Generalization of the Particle Spin as it Ensues from the Ether Theory

Transcription

1 Applied Physics Research; ol. 8, No. ; 6 ISSN E-ISSN Published by Canadian Center of Science and Education Generaliation of the Particle Spin as it Ensues from the Ether Theory I.A.I, Yehud, Israel David Zareski Correspondence: David Zareski, I.A.I., Yehud, Israel. areski@inter.net.il Received: April, 5 Accepted: May 6, 6 Online Published: July 9, 6 doi:.5539/apr.v8np58 URL: Abstract In previous papers we generalied the ether waves associated to photons, to waves generally denoted ξ, associated to Par ( m,es, (particles of mass m and electric charge e, and demonstrated that a Par ( is a superposition ξˆ of such waves that forms a small glule moving with the velocity m, e of this Par ( m,e. That, at a point near to a moving ξˆ, the ether velocity t ξ, i.e., the magnetic field H, is of the same form as that of a point of a rotating solid. This is the spin of the Par ( m,e, in particular, of the electron. Then, we considered the case where e= and showed that the perturbation caused by the motion of a Par ( m, is also propagated in the ether, and is a propagating gravitational field such that the Newton approximation (NA is a tensor G tained by applying the Loren transformation for m, on the NA of the static gravitational potential of forces G, S. It appeared that G is also of the form of a Lienard-Wiechert potential tensor A created by an electric charge. In the present paper, we generalied the above results regarding the spin by showing that the ether elasticity theory implies also that like the electron, the massive neutral particle possesses a spin but much smaller than that of the electron, and that the photon can possess also a spin, when for example it is circularly polaried. In fact, we show that the spin associated to a particle is a vortex in ether which in closed traectories will take only uantied values. Résumé. Dans nos précédents articles nous avons généralisé les ondes d éther associées aux photons en les ondes généralement dénotées ξ associées aux Par(m, es, (particules de masse m et de charge électriue e, et démontré u une Par(m, e est une superposition ξˆ de ces ondes, formant un petit glule se mouvant a la vitesse m, e de cette Par(m, e. Qu en un point près d un ξˆ, la vitesse t ξ de l éther, i.e., le champ magnétiue H est de la même forme ue celle d un point d un solide en rotation. Ceci est le spin du Par(m, e en particulier, de l électron. Puis nous considérâmes le cas où e= et montrâmes ue la perturbation causée par le mouvement d un Par(m, est propagée dans l éther et est un champ gravitationnel se propageant de telle sorte u à l approximation de Newton (NA, ce champ est un tenseur G tenu en appliuant la transformation de Loren pour m à la NA du potentiel gravitationnel statiue de forces G, S. Il apparaît ue G est aussi de la forme d un potentiel tenseur A de Lienard-Wiechert créé par une charge électriue. Dans le présent article, nous généralisons les ci-dessus résultats en montrant ue la théorie de l élasticité de l éther impliue aussi ue comme pour l électron, la particule massive et neutre possède aussi un spin, mais beaucoup plus petit ue celui de l électron, et ue les photons peuvent posséder aussi un spin, uand par exemple le photon est circulairement polarisé. En fait nous montrons ue le spin associé à une particule est un vortex dans l éther ui dans des traectoires fermées ne pourra prendre ue des valeurs uantifiées. Keywords: The Lagrange-Einstein function, the generalied Lienard-Wiechert potential tensor, the spin of the massive neutral or electrical particle and of the circularly polaried photon, spin and vorticity. Introduction Maxwell and Einstein assumed the existence of an ether, Cf. e.g., Zareski (, Zareski (. In this context, we showed there, that the Maxwell euations of electromagnetism ensue from the case of elasticity theory where the elastic medium is the ether, of which the field ξ of the displacements is governed by the Navier-Stokes- Durand Cf., e.g., Euation ( of Zareski (, or Euation ( of Zareski (. Then we extended this elastic interpretation to the case where the particles were not only photons, but can also be Par(m,es, (particles of mass m and of electrical charge e, that can be submitted to electromagnetic and or a gravitational fields. This extension which is well in accord with Einstein s opinion:... the combination of the idea of a continuous field with the 58

