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SSRG International Journal of Applied (SSRG-IJAP) – Volume 7 Issue 1 – Jan to April 2020

Ether and In biquaternionic presentation Alexeyeva L.A. Institute of Mathematics and Mathematical Modeling, Kazakhstan

Abstract The paper is related to the construction of began with Maxwell himself and has a rather solutions of biquaternionic equation of either. It is extensive bibliography ([6-16] and others). Electro-Gravimagnetic (EGM) equation which Removing the restriction on the zero scalar is generalization of Maxwell equations for EM-field part of the of EM field allows in algebra. The special class of us to give a biquaternion representation of the EGM solutions of this biquaternionic (biwave) field, as well as a biquaternion representation of equation is mono-chromatic wave solutions that charges and currents, which contains the EGM describe periodic oscillations and of a fixed charge in the complex scalar part, and the EGM field . Here, fundamental and generalized strength vector in the complex vector part. The latter solutions of this equation are constructed and studies contains the electric field strength (in the real part) that describe photons as EGM waves of a fixed and the gravimagnetic field strength in the complex frequency, emitted by EGM charges and EGM field. It is the union of the vortex magnetic field with currents. Solutions of the homogeneous biwave the potential into one equation are also constructed that describe free gravimagnetic field. photons as free EGM waves of a fixed frequency. The We name ether the EGM field, which is density and of photons, their - described by the biquaternion of the strength of the are determined. Solutions of the EGM field. Its scalar part is naturally called the homogeneous biwave equation are also constructed density of the ether. And the vector part, by that describe free photons as free EGM waves of a construction, describes the strength of the electric and fixed frequency. Based on them, a biquaternion gravimagnetic fields. The biwave equation in this representation of and its energy-momentum case expresses EGM charges and currents through the density are given. bigradient of the EGM field strength. The present work is related to the Keywords: ether, electro-gravimagnetic field, construction of periodic solutions of the EGM field biquaternion, , Maxwell equation, stationary equation. A special class of solutions of the biwave oscillations, energy-momentum, light equation is monochromatic wave solutions that describe periodic oscillations and waves of a fixed I. INTRODUCTION frequency. The equation for biquaternionic amplitudes of oscillations (biamplitute) is the In [1–5], the author developed a stationary biwave equation, in this case becomes biquaternion model of the electro-gravimagnetic field elliptical. Here, fundamental and generalized (EGM field), EGM charges and EGM currents, and solutions of this equation are constructed that EGM interactions based on biquaternion describe photons as EGM waves of a fixed frequency, generalizations of the Maxwell and Dirac equations. emitted by EGM charges and EGM currents. Note that the biquaternion representation of Solutions of the homogeneous biwave equation are Maxwell's equations, which describes the relationship also constructed that describe free photons as free of the EM field with electric charges and currents, is EGM waves of a fixed frequency. based on the representation of the EM field vectors in the form of one biquaternion with a certain restriction II. SCALAR AND VECTOR COMPLEX on its scalar part, which is zero. The Maxwell CHARACTERISTICS OF EGM FIELD equations in the biquaternionic representation are We introduce the notation for the known and equivalent to one biquaternion wave equation new quantities to describe the EGM field, electric and (biwave equation), which expresses the charge- gravimagnetic charges. current biquaternion through the biquaternionic - Vectors E, H are the electric and gravimagnetic gradient (bigradient) of the EM field. The fields; biquaternion wave equation belongs to the class of - Scalars ρ , ρ are densities of electric and hyperbolic and describes solutions of hyperbolic E H gravimagnetic charges (mass charges); systems of 8 differential equations in partial - Vectors j , j are densities of electric and derivatives of the first order. Note that the E H gravimagnetic currents. quaternionic representation of Maxwell's equations Here we combined a potential gravitational

