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

Author

Listed:
  • 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
    Department of Basic Engineering Science, Faculty of Engineering, Menoufia University, Shebin El-Kom 32511, Egypt)

  • 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)

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 + .

Suggested Citation

  • Ebrahem A. Algehyne & Musaad S. Aldhabani & Mounirah Areshi & Essam R. El-Zahar & Abdelhalim Ebaid & Hind K. Al-Jeaid, 2023. "A Proposed Application of Fractional Calculus on Time Dilation in Special Theory of Relativity," Mathematics, MDPI, vol. 11(15), pages 1-11, July.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:15:p:3343-:d:1206737
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    References listed on IDEAS

    as
    1. Sayed M. Khaled & Essam R. El-Zahar & Abdelhalim Ebaid, 2019. "Solution of Ambartsumian Delay Differential Equation with Conformable Derivative," Mathematics, MDPI, vol. 7(5), pages 1-10, May.
    2. Agarwal, Praveen & Singh, Ram, 2020. "Modelling of transmission dynamics of Nipah virus (Niv): A fractional order Approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 547(C).
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