2 apr.ccsenet.org Applied Physics Research ol. 8, No. ; 6 conception of a material point discontinuous in space appears inconsistent was achieved, in Zareski (3. There we showed that the Lagrange-Einstein function L G of such a Par(m,e not only yields the particle fourmotion euation, but also leads to the fact that ϕ, defined by dϕ dt = LG, is the phase of a wave ξ [also denoted ξ ( me, when there is ambiguity], associated to these particles. Such a ξ is a field of the displacement of the points of the ether, i.e., propagated there, and is a solution of an euation that generalies the Navier-Stokes- Durand euation where now a particle traectory is a ray of such a wave that generalies the light ray. Furthermore we showed there, that a specific sum of waves ξ ( me, forms a glule ξ ˆ ( me, that moves like the Par(m, e and contains all its parameters and that, reciprocally, a wave is a sum of such glules ξ ˆ ( me,, i.e., of particles. Then we showed that in its motion a Par ( m, e, i.e., a ξ ˆ ( me, creates a Lienard-Wiechert potential tensor A from which one deduces the electromagnetic field and in particular the magnetic field H which, in the elasticity theory is the velocity tξ of the ether points, Cf. Euation (6 of Zareski ( or Euation (3 of Zareski (. Then we demonstrated there that at a fixed servatory point r near at a given instant to the moving electron, i.e., to the moving ξ ˆ ( me,, the velocity of the ether denoted there by tξ is of the same form as that of a point of a rotating solid. This phenomenon is the electron spin which, as we showed, in a uantum state of an atom, can take only uantied values. In the present article, we extend these results to the massive neutral particle and to the photon. It appears that the massive neutral particle possesses also a spin and that the circular polariation of the photon can be understood as an angular momentum that in fact is a spin. More generally, we show that this spin is in fact a vortex in the ether.. Notations We denote by x, ( =,,3,, the contravariant four- coordinates of a particle submitted to incident fields, the Greek indices taking the values,,3,, and the Latin, the values,,3, these last refer to spatial uantities, while index refers to temporal uantities. c denoting the light velocity in vacuum, one always can impose x ct. The Einstein summation will be used also with the Latin indices. As usual, we denote by g ν the co-covariant Einstein s fundamental tensor, by ds the Einstein infinitesimal element, by A the Lienard-Wiechert electromagnetic potential tensor, by f the uantity defined by f df dt, in particular x denotes the four ν components of the velocity the particle, the expression for s is then s gν x x. The Newton approximation (NA, is the case where << c. 3. Similitude of the NA of the Field Due to a Moving Neutral Massive Particle and the Lienard-Wiechert Field Due to a Moving Electric Charge 3. Introduction The expression for the Lagrange-Einstein function L GEM, of a particle of mass m, of electric charge and of velocity, submitted only to a Lienard-Wiechert electromagnetic potential tensor A is LGEM, mc c A x c = +. ( where the expression for A created by an electric charge of four-velocity distance R from, is, Cf. Euation (8 of Zareski (a, in the static case, ( becomes A GEM, πε L mc c, ( Rc, located at the vectorial R, ( = πε r. (3 On the other hand, the expression for the Lagrange-Einstein function L GG, of a particle of mass m, submitted to only a gravitational field g ν, is ν L = mc g x x. ( GG, 3. NA of the Lagrange-Einstein Function of a Massive Neutral Particle in a Gravitational Field The explicit form of g x ν ν x is ν 59