field with a vortex magnetic field, which, along with

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electric charges and currents, allows us to introduce a Here we use the conjugated biquaternion: gravimagnetic charge and current. Using these values, AA*   ,where A aA is mutual biquaternion. we introduce the following complex characteristics of Everywhere the line above the symbol means the the EGM field: complex conjugation of the scalar and vector parts of the biquaternion. - ether tension When determining the energy-momentum, A AEH  iA  E  i H, the operation of multiplication is used - ether density according to the rule: i EH//,    - charge density FB()()f  F b  B (1)   EH//,  i    fb( F , B )  { fB  bF  [ F , B ]} - current density J JEHEH  iJ   j  i j . Hereinafter, we use the Hamiltonian scalar-vector notation of biquaternions: Here  EE(x , t ), j ( x , t ) are densities of electric charge and current;  HH(x , t ), j ( x , t ) are densities of BGb  B,  g  G , gravimagnetic charge and current. Constants 휀, 휇are electrical conductivity and magnetic permeability of denoting scalar and vector parts with the same lowercase and capital letters in italics (with the , c 1/  is the of light. exception of basic elements e , m  0,1,2,3, of Note that such complex characteristics for m ). In formulas (1) the EM-field were introduced by Hamilton, which allows a system of 8 Maxwell equations (two vector for the rotors E and H and two scalar for their 3 (,)FBFB , divergence) to be reduced to 4 equations (one vector  jj j1 and one scalar for rotor and divergence of the complex vector of EM-field intensity). Generalized 3 [,]FBFB  e , and fundamental solutions for such a Hamiltonian  klm k l m system of equations were constructed by the author in k, l , m 1 [16]. The system of Maxwell equations allows the scalar and vector product of these vectors, klm further generalization in biquaternions algebra, which is the pseudo-Levi-Civita tensor,. leads it to a single biquaternionic wave equation. In the case of a zero density of the EGM field (α = 0), the energy-momentum biquaternion contains the known density of the EM field, and in III. EITHER AND CHARGE-CURRENT the vector part, the Pointing vector: BIQUATERNIONS WAAAEH0,52  0,5(,)0,5*  2  2 , Let us introduce the following biquaternions   of electro - gravimagnetic field, mass charges and P A  A*1  c E  H currents in Murkowski  (,):, x x  R3 The biquaternion of the energy-momentum of an   ct , t is time, c is light speed: EGM charge-current has the form: tension A()(),,x  i   , x A() , x * ΞΘΘW   iP  0,5  charge-current (2) 22  Θ()(),x  i   , x J() , x , 0,5  J  i Re J  0,5Im J , J  Either energy-momentum IV. THE RELATIONSHIP BETWEEN EITHER AND CHARGE– CURRENT. Ξ(),x  0,5AA* GENERALIZED MAXWELL EQUATION

w()(, x iP , x ) The relationship between the intensity of the EGM field and the density of the EGM of charge- currents has the form of a generalized Maxwell charge-current energy-momentum equation in the biquaternion form. For this, * ΞΘΘ0,5 w   iP  differential operators are introduced - mutual bi-

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gradients  ,  , whose action is determined by the ΘΘ(,)x (,)exp( x  i  t ) . (5) rule of quaternion multiplication: Here is the complex biamplitude from the class of  generalized biquaternions, whose components are B   i   b(,)(,) x  B x   generalized functions of slow growth: Θ(xR ) B '(3 ) .

 b i(,[,]  B   B  i  b   B   Similarly, a solution of Eq. (1) can be represented in a similar form:

 b idiv B   B  i grad b  i rotB ΦΦ(,)x (,)exp( x  i  ) (6) Using them, we represent this relationship in the form of the following postulate [1-5]. In this case, from equation (3) it follows

