The effect of deformation of special relativity by conformable derivative
Keywords:conformable derivative, fractional calculus, special relativity
In this paper, the deformation of special relativity within the frame of conformable derivative is formulated. Within this context, the two postulates of the theory are re-stated. Then, the addition of velocity laws are derived and used to verify the constancy of the speed of light. The invariance principle of the laws of physics is demonstrated for some typical illustrative examples, namely, the conformable wave equation, the conformable Schrodinger equation, the conformable Klein-Gordon equation, and conformable Dirac equation. The current formalism may be applicable when using special relativity in a nonlinear or dispersive medium.
J. D. Jackson, Classical electrodynamics, (1999).
J. H. Smith, Introduction to special relativity. (Courier Corporation, 1995).
W. C. Michels and A. Patterson, Special relativity in refracting media, Physical Review 60 (1941) 589.
M. E. Crenshaw, Reconciliation of the Rosen and Laue theories of special relativity in a linear dielectric medium, American Journal of Physics, 87 (2019) 296.
B. Mashhoon, Nonlocal special relativity: Amplitude shift in spin-rotation coupling, arXiv preprint arXiv:1204.6069, (2012).
I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier, 1998.
K. Oldham and J. Spanier, The fractional calculus theory and applications of differentiation and integration to arbitrary order. (Elsevier, 1974).
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations. elsevier, 204 2006.
K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations. (Wiley, 1993).
E. M. Rabei, I. Almayteh, S. I. Muslih, and D. Baleanu, Hamilton-Jacobi formulation of systems within Caputo’s fractional derivative, Physica Scripta, 77 (2007) 015101. [Online] Available: https://doi.org/10.1088/0031-8949/77/01/015101.
N. Sene, Analytical solutions of Hristov diffusion equations with non-singular fractional derivatives, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019) 023112, publisher: American Institute of Physics. [Online] Available: https://aip.scitation.org/doi/10.1063/1.5082645.
S. I. Muslih, D. Baleanu, and E. Rabei, Hamiltonian formulation of classical fields within Riemann-Liouville fractional derivatives, Physica Scripta, 73 (2006) 436. [Online] Available: https://doi.org/10.1088/0031-8949/73/5/003.
M. Yavuz and N. Sene, Stability Analysis and Numerical Computation of the Fractional Predator-Prey Model with the Harvesting Rate, Fractal and Fractional, 4 (2020) 35, https://www.mdpi.com/2504-3110/4/3/35.
E. M. Rabei, K. I. Nawaeh, R. S. Hijjawi, S. I. Muslih, and D. Baleanu, The Hamilton formalism with fractional derivatives, Journal of Mathematical Analysis and Applications 327 (2007) 891. https://www.sciencedirect.com/science/article/pii/S0022247X06004525.
N. Sene, Theory and applications of new fractional-order chaotic system under Caputo operator, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 12 (2022) 20, http://ijocta.org/index.php/files/article/view/1108.
E. M. Rabei, T. S. Alhalholy, and A. Rousan, Potentials of arbitrary forces with fractional derivatives, International Journal of Modern Physics A 19 (2004) 3083, https://www.worldscientific.com/doi/abs/10.1142/S0217751X04019408.
D. Baleanu, S. I. Muslih, and E. M. Rabei, On fractional EulerLagrange and Hamilton equations and the fractional generalization of total time derivative, Nonlinear Dynamics 53 (2008) 67, https://doi.org/10.1007/s11071-007-9296-0.
R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264 (2014) 65.
T. Abdeljawad, On conformable fractional calculus, Journal of computational and Applied Mathematics, 279 (2015) 57.
A. Atangana, D. Baleanu, and A. Alsaedi, New properties of conformable derivative, Open Mathematics, 13 (2015).
A. Al-Jamel, The search for fractional order in heavy quarkonia spectra, International Journal of Modern Physics A, 34 (2019) 1950054.
V. E. Tarasov, Fractional dynamics of relativistic particle, International Journal of Theoretical Physics, 49 (2010) 293.
W. S. Chung, H. Hassanabadi, and E. Maghsoodi, A new fractional mechanics based on fractional addition, Rev. Mex. Fis 67 (2021) 68
W. S. Chung, S. Zare, H. Hassanabadi, and E. Maghsoodi, The effect of fractional calculus on the formation of quantummechanical operators, Mathematical Methods in the Applied Sciences, (2020).
D. Pawar, D. Raut, and W. Patil, An approach to riemannian geometry within conformable fractional derivative, Prespacetime Journal, 9 (2018)
M. Al-Masaeed, E. M. Rabei, and A. Al-Jamel, WKB Approximation with Conformable Operator, http://arxiv.org/abs/2111.01547.
M. Al-Masaeed, E. M. Rabei, A. Al-Jamel, and D. Baleanu, Extension of perturbation theory to quantum systems with conformable derivative, Modern Physics Letters A, 36 (2021) 2150228 https://www.worldscientific.com/doi/abs/10.1142/S021773232150228X.
M. Al-Masaeed, E. M. Rabei, and A. Al-Jamel, Extension of the variational method to conformable quantum mechanics, Mathematical Methods in the Applied Sciences, https://onlinelibrary.wiley.com/doi/pdf/10.1002/mma.7963.
E. M. Rabei, A. Al-Jamel, and M. Al-Masaeed, The solution of conformable Laguerre differential equation using conformable Laplace transform, arXiv:2112.01322.
M. Al-Masaeed, E. M. Rabei, A. Al-Jamel, and D. Baleanu, uantization of fractional harmonic oscillator using creation and annihilation operators, Open Physics, 19 (2021) 395 https://doi.org/10.1515/phys-2021-0035.
A. A. Abdelhakim, The aw in the conformable calculus: It is conformable because it is not fractional, Fractional Calculus and Applied Analysis 22 (2019) 242, https://www.degruyter.com/document/doi/10.1515/fca-2019-0016/html.
M. Mhailan, M. A. Hammad, M. A. Horani, and R. Khalil, On fractional vector analysis, J. Math. Comput. Sci. 10 (2020) 2320.
L. Corry, Hermann minkowski and the postulate of relativity, Archive for History of Exact Sciences, (1997) 273
F. Mozaffari, H. Hassanabadi, H. Sobhani, and W. Chung, Investigation of the Dirac equation by using the conformable fractional derivative, Journal of the Korean Physical Society, 72 (2018) 987.
J. D. Bjorken and S. D. Drell, Relativistic quantum mechanics, (1964).
H. Nikolic, How (not) to teach Lorentz covariance of the Dirac ´ equation, European Journal of Physics, 35 (2014) 035003
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