A Derivation of GravitoElectroMagnetic (GEM) Proca-type Equations in Fractional Space

Transcription

1 A erivation of GravitoElectroManetic (GEM) Proca-type Equations in Fractional Space Victor Christianto 1, Abstract In a recent paper, M. Zubair et al. described a novel approach for fractional space eneralization of the differential electromanetic equations. A new form of vector differential operator el, and its related differential operators, is formulated in fractional space. Usin these modified vector differential operators, the classical Maxwell equations have been worked out for fractal media. In the meantime, there are other papers discussin fractional Maxwell equations. However, so far there is no derivation of Proca equations and Proca-type equations for GEM in fractional space. Therefore in this paper I present for the first time a derivation of GravitoElectroManetic (GEM) Proca-type equations in fractional space. Considerin that Proca equations may be used to explain some electromanetic effect in superconductor, then fractional GEM Proca-type equations may be expected to explain some ravitomanetic effects of superconductor for fractal media. It is our hope, that this paper may stimulate further investiation and experiments in particular with respect to ravitomanetic effects. Introduction There has been much interests to study different physical phenomenon in fractional dimensional space durin the last few decades. It is also important to mention that the experimental measurement of the dimension of real world is 3 ± 1 6, not exactly 3.[1] In a recent paper, M. Zubair et al. described a novel approach for fractional space eneralization of the differential electromanetic equations.[1] A new form of vector differential operator el, and its related differential operators, is formulated in fractional space. Usin these modified vector differential operators, the classical Maxwell equations have been worked out for fractal media. In the meantime, there are other papers discussin fractional Maxwell equations.[][3] However, so far there is no derivation of Proca equations and Proca-type equations for GEM in fractional space. Therefore in this paper I present for the first time a derivation of GravitoElectroManetic (GEM) Proca-type 1 URL: 1

2 equations in fractional space. Considerin that Proca equations may be used to explain some electromanetic effect in superconductor[4], then fractional GEM Proca-type equations may be expected to explain some ravitomanetic effects of superconductor for fractal media.[5] It is our hope, that this paper may stimulate further investiation and experiments in particular with respect to ravitomanetic effects. Review of previous result: Maxwell equations in fractional space I will not re-derive Maxwell equations here. For a ood reference on Classical Electrodynamics, see for example Julian Schwiner et al. s book [9]. Penrose also discusses Maxwell equations shortly in his book: The Road to Reality. [1] Zubair et al. were able to write a differential form of Maxwell equations in far-field reion in the fractional space as follows [1]: div = ρ, (1) v div B =, () B curle =, (3) t curl H = J +, (4) t and the continuity equation in fractional space as: divj ρ t v =, (5) where divand curlare defined as follows [1]: F 1 ( α 1) F F 1 ( α 1) F F 1 ( α 1) F divf = F = x x y y z z x 1 x y y z 3 z, (6) curl F xˆ yˆ zˆ 1 α 1 1 α 1 1 α3 1, (7) x x y y z z 1 = F = F F F x y z

3 where parameters ( < α1 1, < α 1and < α3 1) are used to describe the measure distribution of space where each one is actin independently on a sinle coordinate and the total dimension of the system is = α1 + α + α3. [1] Proca Equations in Fractional Space Proca equations can be considered as an extension of Maxwell equations, and they have been derived in various ways, see for instance [4][6][7]. It can be shown that Proca equations can be derived from first principles [6], and also that Proca equations may have link with Klein-Gordon equation [7]. In this paper I will not attempt to re-derive Proca equations, instead I will use Proca equations as described in [6]. Then I will derive the Proca equations in fractional space, in accordance with Zubair et al. s approach as outlined above. [1] Accordin to Blacklede [7], Proca equations can be written as follows: where: ρ E = ε κ φ, (8) B =, (9) B E =, (1) t E = + + B µ j ε µ κ A, (11) A φ = E, (1) B = A, (13) mc κ = ħ. (14) Therefore, by usin the definition in equation (6) and (7), we can arrive at Proca equations in fractional space from (8) throuh (13), as follows: 3