Postulate 1. EGM charges-current are a  i  i  ΦΦΘ  i   (,) x bigradient of e EGM field tension: From here we get the stationary equation for  A()(  A,, x)() Θ x (3)  biamplitudes– photon equation: This biwave equation is equivalent to a system of two scalar and vector equations:  ΦΘ()()x i x (7)  (  ,xA )    div ,  (4) Here, the operators     are called  - J( , x ) i grad   A  i rot A   gradients. Their composition is commutative and equal to the scalar Helmholtz operator: Hence, for α = 0, the Hamiltonian form [18] of the Maxwell equations follows:     2             divAx ( , ), (8) (5)  A  irot A  J ( , x ) We use this property to construct solutions (7). Taking from it a mutual - gradient we obtain the from which, writing out the real and imaginary parts, we get the classic Maxwell system. inhomogeneous Helmholtz equation Therefore, we call equation (1) the 2  generalized Maxwell equation (GMEq). And  ΦΘ(,)(,)x   i  x  (9) postulate 1 should be called Maxwell's law for the EGM field. It has a deep physical meaning, namely: to which each component of the biquaternion satisfies. The solution of this equation is easy to construct if to EGM charges-currents (substance) are use the fundamental solution of the Helmholtz derivatives of the EGM field (ether) equation, which satisfies the conditions of Somerfield. It has the form [18]: Those of the two states of (matter and field), the field is primary, and the substance is secondary. eix Substance it is a physical manifestation of the (,)x  (10) heterogeneity and motion of the ether. 4 x Next, we will build monochromatic OAM solutions for describing photons - EGM waves on air Taking into account the time factor, it describes with an oscillation frequency diverging harmonic spherical waves that satisfy the Somerfield radiation conditions [18]. V. BIQUATERNIONS OF PHOTONS From (9), using the properties of the fundamental solution and the properties of We construct and study the properties of the convolution, we obtain a solution, that has the form monochromatic solutions of GMEq that we use to of convolution. That is describe the photons emitted by the EGM charge- currents which are on the right side of this equation. photon biquaternion In particular, we consider the case of harmonic oscillations with a frequency when the right-hand side of (1) has the form:

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ΦΘ(,)xi    the propagations at a fixed point in  space: ix ie  (11)    Θ(.)x  4 x   0211 w  ,  xx222 Definition. Let’s call a photon elementary,  (12) 2 generated by a concentrated EGM charge of the form: 0  Pe  x  x 2 iΘ(,)() x eii x e  . VI. FREE PHOTONS AS PLANE HARMONIC We calculate its biquaternionic EGM WAVES representation using (11): Let’s consider the solutions of the homogeneous ir 1  e photon equation (7): Φ0 (,)()xx      4 r  Φ(x ) 0 (13) i r i  r i  r  11 e e e       grad , 44r r r As follows from (9), its solution satisfies to homogeneous Helmholtz equation rx . As a result, we obtain the biquaternion of an 2 elementary photon, whose biamplitude is  Φ(x , )  0 (14)

eir 1 and can be represented as Φ0(,)x  0   0     ex  i    4rr 4 ei r e i r 1 x  Φ(,)(,)x  aj  j x  e j (15) 00 ,,   i  exx e  44r r r r j0 (12) where the scalar potential 0 (,)x and vector It describes a spherical EGM wave in ether, emitted potential jj(,)xe are arbitrary solutions of the by a concentrated electric charge. It moves at a speed homogeneous Helmholtz equation. of 1 (in the original space-time at the c) A simple solution to this equation is plane and decays at as 1 / r. Amplitudes of its harmonic wave (taking into account the time EGM density and EGM tension are proportional to exponent): its frequency: (x ,  ) exp(i ( k , x )), k  (16)  j (,),x  0 4r which propagate in the direction of the vector k at a 112 (,)x    speed of 1 ( a j are arbitrary complex constants). We 0 4r r2 consider each of these terms separately. The biamplitude of the free planar harmonic The energy-momentum density of an elementary photons generated by potential for j=0 has the form photon is equal to

k  0 0 0  Φ0(x , )   0   0  grad  0  ΞΦΦw   iP  0,5 00  (17) eix(,)kk(1 ie )        1 1   1   exx i        e  i      2 xx    2 x        where ek  k / .  EGM density, EGM tension and their 1122 22   iex amplitudes are  xx2 k kk ei(,) e x ,,   0 00 From here we get photon energy density w and k 0 kk iek i(,) e x ,.    Pointing vector P , which determines the direction of 00

