalars built from them that are not. They are the Lorentz invariants of the EM field:

Every inertial observer measures the same numerical value for $I_1$ and $I_2$. The proof is one line if you know that $I_1 \propto F^{\mu\nu}\,(*F)_{\mu\nu}$ and $I_2 \propto F^{\mu\nu}F_{\mu\nu}$ — both are full contractions of tensors, which by construction are scalars. The Lorentz invariants of the EM field sort every possible field configuration into three buckets:

There is also a knife-edge case, $I_1 = I_2 = 0$ with neither field zero — that is exactly an electromagnetic plane wave, which has $|\mathbf{E}| = c|\mathbf{B}|$ and $\mathbf{E}\perp\mathbf{B}$, and is "null" in every frame. No boost can make a light wave look like a static field, ever.

The point of this whole machinery is the picture below.

A static charge in its rest frame produces, by symmetry, a purely radial electric field and zero magnetic field. To any observer in motion relative to that charge, the same object produces both an electric and a magnetic field, with the magnetic part curling around the boost axis exactly as you would expect from a moving point charge — because that is precisely what the boosted observer sees. The classification of the previous section makes this rigorous: a static point charge has $I_1 = 0$ and $I_2 = E^{2} > 0$, so it is an electric-like field, and the rest frame is the unique frame in which $\mathbf{B}$ vanishes. Any other observer carries a non-zero $\mathbf{B}$, but every observer in the universe agrees on the same numerical value of $|\mathbf{E}|^{2} - c^{2}|\mathbf{B}|^{2}$.

The boost mixing of E and B is the cash value of every other §11 topic. §11.3 packages everything in this section into one antisymmetric tensor $F^{\mu\nu}$ so the bookkeeping above stops being six equations and becomes one. §11.4 turns the same picture into the most surprising result in classical physics: two parallel wires at rest carry no charge, but the moving electrons in one feel a force from the other that the charges in the wire would never produce on their own — the magnetic force on a current is the electric force on a moving charge, viewed through the boost we have just written down. The Lorentz transformation of fields is the door; the rest of §11 is what is on the other side.

The voice payoff sits exactly here: the magnetic and electric fields are not separate things. They are projections of one object — the field tensor $F^{\mu\nu}$ — onto your particular state of motion. Change your motion and the projection rotates. Nothing about the underlying object has changed.