Causal Finite-Tick Dynamics as a Resolution of the Classical Radiation Reaction Problem

Abstract

The radiation-reaction problem in classical electrodynamics has long resisted a consistent solution: the Abraham-Lorentz-Dirac equation admits runaways and pre-acceleration, while the Landau-Lifshitz (LL) equation avoids these pathologies only as a reduction-of-order approximation. We introduce Tick-Tock (TT) dynamics, a causal finite-step formulation in which radiation recoil arises from discrete tick-by-tick updates. In the continuum limit, TT reproduces the LL equation, ensuring consistency with all experimental tests of radiation reaction. Unlike LL, however, TT does not rely on reduction of order: the recoil force is expressed directly through finite-difference changes of the Lorentz acceleration, making the Schott-like energy term explicit as a telescoping boundary contribution. This construction eliminates pre-acceleration and runaway solutions while providing a transparent stepwise energy balance. Moreover, the finite-tick structure introduces a natural high-frequency cutoff, suppressing unphysical growth in discontinuous fields. Taken together, TT offers a stable and causal reformulation of radiation reaction, consistent with LL in smooth regimes but extending its applicability to rapidly varying ones, and it suggests that characteristic timescales such as $\tau_{0}$ may play a special role.