[Paper Review] Derivation of the potential, field, and locally-conserved charge-current density of an arbitrarily moving point-charge

André Gsponer|ArXiv.org|Dec 23, 2006

Quantum and Classical Electrodynamics16 references3 citations

TL;DR

This paper derives the complete electromagnetic potential, field, and charge-current density for an arbitrarily moving relativistic point-charge by showing that the standard Liénard-Wiechert formulation fails to ensure local charge conservation. By treating the four-potential as a nonlinear generalized function involving Dirac delta functions and their derivatives, the authors derive a unique, locally conserved current density that includes terms depending on the second and third proper-time derivatives of the charge's position, resolving long-standing inconsistencies in classical electrodynamics.

ABSTRACT

The complete charge-current density and field strength of an arbitrarily accelerated relativistic point-charge are explicitly calculated. The current density includes, apart from the well-established three-dimensional delta-function which is sufficient for its global conservation, additional delta-contributions depending on the second and third proper-time derivatives of the position, which are necessary for its local conservation as required by the internal consistency of classical electrodynamics which implies that local charge-conservation is an {identity}. Similarly, the field strength includes an additional delta-contribution which is necessary for obtaining this result. 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 necessary to verify charge-current conservation. The four-potential from which this field and the conserved charge-current density derive is found to be unique in the sense that it is the only one reducing to an invariant scalar function in the instantaneous rest frame of the point-charge that leads to a point-like locally-conserved charge-current density.

Motivation & Objective

• To resolve the inconsistency in classical electrodynamics where the standard Liénard-Wiechert current fails to satisfy local charge conservation.
• To derive a four-potential that reduces to a scalar function in the instantaneous rest frame of the point-charge and yields a point-like, locally conserved current density.
• To establish the mathematical necessity of including higher-order delta-function contributions in the current and field strength for internal consistency of Maxwell's equations.
• To demonstrate that the standard formulation of the Liénard-Wiechert potential and field is insufficient for local conservation and must be generalized using distribution theory.

Proposed method

• The four-potential is postulated in the form $ A_{ u} = e \frac{\dot{Z}_{\nu}}{\xi} \Upsilon(\xi) \big|_{\tau=\tau_r} $, where $ \Upsilon(\xi) $ characterizes the singularity at $ \xi = 0 $, ensuring proper differentiation of distributions.
• The field strength $ F_{\mu\nu} $ is derived from the generalized potential using standard definitions, retaining all delta-function terms from differentiation.
• The charge-current density $ J_{\mu} $ is computed via $ -4\pi J_{\mu} = \partial^{\nu} F_{\mu\nu} $, with all distributional terms preserved until the final step.
• The structure theorem of distribution theory is applied to rigorously justify the appearance of $ \delta $, $ \delta' $, and $ \delta'' $ terms in the current and field.
• A unicity theorem is proven: the only potential of the form $ A = e\varphi(\xi)\dot{Z}(\tau)\big|_{\tau=\tau_r} $ yielding a locally conserved $ \delta $-like current is $ \varphi(\xi) = \frac{1}{\xi}\Upsilon(\xi) $.
• The biquaternion formalism is used to simplify four-dimensional integrations and maintain generality in calculations.

Experimental results

Research questions

• RQ1 Why does the standard Liénard-Wiechert formulation fail to ensure local charge conservation in classical electrodynamics?
• RQ2 What is the correct mathematical structure of the four-potential for an arbitrarily moving point-charge that ensures local conservation of the current density?
• RQ3 How do higher-order derivatives of the worldline (acceleration and jerk) contribute to the charge-current density and field strength?
• RQ4 Can a unique four-potential be derived that reduces to a scalar in the rest frame and yields a locally conserved, point-like current?
• RQ5 What role do nonlinear generalized functions (beyond standard distributions) play in resolving self-force and consistency problems in classical electrodynamics?

Key findings

• The locally conserved charge-current density is given by $ J_{\mu} = \frac{e}{4\pi} \left( \frac{\dot{Z}_{\mu}}{\xi^2} + \frac{\ddot{Z}_{\mu} + 2\kappa K_{\mu}}{\xi} - (2\kappa^2 + \chi)K_{\mu} \right) \delta(\xi) \big|_{\tau=\tau_r} $, which explicitly depends on the second and third proper-time derivatives of the position.
• The field strength must include additional $ \delta $-function contributions to ensure consistency with the locally conserved current, confirming that the Liénard-Wiechert field is incomplete as a distribution.
• The four-potential $ A_{\mu} = e \frac{\dot{Z}_{\mu}}{\xi} \Upsilon(\xi) \big|_{\tau=\tau_r} $ is the unique solution that yields a locally conserved $ \delta $-like current, as proven by a unicity theorem.
• All distributional terms—especially $ \delta $, $ \delta' $, and $ \delta'' $—must be retained throughout calculations; discarding them leads to incorrect, non-conserved currents.
• The formulation resolves the self-force problem: the correct self-force is $ F_{\mu\nu} J^{\nu} $, where $ J^{\nu} $ is the full conserved current, not the standard $ J^{\rm S} $.
• The results are consistent with all physical principles and do not contradict standard applications, as differences vanish upon integration in physical observables.