SSRG International Journal of Applied Physics (SSRG-IJAP) – Volume 7 Issue 1 – Jan to April 2020 Ether and photons 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) field equation which Removing the restriction on the zero scalar is generalization of Maxwell equations for EM-field part of the biquaternion of EM field intensity allows in biquaternions algebra. The special class of us to give a biquaternion representation of the EGM solutions of this biquaternionic wave (biwave) field, as well as a biquaternion representation of mass equation is mono-chromatic wave solutions that charges and currents, which contains the EGM describe periodic oscillations and waves of a fixed charge in the complex scalar part, and the EGM field frequency. 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 gravitational field 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 motion of photons, their energy- described by the biquaternion of the strength of the momentum 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 light 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, photon, 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 ISSN: 2350 - 0301 www.internationaljournalssrg.org Page 96 SSRG International Journal of Applied Physics (SSRG-IJAP) – Volume 7 Issue 1 – Jan to April 2020 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 quaternion 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 BGb 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 vacuum, c 1/ is the speed of light. exception of basic elements e , m 0,1,2,3, of Note that such complex characteristics for m biquaternion algebra). 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 j1 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 WAAAEH0,52 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 space (,):, 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 * ΞΘΘW 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, * ΞΘΘ0,5 w iP differential operators are introduced - mutual bi- ISSN: 2350 - 0301 www.internationaljournalssrg.org Page 97 SSRG International Journal of Applied Physics (SSRG-IJAP) – Volume 7 Issue 1 – Jan to April 2020 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 radiation conditions of Somerfield.