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

Ebrahem A. Algehyne, Musaad S. Aldhabani, Mounirah Areshi, Essam R. El-Zahar, Abdelhalim Ebaid () and Hind K. Al-Jeaid
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Ebrahem A. Algehyne: Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia
Musaad S. Aldhabani: Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia
Mounirah Areshi: Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia
Essam R. El-Zahar: Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, P.O. Box 83, Al-Kharj 11942, Saudi Arabia
Abdelhalim Ebaid: Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia
Hind K. Al-Jeaid: Department of Mathematical Sciences, Faculty of Applied Sciences, Umm Al-Qura University, Makkah 21955, Saudi Arabia

Mathematics, 2023, vol. 11, issue 15, 1-11

Abstract: 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 t 0 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 α . For small v compared with the speed of light c (i.e., v ≪ c ), our results tend to Newtonian mechanics, i.e., t → t 0 . For v comparable to c such as v = 0.9994 c , our numerical results are compared with the experimental ones for the TD of the muon particles μ + . Moreover, the influence of the arbitrary-order α on the FTD is analyzed. It is also declared that at a specific α , there is an agreement between the present theoretical results and the corresponding experimental ones for the muon particles μ + .

Keywords: fractional calculus; time dilation; special theory of relativity; muon decay (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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