Chapter 4: Four-Current and Charge Conservation
The four-current unifies charge density and current density into a single four-vector. Charge conservation follows from gauge invariance and is expressed as a covariant continuity equation.
The Four-Current Jμ
\( J^\mu = (c\rho, \vec{J}) \)
ρ = charge density, J = current density
The four-current transforms as a four-vector under Lorentz transformations, mixing charge and current densities between frames.
Charge Conservation
\( \partial_\mu J^\mu = 0 \)
Equivalent to \( \frac{\partial \rho}{\partial t} + \nabla \cdot \vec{J} = 0 \)
This is the relativistic continuity equation. It's automatically satisfied when Maxwell's equations hold, and is a consequence of gauge symmetry (Noether's theorem).
Covariant Lorentz Force
\( \frac{dp^\mu}{d\tau} = qF^{\mu\nu}u_\nu \)
Four-force = charge × field tensor × four-velocity