A fresh look at the concept of a relativistic oscillator

The theory of a unidimensional relativistic oscillator is developed assuming a frame of reference fixed to the oscillator. This approach is at variance with conventional consideration of the same problem in an inertial frame of reference. It is shown that examination of a freely falling body by an observer sitting on the oscillator is advantageous because it permits to avoid introducing of any additional conditions required to ensure the Lorentz invariance of the theory. The time track of such a body in a rigid accelerated frame of reference is derived in a general manner. The effect of a systematic delay of time during vibrations is unveiled in the ultrarelativistic regime.