Fractional Modified Special Relativity

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1 Absrac: Fracional Modified Special Relaiviy Hosein Nasrolahpour Deparmen of Physics, Faculy of Basic Sciences, Universiy of Mazandaran, P. O. Box , Babolsar, IRAN Hadaf Insiue of Higher Educaion, P.O. Box , Sari, Mazandaran, IRAN Fracional calculus represens a naural ool for describing relaivisic phenomena in pseudo- Euclidean space-ime. In his sudy, Fracional modified special relaiviy is presened. We obain fracional generalized relaion for he ime dilaion. Keywords: Special relaiviy; Time dilaion; Fracional special relaiviy; Fracional calculus 1- Inroducion The Theory of Special Relaiviy was firs published as a paper in 195[1]. Recenly, some modificaions on his heory have been proposed. For example, in ref. [3] some aspecs of nonlocal special relaiviy have been sudied. Dissipaive Lagrangian for a relaivisic paricle also has been discussed in ref. [4]. In more recen years aenion has been araced o he applicaion of Fracional Calculus (FC) o special (and general) relaiviy [9-13]. Fracional calculus can be used o model he nonlocal dissipaive phenomena in space (-ime) [7, 8]. In paricular FC represens a naural ool for describing relaivisic phenomena in pseudo-euclidean space-ime [9, 11]. The aim of his shor paper is o inroduce fracional modified special relaiviy as a new nonlocal dissipaive model for he special heory of relaiviy. In he following, fracional calculus [5, 6] is briefly reviewed in Sec.. Fracional generalized relaion for he ime dilaion is presened in Sec. 3. Finally, in Sec. 4, we will presen some conclusions and discussion. -Fracional calculus: Riemann-Liouvill inegral operaor Fracional calculus is he calculus of derivaives and inegrals wih arbirary order, which unify and generalize he noions of ineger order differeniaion and n-fold inegraion. In conras whi he definiion of he ordinary derivaive and inegraion, he definiion of he fracional ones is no unique. In fac, here exiss several definiions including, Grünwald - Lenikov, Riemann - Liouville, Weyl, Riesz and Capuo for fracional derivaive and inegral. For he purpose of his sudy we use he lefsided Riemann-Liouville fracional inegral operaor. The lef-sided Riemann -Liouvill fracional inegral of order of a funcion f () is defined by [5, 6] RL 1 1 I f ( ) ( ) f ( ) d ( ), (1) address: hnasrolahpour@gmail.com ; h.nasrolah-pour@umz.ac.ir

2 Where, is he se of posiive real numbers and ( ) denoes he gamma funcion. 3-Time dilaion for fracional special relaiviy We sar wih he following expression for he fracional generalized posiion x ( ) I ( v ) 1 RL () By use of he above equaion for he fracional generalized posiion we can easily derive dilaed ime formula according o he famous ligh clock experimen [] as below ( I ( v )) ( c ) RL c (3) Where c is he speed of ligh in vacuum and is proper ime. By use of Eq. (1), we can obain he following relaionship beween (proper ime) and (dilaed ime): ( ) (4) We can wrie Eq. (4) as ( 1) v where. Now we can see ha as 1, Eq. (4) gives c (5) (6) his leads o well-known ime dilaion formula of ordinary special relaiviy: 1. (7) If we plo (dilaed ime) wih respec o for he special case of v.99c we will have Definiion of Gamma funcion is given by ( 1) ( ). 1 ( ) e d,.for his funcion we have

3 Fig. 1. wih respec o for he special case of v.99 c, 1 Wih he aid of expression (5) we can find he (dilaed ime) versus (proper ime) for he arbirary case of. In paricular, for he special case of 1 we can draw versus Fig.. wih respec o for he special case of v 1.99 c, As shown in his figure, here is a major difference beween Fig.1 and Fig. near he origin of coordinae sysem. We can inerpre his anomalous behaviour as an immediae consequence of applicaions of fracional calculus in our model.

4 4- CONCLUSION AND DISCUSSION Fracional calculus can be used o model he nonlocal dissipaive phenomena in space (-ime). In paricular FC represens an effecive ool for describing relaivisic phenomena in pseudo-euclidean space-ime. Wih his moivaion we obain fracional generalized relaion for he well-known ime dilaion formula, o show he possibiliies for fuure sudies on fracional modified special heory of relaiviy heory. As we shown in Fig. (), here is a anomalous behaviour when we plo wih respec o for he arbirary case of ( 1) near he origin of coordinae sysem. We hope o sudy some oher aspecs of Fracional modified special relaiviy in fuure. Rrefrences [1] A. Einsein, Zur Elekrodynamik bewegeer Körper. Annalen der Physik 195, 17. (English ranslaion: On he elecrodynamics of moving bodies, reprined in Lorenz e al., 193.) [] Rober Resnick, Inroducion o Special Relaiviy (John Wiley and Sons,Inc. New York, London, Sydney 1968.) A.P.French, Special Relaiviy (Nelson, London, 1968). Miller, A.I., Alber Einsein's Special Theory of Relaiviy, Springer, [3] B. Mashhoon, Nonlocal special relaiviy, Ann. Phys. (Berlin) 17(8)1175. [4] G. Gonzalez, Relaivisic moion wih linear dissipaion. In. J. Theor. Phys. 46 (7) [5] I. Podlubny, Fracional Differenial Equaions Academic Press, New York, [6] R. Herrmann, Fracional Calculus: An Inroducion for Physiciss, World Scienific Press, 11. K. S. Miller, B. Ross, An inroducion o he fracional calculus and fracional differenial equaions, John Wiley & Sons, Inc., K. B. Oldham, J. Spanier, he Fracional Calculus, Academic Press, NewYork, S. G.Samko, A. A. Kilbas, O. I. Marichev, Fracional Inegrals and derivaives, Theory and Applicaions, Gordon and Breach, Yverdon, [7] L. Vázquez, "A Fruiful Inerplay: From Nonlocaliy o Fracional Calculus" en "Nonlinear Waves, Classical and Quanum Aspecs", , Ed.: V.V. Konoop y F. Abdullaev, NATO Science Series, Kluwer Academic Publishers 4. [8] V.E.Tarasov, Fracional Vecor Calculus and Fracional Maxwell s Equaions, Ann. Phys. 33 (8) [9] E. Goldfain, Fracional dynamics and he TeV regime of field heory, Comm. Non. Sci. Num. Siml. 13 (8) [1] E. Goldfain, Fracional dynamics, Canorian space ime and he gauge hierarchy problem, Chaos, Solions & Fracals, (4) [11] V. E. Tarasov, Fracional Dynamics of Relaivisic Paricle, In. J. Theor. Phys. 49(1) [1] R. A. EL-Nabulsi, Fracional Noale s Scale Relaiviy and emergence of complexified graviy, Chaos, Solions and Fracals 4 (9)

5 [13] R. A. EL-Nabulsi, Fracional illusion heory of space: Fracional graviaional field wih fracional exra-dimensions, Chaos, Solions and Fracals 4 (9)