3 apr.ccsenet.org Applied Physics Research ol. 8, No. ; 6 where Δ i is defined by and g as following g x x g x x g x x, (5 ν ν = + + Δi 3 3 Δi gxx + g3xx + g3xx, (6 g = g + δ g, (7, where g, denotes the free value of g. With these notations (5 becomes g x x = c + δ g x x + g x x + Δ, (8 ν ν i and ( can be written as following LGG, mc c mc ( δ g x x + g x x + Δi ( c = + It appears that the NA of L GG, denoted then L GGNA,, is (9 LGGNA,, mc c mg x c = +, ( where G is the tensor defined by G c δ g, G, ( c g and that Euations ( and ( are of the same form. Therefore, one can suppose that the tensor A plays the same role as the tensor mg. This supposition is reinforced by considering that in the static case then δg = α r, whereα m k c, and g =, i.e., the expression for G,S, (S for stationary, is G = m k r. (, S Therefore, in the static case L GGNA,, denoted then L GGNAS,, becomes L = mc c + mm k r. (3 GGNAS,, One sees that Euation (3 is of the same form as Euation (3, and that πε and mmk are euivalent. So, the electrostatic field created by an immile electric charge, i.e., Coulomb s field, differs from the NA of the gravitational field created by an immile point mass, i.e., from the NA of Schwarschild s field, by only a constant coefficient of proportionality. It appears that: Electrostatic is of the same form as stationary gravitation where πε and mmk are analogous. Now, as shown in Euations (3-(7 of Zareski (a, the expression for G created by a neutral massive particle of mass m of covariant four-velocity m, located at the vectorial distance R from m, is m, G mk. ( Rc R ( m One sees that G is of the same form as the Lienard-Wiechert potential A, Cf. Euation (, therefore G will be called: gravitational Lienard-Wiechert potential. It is the gravitational potential field seen at an servation point R, due to the particle of mass m that moves with the velocity m, and R is the distance between the position of m at the time t where the signal was emitted and reaches the point R at the time t such that ( t t' c = R. One sees the similarity of the electromagnetic Lienard-Wiechert potential tensor A ( with the gravitational Lienard-Wiechert potential tensor G (, that is the similarity of the electromagnetic Lagrange-Einstein function L GEM, ( with the "NA" of the gravitational Lagrange-Einstein function L GGNA,, (. 6

4 apr.ccsenet.org Applied Physics Research ol. 8, No. ; 6 Since an electric charge possesses also a mass it follows that, in the NA, the expression for the total Lagrange- Einstein function L GGNAEM,, of a particle of mass m and electrical charge submitted to the potential tensors G and A due respectively to m and to, is ( LGGNAEM,, mc c A mg x c = + +. (5. Review of the Ether Elastic Property of the Electron Spin We recall how in Zareski ( we demonstrated that the spin of the moving electric charge is due to the ether elastic property. At the servatory point R, let consider the electromagnetic field created by a moving electric charge of velocity. This field derives from the Lienard-Wiechert potential tensor A for which the expression is given in (. If A denotes the vector of defined by the spatial components of the tensor A, then the expression for the magnetic field H created by is ( ρ H curl A. (6 where ρ denotes the density of the ether, one remind that ρ is classically called, Cf. Zareski (, coefficient of magnetic induction. On account of Euations ( and (6 defined here above, and of Euations (63.8 and (63.9 of Landau and Lifshit (96, expressed in MKSA units, the explicit expression for H at the servation point R of distance vector R from the charge at the retarded time, is ( R R c ( R c R R R H = 3 +, (7 πcρεr( R R c β c where β ( c, and where d dt. As shown in Zareski (, when R is very close to and then denoted r, i.e., where R, denoted then r, is very small, then at r, the expression for H denoted then H, considering also that c ρε =, Cf. Zareski (, is the following: where ρ, for which the expression is H = ρ r, (8 3 ( π ρ r, (9 denotes the volumetric density of electrical charge. Yet, as shown in Zareski (,, the magnetic field H is the field of the relative velocities tξ of the points of the ether, therefore, since at r the expression for tξ denoted more specifically by tξ, is ξ = ρ r, ( t it follows that the velocity tξ of a point of the ether at r, i.e., of a vortex, Cf. Durand (963, defined by (, is of the same form as the velocity Ω of a point on a rotating solid of rotation vector Ω and of radius r, since Ω is of the form = Ω r. ( Ω One sees, by considering (8 and (, that ρ = Ω, ( that is to say that ρ is a rotation vector, on can remark that this vector has the same dimension [/T] as [ ] indeed in Zareski (, we have shown that the ether elasticity theory implies that dim[ ] dim[ / L] Ω, ρ =. Now let us take r = r, where r is the radius of, in this case Ω denotes the spin Ω of the electrically charged particle. And this is what we wanted to demonstrate, (CQFD. The uantum spin, e.g., of an electron in hydrogenous atom, is treated in Zareski (b. 5. The Ether Elasticity Implying that Like the Electron, the Massive Neutral Particle Possesses a Spin but much Smaller From (, (, (, and (, it appears that ( πε and mmkplay the same role and as one can verify, 6

