AbrahamLorentzDirac

The Abraham-Lorentz-Dirac equation is a classical equation of motion for a point charge that includes radiation reaction—the self-force arising from the emission of electromagnetic radiation by the accelerating charge. It describes how a charged particle’s trajectory is modified by the back-reaction of its own field in addition to the external electromagnetic forces.

The equation bears the names of the early 20th-century contributors: Max Abraham and Hendrik Antoon Lorentz developed the nonrelativistic radiation-reaction description, while Paul Dirac provided a covariant relativistic generalization in 1938. The combined term is commonly referred to as the Abraham-Lorentz-Dirac (ALD) equation.

In nonrelativistic form, the equation reads m dv/dt = F_ext + (q^2 / 6π ε0 c^3) da/dt, where m is the particle mass, v its velocity, F_ext the external Lorentz force, q the charge, a the acceleration, ε0 the vacuum permittivity, and c the speed of light. The radiation term is proportional to the time derivative of acceleration.

In covariant form, the Lorentz-Abraham-Dirac equation is written as m a^μ = q F^μν u_ν + (q^2 / 6π ε0 c^3) [ d a^μ/dτ + (a_ν a^ν) u^μ / c^2 ], where u^μ is the four-velocity, a^μ the four-acceleration, F^μν the external field tensor, and τ the proper time. This relativistic expression reduces to the nonrelativistic form in the appropriate limit.

A notable feature of the ALD equation is the appearance of third-derivative terms, which lead to problematic behavior such as pre-acceleration and runaway solutions. Practical use often employs the Landau-Lifshitz approximation, which yields a well-behaved second-order equation that captures the leading radiation-reaction effects. The ALD equation remains a foundational topic in classical electrodynamics and the study of radiation reaction.