Abraham-Lorentz equation

The Abraham-Lorentz equation is the classical equation of motion for a point charge radiating electromagnetic energy while subject to an external force F_ext: m·a = F_ext + (μ₀q²/6πc)·ȧ, where ȧ = da/dt is the jerk (third time-derivative of position). The extra term is the **Abraham-Lorentz radiation-reaction force** F_rad = (μ₀q²/6πc)·ȧ, whose coefficient is set by demanding that its time-averaged work balance the Larmor-formula energy-loss rate −P = −q²a²/(6πε₀c³). Max Abraham derived it in 1903 by computing the self-force of a small charged sphere on itself and taking the point-particle limit; Hendrik Antoon Lorentz wrote it into its modern form in 1904.

The equation is structurally anomalous in two ways. First, it is third-order in time (through the jerk term), while Newton's second law is second-order — solving an Abraham-Lorentz problem therefore requires specifying three initial conditions rather than two. Second, the third-order character admits runaway solutions: a free charge with any initial jerk sees its acceleration grow exponentially as exp(t/τ₀), where τ₀ = q²/(6πε₀mc³) ≈ 6.3×10⁻²⁴ s for an electron, without any external energy source. To kill the runaway one imposes a boundary condition at future infinity, which has the equally unphysical consequence of pre-acceleration — the charge begins responding a time ∼ τ₀ before the external force is applied. Landau and Lifshitz proposed a reduced-order regularisation (substitute Ḟ_ext/m for the jerk term, yielding an ordinary second-order equation) that is accurate whenever radiation reaction is a small perturbation. Paul Dirac extended the equation to a Lorentz-covariant relativistic form in 1938 — the ALD equation — whose pathologies are the same. The quantum-field-theoretic treatment of electron self-interaction (QED) resolves the divergences via renormalisation, but at the classical level radiation reaction remains an open seam in the theory.