Runaway solution

A runaway solution is a solution of the Abraham-Lorentz equation of motion m·a = F_ext + (μ₀q²/6πc)·ȧ in which the acceleration of a free charge grows exponentially in time, a(t) = a₀ exp(t/τ₀), with τ₀ = q²/(6πε₀mc³) ≈ 6.3×10⁻²⁴ s for an electron. In the runaway branch the charge spontaneously accelerates faster and faster forever with no external force and no external energy source — a gross violation of energy conservation.

The runaways arise from the third-order structure of the Abraham-Lorentz equation: demanding that m·a match the radiation-reaction-force jerk term at every instant admits two independent solutions, one in which a decays to match F_ext (physically reasonable) and one in which a grows exponentially (pathological). Mathematically the pathology is eliminated by imposing a boundary condition at t → +∞ that acceleration must approach zero when F_ext is switched off, which selects the non-runaway branch. Physically, this boundary-condition trick buys conservation at the price of causality: the charge begins responding to F_ext a time ∼ τ₀ before the force is applied — the notorious pre-acceleration problem. Both failure modes (runaway acceleration and pre-acceleration) persist in Paul Dirac's 1938 relativistic extension (the Abraham-Lorentz-Dirac equation). The quantum-field-theoretic treatment of electron self-interaction (QED) resolves the issue via renormalisation — infinite bare quantities compensate the self-energy divergence — but at the classical level the runaway solution remains the cleanest demonstration that classical electrodynamics does not close as a self-consistent theory of point charges. The practical resolution for almost all engineering and astrophysical purposes is to treat radiation reaction as a small perturbation (Landau-Lifshitz reduced-order form), which is well-defined and free of pathologies whenever τ₀·|ȧ|/|a| ≪ 1.