Accelerating electrons are known to radiate electromagnetic waves, a property that is central to the concept of many devices, from antennas to synchrotrons. While the electrodynamics of accelerating charged particles is well understood, the same is not true for charged matter waves: would a locally accelerating charged matter wave, like its particle counterpart, radiate? Here we construct a novel class of matter waves, angular accelerating electron waves, by superpositions of twisted electrons carrying orbital angular momentum. We study the electrodynamic behaviour of such accelerating matter waves and reveal the generation of a solenoidal magnetic field in each component, and an accelerating electron wave that does not radiate. These novel properties will have practical impact in spin flipping of qubits for quantum information processing, have been suggested for control of time dilation and length contraction, and raise fundamental questions as to the nature of wave-particle duality in the context of radiating charged matter.
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