In the R-Minkowski space-time, which we recently defined from an appropriate deformed Poisson brackets that reproduce the Fock coordinate transformation, we derive an extended form for Maxwell's equations by using a generalized version of Feynman's approach. Also, we establish in this context the Lorentz force. As in deformed special relativity, modifying the angular momentum in such a way as to restore the R-Lorentz algebra generates the magnetic Dirac monopole.

After having obtained previously an extended first approximation of Maxwell's equations in Fock's nonlinear relativity, we propose here the corresponding exact form. In order to achieve this goal, we were inspired mainly by the special relativistic version of Feynman's proof from which we constructed a formal approach more adapted to the noncommutative algebra. This reasoning lets us establish the exact form of the generalized first group of Maxwell's equations. To deduce the second one, we have imposed the electric-magnetic duality. As in the k-Minkowski space-time, the generalized Lorentz force depends on the mass of the particle. After having restored the R-Lorentz algebra symmetry, we have used the perturbative treatment to find the exact form of the generalized Dirac's magnetic monopole in our context. As consequence, the Universe could locally contain the magnetic charge but in its totality it is still neutral.

On the basis of all commutation relations of the κ-deformed phase space incorporating the κ-Minkowski space-time, we have derived in this paper an extended first approximation of both Maxwell's equations and Lorentz force in doubly (or deformed) special relativity (DSR). For this purpose, we have used our approach of the special relativistic version of Feynman's proof by which we have established the explicit formulations of electric and magnetic fields. As in Fock's nonlinear relativity (FNLR), the laws of electrodynamics depend on the particle mass which therefore constitutes a common point between the two extended forms of special relativity. As one consequence, the corresponding equation of motion contains two different types of contributions. In addition to the usual type, another one emerges as a consequence of the coexistence of mass and charge which are coupled with the κ-deformation and electromagnetic field. This new effect completely induced by the κ-deformed phase space is interpreted as the gravitational-type Lorentz force. Unlike FNLR, the corrective terms all depend on the electromagnetic field in DSR.

This work completes a series of two papers devoted to the extension of the fundamental laws of electrodynamics in the context of Fock’s nonlinear relativity (FNLR). Indeed, after having established in the previous study the exact generalizations of both Maxwell’s equations and Dirac’s magnetic monopole, we present here the remaining exact Lorentz force. As in [Formula: see text]-Minkowski space–time, two different contributions appear in the corresponding equation of motion where the new effect is interpreted as the gravitational-type Lorentz force. This common point separately induced by the radius of the universe in our case, or Planck energy in other works, reinforces once more the analogy between electromagnetism and gravity in two different scientific approaches. As a relative difference, it is very important to highlight that more homogeneity characterizes our results where each effect is exclusively generated by mass or charge but not both at the same time. Even more, the new effect emerges as the result of the triple effect of the R-deformation, mass and the square of the velocity but completely independent of electromagnetic field.