5 apr.ccsenet.org Applied Physics Research ol. 8, No. ; 6 ( πε = [ mmk] dim dim. (3 It follows that for the neutral massive particle submitted to gravitation the coefficient which is euivalent to π of (9 is ( m ( ε k π. ( Therefore, if AG, denotes the gravitational Lienard-Wiechert potential created by m to which is submitted the massive neutral particle of mass m, then, considering ( and (, one has A m G, k πε m, ( Rc R m, (5 and by the same reasoning as made in Sec. I, one has, denoting H G, the euivalent of H, H = ρ πε k r, (6 G, m m where ρ m is the mass volumetric density of the neutral massive particle defined by m ρm. (7 3 π r As shown above, H G, is the velocity tξ G, of the points of the ether at r. Therefore, one has ξ = ρ πε k r, (8 t G, m m and if we denote Ω = ρ πε k, then, m m m ξ =Ω r. (9 t G, m That is, ρm πε km denotes a rotation vector, i.e., a spin that plays the same role as the spin ρ. We show now that for the same velocity the proton spin is much smaller than that of the electron. Indeed, let us compare the coefficients ρ of (9 and ρm πε k of (8 for the same r, that is, let us compare m πε k and. If m is the mass of the proton and is the electric charge of the electron, then, in MKSA units, one has: m =.6, =.6, ε = 8.9, and k = 7, it follows that m πε k = 5 9 which is much smaller than =.6. This shows that the for the same velocity the spin of the electron is very very much greater than the spin of the proton, in fact the proton spin, i.e., the vorticity of the ether that it creates is negligible in front of that of the electron. 6. The Photon Circular Polariation as Identical to a Spin Let e x, e y, e denote the unitary vectors along Resp. the x, y, axes, and E the electrical field of an electromagnetic plane wave propagated along the x axe for which the expression is ( y cos sin where E E, and where the expression for the phase θ is Since the vector potential A is related to E by the relation E= E e θ + e θ, (3 ( t x c θ ω +. (3 E= ta, it follows that ( E ω( ysinθ cosθ A = e e. (3 Now, as one can verify ( ( ysin cos curla = E c e θ + e θ, (33 6

6 apr.ccsenet.org Applied Physics Research ol. 8, No. ; 6 therefore, t = = ( ρ curl = E ( ρc ( ysinθ + cosθ But since, considering (3, ex E= E ( eysinθ + e cosθ, it follows that that can be written Let us show now that the dimension of ( ρ rc ξ H A e e. (3 ( ρ c ξ t = H = e x E (35 t = E ( ρrc E is /T that is, the vector E ( ρrc ξ c r. (36 c is a rotation vector Ω. M Indeed, as shown in Sec. I of Zareski (, dim E = which implies dim E =, that is to say that LT ρrc T ( ρ rc E c = Ω, (37 and tξ = Ω r. (38 This shows that the photons created by the moving electron that itself possesses a spin, can possess also a spin this is the case where the photon is circularly polaried. Finally, an EM field is a vibration of the ether, this vibration can be linear, elliptic or circular, in this last case the photon possesses a spin, which in fact is a vortex in the ether. 6. Conclusions We have shown that like the electron, the massive neutral particle possesses a spin, that the circular polariation of the photon is also a spin, and that the particle spin is in fact a vortex in the ether that in closed traectories can take only uantied values. Acknowledgments I thank Professor Lawrence Horwit from Tel-Aviv and Bar-Ilan Universities Israel, for his encouragements and his precious counsels. References Durand, W. F. (Ed.. (963. Aerodynamic Theory. New York: Dover. Landau, L. D., & Lifshit, E. M. (96. The classical theory of field. Pergamon Press. Zareski, D. (. The elastic interpretation of electrodynamics. Foundations of Physics Letters, (5, Zareski, D. (. On the elasto-undulatory interpretation of fields and particles. Physics Essays, 5(. Zareski, D. (3. Fields and wave-particle reciprocity as changes in an elastic medium: The ether. Physics Essays, 6(, Zareski, D. (. The Electron Spin as Resulting From the Ether Elasticity. Applied Physics Research, 6(5,. Zareski, D. (a. The ether theory as implying that electromagnetism is the Newtonian approximation of general relativity. Physics Essays, 7(, Zareski, D. (b. The Quantum Mechanics as Also a Case of the Ether Elasticity Theory. Applied Physics Research, 6(, 8. Copyrights Copyright for this article is retained by the author(s, with first publication rights granted to the ournal. This is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license ( 63