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Note, in this EGM wave vectors of electric and If e[]j perpendicular to k ( cos( j ) 0 ), it is gravimagnetic fields are parallel : EH. The density transversely polarized harmonic EGM wave with of its energy-momentum is zero EGM density and Pointing vector. Such wave is named a tensional wave. k k k2 k ΞΦΦ0(x , ) 0,5 0 0 *  (1  ie ) But if (k ,ej ) 0 this EGM wave has longitudinal part in electric and gravimagnetic Its energy density and Pointing vector are constant, tensions. Using these free harmonic photons and independent of x: formula (14), we obtain a biquaternionic representation of plane harmonic photons: k22 k k w00, P e (18) 4 kk Φ (,)()xa  jjk Φ It is plane longitudinal harmonic EGM wave. j0 (17) The biamplitude of a free photon generated 4 kk by the j-th vector potential (j=1, 2, 3) has the next aj()(,)(,)k  j x    j x   form j0

k  where ax() are arbitrary functions or constants. Φ j(,)xe   j[] j   j i(,)(,)(,)k x i k x i k x n They are propagate in the direction of a wave vector  ikj e  e e[] j rot  e e n j   k at the speed of 1 (in the original space-time with ei(,)(,)(,)(,) k x  ik e ik x  e e i k x   e  e i k x  n  the speed of light c and frequency ωc). Its  j[] j lmn l m j  longitudinal part in a vector addend is a light pressure ix(,)k e  ikj  e[] j  i mjl k m e l   p(x,ω) which we observed in practice: ix(,)k  e  ikj  e[] j  i ke, j  4 kk p( x , ) ajj (k ) ( x , ), e  . j0 (19)

The EGM density and EGM tension and their VII. BIQUATERNIONIC REPRESENTATION amplitudes are OF LIGHT

kix(,) k k Consider the photons emitted by j(,)x   ik j e ,  j (,) x    cos(),  j monochromatic charges-currents. Using the property k kix(,) k of convolution differentiation, from formula (11) we (,)[,],x  e  i e e e j [] j j  obtain the following biquaternionic representation k 2 through elementary photons jj(x , )   2  cos (  ),

ΦΘΦ(,)(.)(,)x i x 0 x  k  (18) where cos( jj ) (ee[] , ) . Note, in this EGM wave the vectors of Here, the convolution for regular biquaternions has electric and gravimagnetic tensions (E and H) are the next integral representation: perpendicular as in classic electrodynamics. The density of its energy-momentum is 0 ΦΘΦ(,)(,)(,)x i xy  ydydydy  1 2 3 3 k k k R ΞΦΦj(x , ) 0,5 j j * Тноsе EGM charges and currents are emitters of 0,5ikj  e[][] j  ik , e j   ik j  e j  i  k , e j   elementary photons, the intensity of which is 2 0.5k22  ke ,  i k e  determined by their biquaternion densities.  j j j[] j The energy-momentum density of such 2 2 photons is equal to: 0.5 k  i kjj e[]  2  1ie cos(jj ) [] ΞΦΦ(x , ) 0,5 

00* Its energy density and Pointing vector also are 0,5ΘΦΘΦ     (19) constant, independent of x: 0 0* * 0,5ΘΦΦΘ     kk22 wj(,) x  , P j (,) x   cos()  j e[] j (20)

