On the effect of the maximal proper acceleration in the inertia

The effect of a hypothetical maximal proper acceleration on the mass of a charged particles is investigated in the context of particle accelerators. In particular, it is shown that maximal proper acceleration implies an increase in the kinetic energy of the particle being accelerated with respect to the relativistic energy. Such an increase in kinetic energy leads to a reduction of the luminosity of the bunches with respect to the expected luminosity in the relativistic models of the bunches. This relative loss in luminosity is of the order $10^{-3}$ to $10^{-5}$ for the LHC bunches and can be of order up to $10^{-3}$ for certain laser plasma accelerator facilities. Although the effect is small, it increases with the square of the bunch population.