Beyond the Fundamentals of Special Relativity: Full Lorentz gamma factor

Special relativity calculates, by means of the Lorentz gamma factor, the proper time of all inertial systems from the observer proper time, which is taken as a time standard. So, any temporal inference relies in first instance on the observer own time. The question is thus: what fixes the observer proper time? This will be the crucial point debated here. This implies analyzing at the very first why the observer can be taken as a motionless reference in spite of being himself inertial. Is this just an approximation, and if so, up to what extent can it be applied? The framework of special relativity is compared to an amended form in which the fact of taking himself as a reference does not allow the observer to overlook its own kinetics. So, the issue stands on which of two formulations of the Lorentz gamma factor is the most accurate one: its standard expression or an amended one which takes into account the fact that the observer is himself inertial, while the former disregards it. When the observer speed is ignored, the two formulations become identical. Hence, the standard relativistic expression of gamma can be seen as an approximation applicable when the observer motion is null or low, such as it is the instance on Earth.