The locally-conserved current density of the Lienard-Wiechert field

Abstract

The complete charge-current density and field strength of an arbitrarily accelerated relativistic point-charge are explicitly calculated. That current includes, apart from the well-established delta-function term which is sufficient for its global conservation, additional contributions depending on the second and third proper-time derivatives of the position. These extra contributions are necessary for the local conservation of that current, whose divergence must vanish {identically} even if it is a distribution, as is the case here. Similarly, the field strength includes an additional delta-like contribution which is necessary for obtaining this result. Altogether, the Lienard-Wiechert field and charge-current density must therefore be interpreted as nonlinear generalized functions, i.e., not just as distributions, even though only linear operations are needed to verify local charge-current conservation.