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

In previous papers we generalized the ether waves associated to photons, to waves generally denoted ξ , associated to ( )s e , m Par , (particles of mass m and electric charge e), and demonstrated that a ( ) e , m Par is a superposition ξ̂ of such waves that forms a small globule moving with the velocity e , m V of this ( ) e , m Par . That, at a point near to a moving ξ̂ , the ether velocity ξ t ∂ , 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 ( ) e , m Par , in particular, of the electron. Then, we considered the case where e=0 and showed that the perturbation caused by the motion of a ( ) 0 , m Par is also propagated in the ether, and is a propagating gravitational field such that the Newton approximation (NA) is a tensor μ G obtained by applying the Lorenz transformation for 0 , m V on the NA of the static gravitational potential of forces S , Gμ . It appeared that μ G is also of the form of a Lienard-Wiechert potential tensor μ A created by an electric charge. In the present paper, we generalized 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 polarized. In fact, we show that the spin associated to a particle is a vortex in ether which in closed trajectories will take only quantized 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, e)s, (particules de masse m et de charge électrique e), et démontré qu’une Par(m, e) est une superposition ξ̂ de ces ondes, formant un petit globule se mouvant a la vitesse e , m V de cette Par(m, e). Qu’en un point près d’un ξ̂ , la vitesse ξ t ∂ de l’éther, i.e., le champ magnétique H est de la même forme que 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=0 et montrâmes que la perturbation causée par le mouvement d’un Par(m, 0) est propagée dans l’éther et est un champ gravitationnel se propageant de telle sorte qu’à l’approximation de Newton (NA), ce champ est un tenseur μ G obtenu en appliquant la transformation de Lorenz pour 0 m V à la NA du potentiel gravitationnel statique de forces S , Gμ . Il apparaît que μ G est aussi de la forme d’un potentiel tenseur μ A de Lienard-Wiechert créé par une charge électrique. Dans le présent article, nous généralisons les ci-dessus résultats en montrant que la théorie de l’élasticité de l’éther implique aussi que comme pour l’électron, la particule massive et neutre possède aussi un spin, mais beaucoup plus petit que celui de l’électron, et que les photons peuvent posséder aussi un spin, quand par exemple le photon est circulairement polarisé. En fait nous montrons que le spin associé à une particule est un vortex dans l’éther qui dans des trajectoires fermées ne pourra prendre que des valeurs quantifiées.

. That, at a point near to a moving ξ ˆ, the ether velocity ξ t ∂ , 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 ( ) , in particular, of the electron.Then, we considered the case where e=0 and showed that the perturbation caused by the motion of a ( ) 0 , m Par is also propagated in the ether, and is a propagating gravitational field such that the Newton approximation (NA) is a tensor μ G obtained by applying the Lorenz transformation for 0 , m

V
on the NA of the static gravitational potential of forces S , G μ .It appeared that μ G is also of the form of a Lienard-Wiechert potential tensor μ A created by an electric charge.
In the present paper, we generalized 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 polarized.In fact, we show that the spin associated to a particle is a vortex in ether which in closed trajectories will take only quantized values.

