On the Lorentz Transformation of Charge and Current Densities

The components of ρv / c and iρ must transform as components of a four-vector, so that if measured in one coordinate system they are known in all coordinate systems. On the other hand, any operational definition of ρ(r,t) must take account of the positions of all particles at the same time t, that of the s-th particle being r_s(t). Upon performing the Lorentz transformation these will be r_s^′(t_s^′), and the transformed time t_s^′ will be different for each particle. Another observer, in measuring ρ^′, would use r_s^′(t^′), t^′ being the same for all particles. As particles are in motion r_s^′(t_s^′)≠r_s^′(t^′), and there appears to be no necessary relation between ρ(r,t) and ρ^′(r^′,t^′), operationally defined in each coordinate system. It turns out, however, that if in each coordinate system the charge density is defined by ρ(r,t)=Σse_sδ(r-r_s(t)), then relativistic equations of transformation hold.