Einstein at a Crossroads: The Lorentz Force and Time Dependence of a Charged Particle Mass

In his first paper on the special theory of relativity, Einstein derived, inter alia, relativistic equations of motion of a charged particle in an external electromagnetic field, employing transformation equations for acceleration and fields components. While correct, the equations he proposed lack conceptual simplicity and generality. A satisfactory, simple and general relativistic equation of motion was first derived by Planck in 1906, who in the next year also presented transformation equations for the Lorentz force expression, in the framework of relativistic electrodynamics. The Planck's equation of motion was promoted by Einstein in his 1907 review paper on special relativity. In the present paper, we give a detailed account of Planck's 1906 succinct derivation which, despite its significance for modern physics, is not expounded in the literature; we reconstruct the missing steps, and outline a direct proof of Lorentz-covariance of the Planck's equation, recalling Tolman's contribution. Both Planck's and Einstein's arguments rest on the assumption that the mass m of the charged particle is time-independent Lorentz scalar. We speculate what Einstein, as the protagonist of the theory of relativity, could have inferred about the sought equation of motion if he had employed the weaker assumption that the mass m is Lorentz scalar that may depend on the time. We discuss the issue not only without four-tensors, but also from tensorial perspective, employing the generalized relativistic force transformations. While the constancy of m remains the fundamental assumption of modern classical electrodynamics, for both pointlike and extended classical charged particle models, the analysis presented might be helpful even if only in clarifying basic concepts.