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Relativistic Formulae for the Biquaternionic Model of Electro-Gravimagnetic Charges and Currents

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Relativistic Formulae for the Biquaternionic Model of Electro-Gravimagnetic Charges and Currents Journal of Modern Physics, 2017, 8, 1043-1052 http://www.scirp.org/journal/jmp ISSN Online: 2153-120X ISSN Print: 2153-1196 Relativistic Formulae for the Biquaternionic Model of Electro-Gravimagnetic Charges and Currents Lyudmila Alexeyeva Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan How to cite this paper: Alexeyeva, L. Abstract (2017) Relativistic Formulae for the Biqua- ternionic Model of Electro-Gravimagnetic Here the biquaternionic model of electro-gravimagnetic field (EGM-field) has Charges and Currents. Journal of Modern been considered, which describes the change of EGM-fields, charges and cur- Physics, 8, 1043-1052. rents in their interaction. The invariance of these equations with respect to the https://doi.org/10.4236/jmp.2017.87066 group of Poincare-Lorentz transformations has been proved. The relativistic Received: May 3, 2017 formulae of transformation for density of electric and gravity-magnetic Accepted: June 11, 2017 charges and currents, active power and forces have been obtained. Published: June 14, 2017 Keywords Copyright © 2017 by author and Scientific Research Publishing Inc. Electro-Gravimagnetic Field, Electric Charge, Gravimagnetic Charge, Current, This work is licensed under the Creative Poincare-Lorentz Transformation, Relativistic Formula Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access 1. Introduction The mathematical representation of physical processes in various media is al- ways connected with the choice of the coordinate system. For example, the coordinates of vectors, gradients, and rotors of physical fields change by transfer from one coordinate system to another. Therefore, in the construction of ma- thematical models of physical processes, in addition to the various equations and relationships which connect their physical characteristics in some coordinate system, the laws of transformation of these quantities and relations must be de- termined when passing from one coordinate system to another. The most common one is the Cartesian coordinate system. Since its choice in Euclidean space is quite arbitrary both with respect to the origin of coordinates and with respect to the orientation of its base frames, therefore, for any mathe- matical model it is necessary to determine the transformation formulas of all re- lations and magnitudes for orthogonal group of coordinate transformations and DOI: 10.4236/jmp.2017.87066 June 14, 2017 L. A. Alexeyeva for frame shift. It is well known that the equations of motion of homogeneous media in Newtonian mechanics are invariant with respect to these transforma- tions and Galilean transformations in coordinate systems moving relative to each other at a constant speed in a fixed direction. For isotropic media, the equ- ations of motion even retain their form, for anisotropic media they are trans- formed in accordance with the rules of tensor analysis. However in dynamics of electromagnetic media these properties don’t remain. In particular, H.A. Lorentz proved that Maxwell equations are not invariant under Galilean transformation, and he constructed the linear transformation keeping their invariance in Minkowski space [1]. This transformation began to be called Lorentz’s transformation, and transformation formulae (relativistic formulae) have formed the basis of the relativity theory of A. Einstein. Poincare, investigating Maxwell’s equations, constructed the group of linear transforma- tions keeping their invariance in Minkowski space [2]. They represent superpo- sition of orthogonal transformations and Lorentz’s transformation. In the real work the biquaternionic model is considered, which is earlier of- fered by the author for electro-gravimagnetic (EGM) fields, charges and cur- rents, and their invariance relative to the group of Poincare-Lorentz transforma- tions on Minkowski space is investigated. In the papers [3] [4] the biquaternionic model of electro-gravimagnetic (EGM) field has been considered which is designated as biquaternionic A-field. For this the Hamilton form of Maxwell’s equations has been used [5] and its qu- aternionic record [6]. Based on the hypothesis of a magnetic charge, it’s carried out field complexification with the introduction of the gravimagnetic density to the Maxwell equations. The closed system of equations of EGM-charges and currents interaction has been constructed, which is the field analogue of the three famous Newton’s laws for mechanics of material points. Here we prove an invariance of these equations with respect to Poincare-Lo- rentz transformation. We obtain the relativistic formulas for the densities of electric and gravimagnetic charges and currents, active power and forces by their interaction. At first we give some definitions of differential algebra of biquater- nions, for not to refer the reader to other author’s works to read this article. Then we consider how biquaternionic differential operators-mutual bigradients are changing by these transformations. After that we’ll consider equations of EGM-charge and current transformation under action of external EGM-field and construct relativistic formulas for all introduced values. 