Introduction
Maxwell and Einstein assumed the existence of an ether, Cf. e.g., Zareski (2001), Zareski (2014).In this context, we showed there, that the Maxwell equations 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., Equation (4) of Zareski (2001), or Equation (1) of Zareski (2014).Then we extended this elastic interpretation to the case where the particles were not only photons, but can also be Par(m,e)s, (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 conception of a material point discontinuous in space appears inconsistent…was achieved, in Zareski (2013).There we showed that the Lagrange-Einstein function G L of such a Par(m,e) not only yields the particle fourmotion equation, but also leads to the fact that ϕ , defined by , is the phase of a wave ξ [also denoted ( ) , m e ξ 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 equation that generalizes the Navier-Stokes-Durand equation where now a particle trajectory is a ray of such a wave that generalizes the light ray.Furthermore we showed there, that a specific sum of waves ( ) and contains all its parameters and that, reciprocally, a wave is a sum of such globules ( ) ˆ, m e ξ , i.e., of particles.Then we showed that in its motion a ( ) Par m e , i.e., a ( ) creates a Lienard-Wiechert potential tensor A μ from which one deduces the electromagnetic field and in particular the magnetic field H which, in the elasticity theory is the velocity t ∂ ξ of the ether points, Cf.Equation (6) of Zareski (2001) or Equation (3) of Zareski (2014).Then we demonstrated there that at a fixed observatory point ob r near at a given instant to the moving electron, i.e., to the moving ( ) ˆ, m e ξ , the velocity of the ether denoted there by t ob ∂ ξ 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 quantum state of an atom, can take only quantized 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 polarization 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 μ , ( 1, μ = 2,3,4), the contravariant four-coordinates of a particle submitted to incident fields, the Greek indices taking the values 1,2,3,4, and the Latin, the values 1,2,3, these last refer to spatial quantities, while index 4 refers to temporal quantities.c denoting the light velocity in "vacuum", one always can impose 4 x ct ≡ .The Einstein summation will be used also with the Latin indices.As usual, we denote by g μν the co-covariant Einstein's fundamental tensor, by ds the Einstein infinitesimal element, by A μ the Lienard-Wiechert electromagnetic potential tensor, by f  the quantity defined by f df dt ≡  , in particular x μ  denotes the four components of the velocity the particle, the expression for s  is then s g x x μ ν μν ≡    .The Newton approximation (NA), is the case where V c << .

Introduction
The expression for the Lagrange-Einstein function , G EM L of a particle of mass m, of electric charge q and of velocity V, submitted only to a Lienard-Wiechert electromagnetic potential tensor where the expression for A μ created by an electric charge 0 q of four-velocity 0 , q V μ located at the vectorial distance R from q, is, Cf.Equation (28) of Zareski (2014a), ( ) in the static case, (1) becomes On the other hand, the expression for the Lagrange-Einstein function , G G L of a particle of mass m, submitted to only a gravitational field

"NA" of the Lagrange-Einstein Function of a Massive Neutral Particle in a Gravitational Field
The explicit form of g x x where i j ≠ Δ is defined by and g μμ as following 0, where 0, g μμ denotes the free value of g μμ .With these notations (5) becomes and ( 4) can be written as following where G μ is the tensor defined by and that Equations ( 1) and ( 10) are of the same form.Therefore, one can suppose that the tensor qA μ plays the same role as the tensor mG μ .This supposition is reinforced by considering that in the static case then Therefore, in the static case One sees that Equation ( 13) is of the same form as Equation (3), and that 0 0 4 qq πε − and 0 mm k are equivalent.So, the electrostatic field created by an immobile electric charge, i.e., Coulomb's field, differs from the "NA" of the gravitational field created by an immobile point mass, i.e., from the "NA" of Schwarzschild's field, by only a constant coefficient of proportionality.It appears that: Electrostatic is of the same form as stationary gravitation where V μ located at the vectorial distance R from m, is One sees that G μ is of the same form as the Lienard-Wiechert potential A μ , Cf. Equation (2), therefore G μ will be called: gravitational Lienard-Wiechert potential.It is the gravitational potential field seen at an observation point ob R , due to the particle of mass 0 m that moves with the velocity 0 m V , and R is the distance between the position of 0 m at the time t' where the signal was emitted and reaches the point ob R at the time t such that ( ) Since an electric charge possesses also a mass it follows that, in the NA, the expression for the total Lagrange-Einstein function , , G G NA EM L of a particle of mass m and electrical charge q submitted to the potential tensors G μ and A μ due respectively to 0 m and to 0 q , is ( )

