Finding the exact equation of motion for a moving charged particle is one of the oldest open problems in physics. The problem originates in the emission of radiation by an accelerated charge, which must result with a loss of energy and recoil of the charge, adding a correction to the well-known Lorentz force. When radiation reaction is neglected, it is well known that the dynamics of a charge in a plane-wave laser field are inevitably periodic. Here we investigate the long-time dynamics of a charge in a plane wave and show that all current models of radiation reaction strictly forbid periodic dynamics. Consequently, we find that the loss of energy due to radiation reaction actually causes particles to asymptotically accelerate to infinite kinetic energy. Such a phenomenon persists even in weak laser fields and puts forward the possibility of testing the open problem of radiation reaction through long-duration weak-field precision measurements, rather than through strong-field experiments. Our findings suggest realistic conditions for such measurements through the asymptotic frequency shift and energy loss of a charge, which for example can be detected in electron energy loss spectrometers in electron microscopes.
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