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Light contains a whole spectrum of and Also note that he proposed biquaternionic can be represented as a cloud of elementary photons theory of ether and photons is very constructive. It in the form of the Fourier integral: allows determining its characteristics at any point in space-time what is impossible in models of quantum 2 . i ΛΦ(,)(,)x x  e d  REFERENCES 1 (20) [1] Alexeyeva L.A. The equations of interaction of A-fields and 2 the laws of Newton. News of National Academy of iΘΦ(.)(,) x0 x  ei d  Sciences of Kazakhstan. A physical and mathematical   1 series, 2004, No. 3, 45-53 (in Russian). [2] Alexeyeva L.A. Field analogues of Newton's laws for one ( 1, 2) model of the electro-gravimagnetic field. Hypercomplex where is the spectral interval of light (visible in geometry and physics, 6(2009), No. 1, 122-134 photons). Accordingly, it is also possible to (in Russian). determine the density of its energy-momentum: [3] Alexeyeva L.A. Newton’s laws for a biquater-nionic model of electro-gravimagnetic fields, charges, currents, and their interactions, Journal of Physical Mathematics, 2009, issue Λ  ΞΛΛ(,)(,)(,)x t x t x t (21) 1. Article ID S090604 , 5 pages, doi:10.4303/jpm/S090604 [4] Alexeyeva L. A. Biquaternionic Form of Laws of Electro- Gravimagnetic Charges and Currents Interactions. Journal In a similar way, we can build a cloud of free photons: of , 2006, issue 7, 1351–1358. http://dx.doi.org/10.4236/jmp.2016.711121. Ο(,)x   [5] Alexeyeva L.A. Biquaternionic model of electro- gravimagnetic fields an interactions. Advances in 4 2  i k theoretical and computational physics, 2(2019), issue 6, 1-8 eΩ (,)(,)(,)(), x bk Φ x  dSk d  nd  j j j (presented, Abstracts of 2 Global Summit on Physics. j0 1 k  Plenary forum, Paris. 26-27 September 2019, 19). ΞΟΟΟ (,)x 0,5(,) x * (,) x  [6] Hamilton W. R. On a new Species of Imaginary Quantities connected with a theory of . Proceedings of the (22) Royal Irish Academy, Nov 13, 1843, 424–434. where bjj( x , ),Ω ( x , ) are arbitrary regular [7] Edmonds J.D. Eight Maxwell equations as one quaternionic. Amer. J. Physics, 46(1978), No. 4, 430. functions and biquaternions that admit such a [8] Shpilker G.L. Hypercomplex solutions of Maxwell's convolution. Apparently, such biquaternions describe equations. Reports of Academy of Sciences of USSR. ball . 272(1983), No 6, 1359-1363. [9] Rodrigues, W. A., Jr., Capelas de Oliviera E. Dirac and Note that these formulas can also be used for Maxwell equations in the Clifford and Clifford singular biquaternions, such as simple and dabble bundles. Int. Journal of Theoretical Physics, 29(1990), 397– layers. Only in this case, convolutions must be taken 412. [10] Finkelstein D., Jauch J. M., Schiminovich S., Speiser D. according to the convolution rules for generalized Foundations of quaternion . J. Math. functions [18]. Phys, 3(1992), No. 2, 207–220. [11] Adler S. L. Quaternionic quantum mechanics and quantum

VIII. CONCLUSION fields. New York: Oxford University Press, 1995. [12] De Leo S., Rodrigues Jr. W. A. Quaternionic quantum The arbitrariness of the functions included in mechanics: from complex to complexities quaternions. Int. J. Theor. Phys. -36 (1997), 2725–2757. the determination of the solutions of the photon [13] Efremov A.P. Quaternions: Algebra, Geometry and equation allows us to construct an infinite of Physical Theories. Hypercomplex Numbers in Geometry the most diverse solutions for photons, light, photon Physics, 2004, No.1, 111-127. clouds from the formulas presented, which can be [14] Acevedo M., Lopez-Bonilla J., Sanchez - Meraz M. done by an interested reader. It would be useful to Quaternions, Maxwell Equations and Lorentz give such a construction to students and Transformations, Apeiron, 12(2005), No. 4, 371-379. [15] Marchuk N.G. Clifford Field Theory and Algebra undergraduates as programming exercises, which Equations. Moscow-Izhevsk, 2009. would allow them to show their ingenuity and skills [16] Alexeyeva L.A. Biquaternions algebra and its applications in the study of a wide variety of light phenomena. by solving of some theoretical physics equations. Clifford At the end of the article, a few words about Analysis, Clifford Algebras and their Applications, 7(2012), gravitational waves, for which so hunt for physicists- No 1, 19-39. experimenters. As follows from the formulas for [17] Alexeyeva L.A. Hamiltonian form of Maxwell equations and its generalized solutions. Differential Equations, ether and photons, no pure gravitational waves exist. 39(2003), No. 6, 769-776. Any changes of gravitational field lead to changing [18] Vladimirov V.S. The equations of , of . Moscow:Nauka, 1976 (in Russian). Photons themselves are electro-gravimagnetic waves. They contain a as a This work was supported by the Ministry of component of the EGM wave. It is the gravitational Education and Science of the Republic of Kazakhstan component that determines the light pressure, well (Grant ARO 05132272) known in practice.

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