Review of the Ether Elastic Property of the Electron Spin
We recall how in Zareski (2014) we demonstrated that the spin of the moving electric charge is due to the ether elastic property.At the observatory point ob R , let consider the electromagnetic field created by a moving electric charge 0 q of velocity 0 q V .This field derives from the Lienard-Wiechert potential tensor A μ for which the expression is given in (2).If A denotes the vector of defined by the spatial components of the tensor A μ , then the expression for the magnetic field H created by 0 q is ( ) where 0 ρ denotes the density of the ether, one remind that 0 ρ is classically called, Cf.Zareski (2001), "coefficient of magnetic induction".On account of Equations ( 2) and ( 16) defined here above, and of Equations (63.8) and (63.9) of Landau and Lifshitz (1962), expressed in MKSA units, the explicit expression for H at the observation point ob R of distance vector R from the charge at the retarded time, is where ( ) , and where . As shown in Zareski (2014), when ob R is very close to 0 q and then denoted ob r , i.e., where R, denoted then r, is very small, then at ob r , the expression for H denoted then ob H , considering also that 2 0 0 1 c ρ ε = , Cf. Zareski (2014), is the following: where 0 q ρ , for which the expression is ( ) denotes the volumetric density of electrical charge.Yet, as shown in Zareski (2001Zareski ( , 2014)), the magnetic field H is the field of the relative velocities t ∂ ξ of the points of the ether, therefore, since at ob r the expression for t ∂ ξ denoted more specifically by t ob it follows that the velocity t ob ∂ ξ of a point of the ether at ob r , i.e., of a vortex, Cf.Durand (1963), defined by (20), is of the same form as the velocity Ω V of a point on a rotating solid of rotation vector Ω and of radius r, since Ω V is of the form ) One sees, by considering ( 18) and ( 21), that 0 0 that is to say that 0 0 q q ρ V is a rotation vector, on can remark that this vector has the same dimension [1/T] as [ ] Ω , indeed in Zareski (2012), we have shown that the ether elasticity theory implies that . Now let us take 0 q r r = , where 0 q r is the radius of 0 q , in this case Ω denotes the spin 0 q Ω of the electrically charged particle.And this is what we wanted to demonstrate, (CQFD).The quantum spin, e.g., of an electron in hydrogenous atom, is treated in Zareski (2014b).

The Ether Elasticity Implying that Like the Electron, the Massive Neutral Particle Possesses a Spin but much Smaller
From (1), ( 10), (2), and ( 14), it appears that ( ) 0 0 4 q q πε and 0 m mk play the same role and as one can verify, It follows that for the neutral massive particle submitted to gravitation the coefficient which is equivalent to ( ) 0 4 q π of ( 19) is ( ) Therefore, if , G A μ denotes the gravitational Lienard-Wiechert potential created by 0 m to which is submitted the massive neutral particle of mass m, then, considering ( 24) and ( 14), one has q q ρ V .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 0 q ρ of (19) and .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.

The Photon Circular Polarization as Identical to a Spin
Let x e , y e , z e denote the unitary vectors along Resp. the x, y, z axes, and E the electrical field of an electromagnetic plane wave propagated along the x axe for which the expression is ( ) where E ≡ E , and where the expression for the phase θ is ( ) (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 polarized.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.

Conclusions
We have shown that like the electron, the massive neutral particle possesses a spin, that the circular polarization of the photon is also a spin, and that the particle spin is in fact a vortex in the ether that in closed trajectories can take only quantized values.
ξ ˆ of such waves that forms a small globule moving with the velocity e

.
Dans nos précédents articles nous avons généralisé les ondes d'éther associées aux photons en les ondes généralement dénotées ξ associées aux Par(m, e)s, (particules de masse m et de charge électrique e), et démontré qu'une Par(m, e) est une superposition ξ ˆ de ces ondes, formant un petit globule se mouvant a la vitesse e , m V de cette Par(m, e).Qu'en un point près d'un ξ ˆ, la vitesse ξ t ∂ de l'éther, i.e., le champ magnétique H est de la même forme que 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=0 et montrâmes que la perturbation causée par le mouvement d'un Par(m, 0) est propagée dans l'éther et est un champ gravitationnel se propageant de telle sorte qu'à l'approximation de Newton (NA), ce champ est un tenseur μ G obtenu en appliquant la transformation de Lorenz pour 0 m V à la NA du potentiel gravitationnel statique de forces analogous.Now, as shown in Equations (23)-(27) of Zareski (2014a), the expression for G μ created by a neutral massive particle of mass 0 m of covariant four-velocity 0 , m with the "NA" of the gravitational Lagrange-Einstein function )and by the same reasoning as made in Sec.IV, one has, denoting , a rotation vector, i.e., a spin that plays the same role as the spin 0 0

m
is the mass of the proton and 0 q is the electric charge of the electron, then, in MKSA units, one has:

)
Since the vector potential A is related to E by the relation t