A Proposed Application of Fractional Calculus on Time Dilation in Special Theory of Relativity

Time dilation (TD) is a principal concept in the special theory of relativity (STR). The Einstein TD formula is the relation between the proper time (Formula presented.) measured in a moving frame of reference with velocity v and the dilated time t measured by a stationary observer. In this paper, an integral approach is firstly presented to rededuce the Einstein TD formula. Then, the concept of TD is introduced and examined in view of the fractional calculus (FC) by means of the Caputo fractional derivative definition (CFD). In contrast to the explicit standard TD formula, it is found that the fractional TD (FTD) is governed by a transcendental equation in terms of the hyperbolic function and the fractional-order (Formula presented.). For small v compared with the speed of light c (i.e., (Formula presented.)), our results tend to Newtonian mechanics, i.e., (Formula presented.). For v comparable to c such as (Formula presented.), our numerical results are compared with the experimental ones for the TD of the muon particles (Formula presented.). Moreover, the influence of the arbitrary-order (Formula presented.) on the FTD is analyzed. It is also declared that at a specific (Formula presented.), there is an agreement between the present theoretical results and the corresponding experimental ones for the muon particles (Formula presented.).