CHARGE INVARIANCE

§11 opens with a question that should already feel uncomfortable. Length contracts. Time dilates. Mass and energy mix. The electric and magnetic fields, as we will see in §11.3, rotate into each other under a boost. Every quantity in this branch can rotate, mix, change sign — except this one. Charge is a Lorentz scalar. A box of N coulombs in the lab frame contains N coulombs in any inertial frame.

To see why, we need the right packaging. Charge density ρ (C/m³) and current density J (A/m²) look like separate quantities in the three-dimensional vocabulary of §01–§07. Special relativity insists they are not separate; they are the four components of a single object, the four-current:

The factor of c on the time component is a unit-matching convention so that all four entries carry units of A/m². The physics is in how the four components transform. Boost along +x with velocity βc and you find

— exactly the same Lorentz transform the spacetime pair (ct, x) obeys. So J^μ is a genuine four-vector, not a glued-together pair of three-vectors. Charge density and current density are the time and space parts of one object; observers in different frames see them mix.

Take a closed region V in some frame and integrate ρ over it. The total enclosed charge is

Boost to a new frame and naively both factors change. The charge density goes up — ρ' = γρ, because the time component picks up a γ from mixing with J_x. The volume goes down — V' = V/γ, because lengths along the boost contract. The two transformations are exactly reciprocal, so the product ρ' V' = ρ V is invariant. The γ that compresses the volume is the same γ that concentrates the charge inside it; nothing has been gained or lost.

A cleaner way to see the same result is geometric. Q is the flux of J^μ through a constant-time hypersurface — a 3-surface of simultaneity in your frame. Different observers pick different 3-surfaces, but the four-current is divergence-free,

That is local charge conservation, the four-dimensional version of the everyday continuity equation. By the four-divergence theorem, the flux of a divergence-free four-current through any two 3-surfaces bounding the same world-tube is the same. Pick any slicing; you get the same Q.

The cleanest demonstration is to count. Bound charges sit on the plates of a capacitor; counting them is a frame-independent operation, because dots are dots. Slide the boost; the geometry visibly contracts; the dot count holds.

In the lab frame the capacitor plates have length L₀ and carry +Ne and −Ne. In the boosted frame, lengths along the boost contract to L₀/γ, so the charge density per unit length on each plate goes up by γ (cramming the same charge into a shorter plate). But the total charge — what the meter on the side of the capacitor reads — is +Ne and −Ne in both frames. The integrated quantity does not change; only its distribution does.

This is what the four-current packaging buys you. The "increase in ρ" you would measure on a charged wire that streams past you is real (it's the famous mechanism behind §11.4's derivation of magnetism from a boost of an electrostatic line), but it is exactly compensated by the contracted volume so the enclosed charge is the same. You can double-book ρ' = γρ as long as you also book V' = V/γ; the two factors are the same γ.

Six years before Maxwell predicted that light is an electromagnetic wave, electricity itself had already produced the speed of light as a number. In 1856 Wilhelm Weber and Rudolf Kohlrausch measured the ratio of the electrostatic unit of charge (defined by the force between two static charges via Coulomb's law) to the electromagnetic unit of charge (defined by the force between two parallel current-carrying wires via Ampère's law). They found

That is c. The constant of nature governing how light propagates was already lurking in the SI unit system, sitting between two definitions of charge that no nineteenth-century physicist had connected to optics.

Maxwell saw what Weber and Kohlrausch had not. In his 1862 paper On Physical Lines of Force he wrote down the wave equation for the electromagnetic field and noticed that its propagation speed was $1/\sqrt{\mu_0\varepsilon_0}$ — the same number Weber & Kohlrausch had measured. The first quantitative hint that c was lurking in EM constants. Maxwell did not pull c out of nowhere — it was already in the ratio of two charge-unit definitions, decades before anyone could say why.

Charge invariance is the entry point to the four-vector vocabulary the rest of §11 will speak. Three of the central objects the next four topics cover are all four-vectors of the same kind:

Charge is the time component of J^μ that no boost can mix away — because the "spatial mixing partners" of ρ are the J components, and ∇·J integrates to zero on any boundary at infinity. The flux of a conserved four-current through a closed 3-surface enclosing the entire universe is identically zero, which is why total charge does not change as you re-slice spacetime.

The next four topics specialise this picture. §11.2 folds Maxwell's equations into four-vector form. §11.3 builds the field tensor F^ sourced by J^μ and watches E and B mix. §11.4 derives the magnetic force on a moving wire purely from a boost of an electrostatic configuration. §11.5 closes the module with the four-potential A_μ coupled to J^μ in the action.

Hold onto the one fact: of every quantity in electromagnetism, total electric charge is the only Lorentz scalar that appears at the level of a single particle. Charge stands alone. Every observer agrees.