2. Biquaternions in the Minkowski space Some Definitions and Designations We consider the linear functional space of biquaternions F =fF + , where f is complex function , F is complex vector-function on Minkowski space M . It is linear space: aFG+ b = a( f ++ F) b( g += G) ( af + bg) +( aF + bG) (1.1) with the known operation of quaternionic multiplication 1044 L. A. Alexeyeva FG=+( f F) ( g += G) ( fg −( F,, G)) +( fG ++ gF[ F G]) (1.2) 3 Here scalar product (F, G) = ∑ FGkk, vector product k=1 3 3 [F, G] = ∑ ε klj FGe k l j , ε klj is Leavy-Chivitta symbol, x= ∑ xejj. We use the kl,, j= 1 j=1 following notations. F− =fF − , FF−−12=( f + ( FF, )) when it exists. Complex conjugate Bq. to F is F =fF + , where the bar over symbol denotes the complex conjugate number. Unitary Bq F if FF= FF = 1 . Conjugate Bq. F* =fF − . If FF* = , then F is self-conjugated Bq. Scalar production of biquaternions is the bilinear operation (FF1,, 2) =ff 12 + ( F 1 F 2) The norm of F is 22 F=( FF,,) =f ⋅+ f( FF) = f + F . The pseudo norm of F is 22 F =f ⋅− f( FF,.) = f − F We use biquaternionic differential operators which are named mutual bigra- dients: +− ∇=∂+∇ττii,, ∇=∂−∇ ± ∇F =( ∂τ ±∇i) ∓( fF +) =∂ ττ fiF( ∇,,) +∂ FifiF ±∇ ±[ ∇ ] ± ↔∇F =( ∂ττf∓ idiv F) +( ∂ Fi ±grad fi ± rot F) Here ∇=grad =( ∂123 , ∂ , ∂ ) . Their superposition has the useful property: 223 −+ +−∂∂ ∇ ∇=∇∇=22 −∑ =, (1.3) ∂∂τ j=1 x j where is classic wave operator (d’alambertian). 3. The Poincare-Lorentz Transformation on M We can quaternize Minkowski space if to enter complex conjugate Bqs: ZZ=+=−ττix,. ix They are self-conjugated: ZZ=**, ZZ = and 1045 L. A. Alexeyeva 2 2 Z= Z = ( ZZ,,) Z= Z = Z Z (3.1) If to enter the Bqs UU(θθθθθθ,e) =+=−= ch ie sh,( , e) ch ie sh, e 1 UU= UU =−=ch22θθ sh 1 UUUU=**, = then we have the biquaternionic form of Lorentz transformation Z′′= UZU**,. Z= UZ U (3.2) This implies ττ′′=+ch2 θ(ex ,) sh2 θ , x =++ x τ esh2 θ 2 eex( ,) sh2 θ (3.3) ττ=−′ch2 θ(ex , ′) sh2 θ , x =−+ x ′′ τ esh2 θ 2 eex( , ′) sh2 θ If to denote 1 v ch2θθ= , sh2= ,v < 1 (3.4) 11−−vv22 then we have well known formulas of Lorentz transformation: ττ++vex( ,,) eex( ) v τ ′′=,,x =−+( x eex( )) (3.5) 11−−vv22 ττ′−−vex( ,,) eex( ′′) v τ =,,x =−+( x′ eex( )) , 11−−vv22 which corresponds to the motion of the system in the direction of the vector e with a dimensionless velocity v < 1 . The pseudo norm is preserved as 22 Z′ = UZUUZU = Z Similarly, with the help of conjugated biquaternions W(φφφφφφφ,e) =−= cos e sinUW( ie ,,) * ( , e) =+ cos e sin, e = 1, WW**= W W = 1, we can write a group of orthogonal transformations on the vector part of Z : Z′′= WZW **, Z= W Z W (3.6) This implies ττ′′==,x sin2( φ)[ xe ,] + cos2( φ)( x −+ eex( ,)) eex( , ) ττ==−+′,x sin2( φ)[ x ′ ,cos2 e] ( φ)( x ′′ −+ eex( ,)) eex( , ′) It is the turn of R3 around the vector e at the angle 2φ . The time τ does not change. This transformation preserves the norm and the pseudo-norm Z . Lemma1. Poincare-Lorentz transformation on M has the form: 1046 L. A. Alexeyeva − − Z′′= LZL**, Z= L Z ( L) (3.7) It is equal to ττ′ =ch2 θ + (ex ,) sh2 θ , x′ =τθ esh2 + eex( ,) ch2 θ +−( x eex( ,)) cos2φ +[ xe ,sin2;] φ (3.8) ττ=′′ch2 θ − (ex ,) sh2 θ , x=−τθ′ esh2 + eex( , ′) ch2 θ +−( x ′′ eex( ,)) cos2φ −[ x ′ ,sin2 e] φ and −− −− LL= L L = 1, L** ( L) =( L *) L = 1. Proof. By use of associativity of biquaternionic product we have * Z′ = LZL −−−− ⇒=Z L LZL** ( L) = L Z′ ( L*) After calculating the scalar and vector part we obtain (3.8). Here the unit vec- tor e determines the direction of motion of the new coordinate system, and other parameters define the speed of motion and the angle of rotation, as has been shown above. 4. Biquaternionic Representation of EGM-Field Characteristics For EGM-field description we entered next biquaternions [3] [5]: EGM-potential Φ =iφ −Φ EGM-intensity A =iAα + Charge-current Θ =−+(iJρ ) Energy-impulse of EGM-field * Ξ ==−=+0.5AA 0.5( A , A) 0.5 A , A W iP Here Ax( ,τ ) is the complex vector-function of intensities of EGM-field: A=εµ Ei + H, where EH, are the electric and gravimagnetic field intensities, εµ, are the constants of electrical conductivity and magnetic permeability. The potential part of the vector H describes the intensity of the gravitational field, and the vortex part defines the magnetic field. The charge density of the EGM-field ρ is expressed in terms of the electric charge density ρ E and the gravymagnetic mass density ρ H : ρ=div Ai = ρEH ερ − µ (4.1) Electro-gravimagnetic current J is expressed in terms of the electric current density ( j E ) and it gravymagnetic current density ( j H ): J=µε jijEH − (4.2) The equation of EGM field has the form 1047 L.
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