4 where: ρ dive = ε κ φ, (15) divb =, (16) B curle =, (17) t E curlb = µ j + ε µ + κ A, (18) A φ = E, (19) B = curl A, () and el operator can be defined as follows [1]: 1 α1 1 1 α 1 1 α3 1 = + xˆ + + yˆ + + zˆ. (1) x x y y z z To my best knowlede so far, the above expression of Proca equations in fractional space has not been proposed elsewhere before. GravitoElectroManetic (GEM) Proca-type Equations in Fractional Space The term GravitoElectroManetism (GEM) refers to the formal analoies between Newton s law of ravitation and Coulomb s law of electricity. The theoretical analoy between the electromanetic and the ravitational field equations has been first suested by Heaviside in 1893, see [8]. The fields of GEM can be defined in close analoy with the classical electrodynamics. Therefore, if we can consider Proca equations as extension of Maxwell equations, then we can also find Proca-type equations for GEM. In accordance with emir [8], the Proca-type equations for GEM can be expressed straihtforward from their electromanetic counterpart as follows (Here I use emir s notations instead of Blacklede s notations): E = ρ κ φ, () e 4

5 H =, (3) H E = t, (4) E e e H = J + + κ A, (5) where the fields E and H can be defined in terms of the potentials just as iven in equation (1) and (13), and the term κ represents the inverse Compton wavelenth of the raviton, [8] m c κ =. (6) ħ Now I will present the Proca-type equations for GEM in fractional space usin the same method as described in the previous section and equation (6) and (7), which can be written as follows: div E = ρ κ φ, (7) e div H =, (8) curl E H = t, (9) E e e curlh = J + + κ A, (3) To the best of my knowlede, the above expression of Proca-type equations for GEM in fractional space has not been proposed elsewhere before. Fractional Helmholtz equation and solution of classical wave equation in fractional space It is worth notin here that Zubair also wrote Helmholtz equations in fractional space for E and H field as a consequence of his Maxwell equations in fractional space, as follows [1]: E E µε =, (31) 5

6 H H µε =. (3) In another paper, Zubair, Muhal & Naqvi ive a solution of this kind of Helmholtz equation in fractional space [11]. The Laplacian operator in -dimensional fractional space is defined as follows: α1 1 α 1 α3 1 = x x x y y y z z z. (33) Then they derive a solution of equation (31) with the help of Bessel equation [11]. It is also interestin to note here, that Shpenkov has suested that a classical wave equation which is essentially the same with Helmholtz equation can be used to derive a periodic table of elements which is near to Mendeleyev s periodic table [1][13]. This result is in contradiction to spherical solution of Schrodiner equation which does not explain any atom, except perhaps hydroen[14][15]. Therefore it seems worth to discuss what the effect of an extension of classical wave equation in fractional space to the structure of atoms and molecules is. A review of Shpenkov s interpretation and use of classical wave equation has been submitted by this writer [16]. Since the classical wave equation as described by Shpenkov is the same with the equation of vibratin strin in 1-dimension, then it seems to compare the solution of equation (31) with solution of fractal vibratin strin. A few papers have been written discussin this fractal vibratin strin in detail, which can be found elsewhere. [17][18] It appears to be worth to study spherical solution of this fractal vibratin strin equation in order to verify Shpenkov s results, includin his periodic table of elements. One possible way to find such a solution of fractal vibratin strin is to obtain numerical solution of such an equation, by a method of convertin fractional differential equation to partial differential equation as proposed by He & Li [19]. After a partial differential equation (PE) is obtained then it seems not so difficult to find its numerical solution with computer alebra packaes like Mathematica, Maple or MatLab. Here I also propose to investiate further the spherical solution of Helmholtz equation correspondin to Proca-type equations for GEM in fractional space. This kind of investiation may be useful for the study of ravitomanetic effect and ravitational wave. Concludin remarks In a recent paper, M. Zubair et al. described a novel approach for fractional space eneralization of the differential electromanetic equations. A new form of vector 6

