T. J. BARTON, Y. FRIEDMAN; BOUNDED DERIVATIONS OF JB* - TRIPLES, The Quarterly Journal of Mathematics, Volume 41, Issue 3, 1 September 1990, Pages 255–268, htt

Based on the generalized principle of relativity and the ensuing symmetry, it has been shown that if the Clock Hypothesis is not true, there is a universal maximal acceleration a_m. We present here an extension of relativistic dynamics for which all admissible solutions will have have a speed bounded by the speed of light c and the acceleration bounded by a_m. An additional Doppler type shift for an accelerated source is predicted. The formulas for such shift are the same as for the usual Doppler shift with v/c replaced by a/a_m. The W. Kündig experiment of measurement of the transverse Doppler shift in an accelerated system was also exposed to a longtitudal shift due to the acceleration. This experiment, as reanalyzed by Kholmetskii et al., shows that the Clock Hypothesis is not valid. Based on the results of this experiment, we predict that the value of the maximal acceleration a_m is of the order 10¹⁹m/s². Moreover, our analysis provides a way to measure experimentally the maximal acceleration with existing technology.

We obtain exact solutions in the laboratory frame for the relativistic position, velocity, and energy of a charged particle moving in constant, uniform, and mutually perpendicular electromagnetic fields. We also obtain an exact equation relating the proper time and the laboratory time, and examine all these quantities in the large and small time limits. Working in the laboratory frame makes the accelerated motion that can occur when $\mid \mathbf{E}\mid ⩾c\mid \mathbf{B}\mid$ easier to analyze and understand.

We explore the role of symmetry in the theory of Special Relativity. Using the symmetry of the principle of relativity and eliminating the Galilean transformations, we obtain a universally preserved speed and an invariant metric, without assuming the constancy of the speed of light. We also obtain the spacetime transformations between inertial frames depending on this speed. From experimental evidence, this universally preserved speed is c, the speed of light, and the transformations are the usual Lorentz transformations. The ball of relativistically admissible velocities is a bounded symmetric domain with respect to the group of affine automorphisms. The generators of velocity addition lead to a relativistic dynamics equation. To obtain explicit solutions for the important case of the motion of a charged particle in constant, uniform, and perpendicular electric and magnetic fields, one can take advantage of an additional symmetry—the symmetric velocities. The corresponding bounded domain is symmetric with respect to the conformal maps. This leads to explicit analytic solutions for the motion of the charged particle.