  This research aims to develop a new approach toward a consistent coupling of electromagnetic and gravitational fields by using an electron that couples with a weak gravitational potential by means of its electromagnetic field. We find the value of the tiny coupling constant of gravity with electromagnetic fields, which depends on the speed of light and a universal minimum speed that represents the lowest limit of speed for any particle. Such a minimum speed, unattainable by particles, represents a preferred reference frame associated with a background field that breaks the Lorentz symmetry. The metric of the flat spacetime shall include the presence of a uniform vacuum energy density, which leads to a negative pressure at cosmological scales, i.e., the cosmological anti-gravity. The tiny values of the cosmological constant and the vacuum energy density will be successfully obtained in agreement with the observational data. 
On the electrodynamics of moving particles in a quasi flat spacetime   with Lorentz violation and its cosmological implications