7 differential operator el, and its related differential operators, is formulated in fractional space. Usin these modified vector differential operators, the classical Maxwell equations have been worked out for fractal media. In the meantime, there are other papers discussin fractional Maxwell equations. However, so far there is no derivation of Proca equations and Proca-type equations for GEM in fractional space. Therefore in this paper I present for the first time a derivation of GravitoElectroManetic (GEM) Proca-type equations in fractional space. Considerin that Proca equations may be used to explain some electromanetic effect in superconductor, then fractional GEM Proca-type equations may be expected to explain some ravitomanetic effects of superconductor for fractal media. It is our hope, that this paper may stimulate further investiation and experiments in particular with respect to ravitomanetic effects. Here I also propose to investiate further the spherical solution of Helmholtz equation correspondin to Proca-type equations for GEM in fractional space. This kind of investiation may be useful for the study of ravitomanetic effect and ravitational wave. Acknowledments Special thanks to r. Geore Shpenkov for sendin his papers to this writer. And many thanks to Prof. Akito Takahashi and Prof. L. Wilardjo for their comments on Shpenkov s work. Version 1.: April 13 th, 14 VC, victorchristianto@mail.com References [1] Zubair, M., Muhal, M.J., Naqvi, Q.A., & Rizvi, A.A. 11. ifferential Electromanetic Equations in Fractional Space. Proress in Electromanetic Research, Vol. 114, URL: [] Tarasov, Vasily E. 8. Fractional vector calculus and fractional Maxwell equations. Annals of Physics 33, , URL: [a] also in arxiv: [math-ph], URL: 7

8 [3] Ostoja-Starzewski, Martin. 1. Electromanetism in anisotropic fractal media. Z. Anew. Math. Phys.(ZAMP) OI 1.17/s z. URL: 56b39d64e pdf; [3a] Also in arxiv: [math-ph], URL: [4] Tajmar, M. 8. Electrodynamics in Superconductor explained by Proca equations. arxiv:83.38 [5] de Matos, C.J., & Tajmar, M. 6. Gravitomanetic London Moment and the Graviton Mass inside a Superconductor. arxiv:r-qc/6591 [6] Gondran, Michel. 9. Proca equations derived from first principles. arxiv:91.33 [quant-ph], URL: [7] Blacklede, Jonathan M. 7. An Approach to Unification usin a Linear Systems Model for the Propaation of Broad-band Sinals. ISAST Transaction on Electronics and Sinal Processin, Vol. 1, No. 1, 7. URL: [8] emir, Suleyman. 13. Space-time alebra for the eneralization of ravitational field equations. Pramana Vol. 8 No. 5 (Indian Academy of Sciences), May 13, URL: [9] Schwiner, Julian, eraad, Jr., Lester L., Milton, Kimball A., & Tsai, Wu-yan Classical Electrodynamics. Readin, Massachusetts: Perseus Books. 591 p. [1] Penrose, Roer. 4. The Road to Reality: A Complete Guide to the Laws of the Universe. London: Jonathan Cape. 113 p. URL: [11] Zubair, M., Muhal, M.J., & Naqvi, Q.A. 11. On electromanetic wave propaation in fractional space. Nonlinear Analysis: Real World Applications. doi:1.116/j.nonrwa [1] Shpenkov, Geore P. 13. ialectical View of the World: The Wave Model (Selected Lectures). Volume I: Philosophical and Mathematical Backround. URL: [13]. 6. An Elucidation of the Nature of the Periodic Law, Chapter 7 in "The Mathematics of the Periodic Table", Rouvray,. H. and Kin, R. B., ed., Nova Science Publishers, NY, pp

9 [14] Shpenkov, Geore P. & Kreidik, Leonid G. 5. Schrödiner s error in principle. Galilean Electrodynamics Vol. 16, No. 3, 51-56, (5); URL: [15] Kreidik, Leonid G., & Shpenkov, Geore P.. Important Results of Analyzin Foundations of Quantum Mechanics, Galilean Electrodynamics & QE-EAST, Vol. 13, Special Issues No., 3-3; URL: [16] Christianto, Victor. 14. The Spherical Solution of Schrodiner equation does not aree with any experiment. Submitted to Prespacetime Journal, April 14. Available at: [17] Tarasov, Vasily E. 5. Wave Equation for Fractal Solid Strin. Mod. Phys. Lett. B, Vol. 19, No. 15, URL: [18] Hu, Min-Shen, Aarwal, Ravi P., & Yan, Xiao-Jun. 1. Local Fractional Fourier Series with Application to Wave Equation in Fractal Vibratin Strin. Abstract and Applied Analysis Vol. 1, Article I 56741, 15p. doi:1.1155/1/ URL: [19] He, Ji-Huan, & Li, Zhen-Biao. 1. Convertin Fractional ifferential equations into Partial ifferential Equations. Thermal Science, Year 1, Vol. 16, No., , URL: 9