Relativistic integro-differential form of the Lorentz-Dirac equation in 3D without runaways

It is well known that the third-order Lorentz-Dirac equation admits `runaway' solutions wherein the energy of the particle grows without limit, even when there is no external force. These solutions can be denied simply on physical grounds, and on the basis of careful analysis of the correspondence between classical and quantum theory. Nonetheless, one would prefer an equation that did not admit unphysical behaviour at the outset. Such an equation - an integro-differential version of the Lorentz-Dirac equation - is currently available either in only one dimension or in three dimensions (3D) in the non-relativistic limit.

It is shown herein how the Lorentz-Dirac equation may be integrated without approximation, and is thereby converted to a second-order integro-differential equation in 3D satisfying the above requirement, i.e. as a result, no additional constraints on the solutions are required because runaway solutions are intrinsically absent. The derivation is placed within the historical context established by standard works on classical electrodynamics by Rohrlich, and by Jackson.