Charge invariance

Charge invariance is the experimental and theoretical fact that electric charge is a Lorentz scalar: every inertial observer, in every state of uniform motion, measures the same total charge enclosed in a given closed volume. Length contracts, mass-energy mixes, electric and magnetic field components rotate into each other under boosts — but the integrated charge in a closed region is frame-independent. Formally, Q = ∫ J⁰ d³x is the integral of the time component of the four-current J^μ = (cρ, J) over a constant-time hypersurface, and the local conservation law ∂_μ J^μ = 0 (which expresses ∂ρ/∂t + ∇·J = 0 in 3-space language) means the flux of J^μ through any closed three-surface bounding the same charge content gives the same Q.

The intuition is geometric. In the lab frame, a parallel-plate capacitor of length L₀ holds N magenta dots on one plate. Boost to a frame moving at βc along the plate's length. In the new frame, the plate length contracts to L₀/γ — but the N dots on it stay N dots. You can count them. They sit on a physical plate and counting is frame-independent. The charge per unit length (line density ρ) does change under boost — it scales as γ for a moving lattice — but the total Q = ρL = (γρ₀)(L₀/γ) = ρ₀L₀ is invariant. The 1856 Weber–Kohlrausch measurement of √(1/μ₀ε₀) = c, six years before Maxwell's 1862 prediction that light is an EM wave, was the first quantitative hint that c was lurking inside the unit definition of charge — a fact only fully understood after the four-dimensional packaging of charge and current into J^μ was made explicit.