[Paper Review] The self-interaction force on an arbitrarily moving point-charge and its energy-momentum radiation rate: A mathematically rigorous derivation of the Lorentz-Dirac equation of motion

André Gsponer|ArXiv.org|Dec 18, 2008

Mathematical and Theoretical Analysis9 references3 citations

TL;DR

This paper presents a mathematically rigorous derivation of the Lorentz-Dirac equation of motion for a point charge using Colombeau generalized functions to handle singularities in the self-interaction force. By treating the self-energy as a delta-squared integral rather than a Coulomb self-energy, it resolves the long-standing 4/3 problem and correctly accounts for the Schott term, showing that the radiated energy-momentum flux equals minus the self-force, thus deriving the Lorentz-Dirac equation without modifying Maxwell’s theory.

ABSTRACT

The classical theory of radiating point-charges is revisited: the retarded potentials, fields, and currents are defined as nonlinear generalized functions and all calculations are made in a Colombeau algebra. The total rate of energy-momentum radiated by an arbitrarily moving relativistic point-charge under the effect of its own field is shown to be rigorously equal to minus the self-interaction force due to that field. This solves, without changing anything in Maxwell's theory, numerous long-standing problems going back to more than a century. As an immediate application an unambiguous derivation of the Lorentz-Dirac equation of motion is given, and the origin of the problem with the Schott term is explained: it was due to the fact that the correct self-energy of a point charge is not the Coulomb self-energy, but an integral over a delta-squared function which yields a finite contribution to the Schott term that is either absent or incorrect in the customary formulations.

Motivation & Objective

• To resolve long-standing inconsistencies in classical electrodynamics related to the self-interaction force of a point charge.
• To address the 4/3 problem and the missing Schott term in conventional derivations of the Lorentz-Dirac equation.
• To provide a mathematically rigorous framework using Colombeau generalized functions to handle singular fields and currents.
• To show that the radiated energy-momentum flux is exactly equal to minus the self-interaction force, validating the Lorentz-Dirac equation without additional assumptions.

Proposed method

• The self-interaction force is derived using retarded Liénard-Wiechert potentials and currents defined as nonlinear generalized functions in a Colombeau algebra.
• The energy-momentum tensor and its divergence are computed via Gauss’s theorem in a 4-volume enclosing the world-line, with singularities handled rigorously.
• The self-energy is treated as a delta-squared integral, yielding a finite contribution to the Schott term, unlike the divergent Coulomb self-energy in standard formulations.
• The self-force is calculated as the volume integral of $ F_{\mu\nu}J^\nu $, with $ J^\nu $ represented as a 3D delta-function along the world-line.
• The derivation uses proper-time parametrization and enforces the constraint $ \dot{Z}_\mu \dot{Z}^\mu = 1 $, ensuring relativistic consistency.
• The Schott term arises naturally from a finite $ C_{[1]} $ moment of the mollifier, which is fixed by requiring orthogonality of the force to the 4-velocity.

Experimental results

Research questions

• RQ1 Why do conventional derivations of the self-force and radiated energy-momentum flux fail to agree, particularly regarding the Schott term?
• RQ2 What is the correct mathematical treatment of the self-energy of a point charge that avoids the 4/3 problem and yields a finite Schott term?
• RQ3 How can the Lorentz-Dirac equation be derived without modifying Maxwell’s equations or introducing ad hoc assumptions?
• RQ4 What role does the delta-squared integral play in generating the correct Schott term compared to the standard Coulomb self-energy?
• RQ5 Why do approaches like Barut’s or Dirac’s fail to reproduce the full Schott term without additional assumptions?

Key findings

• The total rate of energy-momentum radiated by a point charge is rigorously equal to minus the self-interaction force due to its own field, resolving a century-old inconsistency.
• The correct self-energy is not the divergent Coulomb self-energy but a finite contribution from a $ \delta^2 $-integral, which supplies the missing Schott term.
• The Schott term arises naturally from the moment $ C_{[1]} = \int x \eta^2(-x) dx = 1 $, which is derived from the orthogonality condition $ \dddot{Z}_\mu \dot{Z}^\mu = -\mathcal{A}^2 $.
• The Lorentz-Dirac equation is derived without modifying Maxwell’s theory, with the self-force correctly including the $ \dddot{Z}_\mu $ term via the $ \delta^2 $-integral.
• The 4/3 problem is resolved because the self-mass term is not $ 2e^2/3\xi $, but a finite $ C_{[0]}/(2\epsilon) $, with $ C_{[0]} $ determined by the mollifier.
• The derivation shows that Barut’s and Dirac’s approaches miss the Schott term because they use the wrong self-energy (Coulomb-like), while the current method correctly identifies the $ \delta^2 $-integral as the source of the missing term.