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Relativistic Medium From Unified Energy Vector

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 Relativistic medium from unified energy vector 

 Part 1 

David Harness 


A photon is postulated to be a wavetrain having a coherence length equal to one wavelength.  The wavepulse is analyzed for a unified flux distribution, in its own inertial frame, based on the symmetric addition to the Poynting-Heaviside energy vector S of a gravitational cross product, as an alternative to that first proposed by Heaviside.  From the mutual restoring force of the cross products a relativistic medium emerges, wherein there is no string, presenting an alternative to string/M-theory. 

Maple 10 is used to analyze the symmetric three dimensional electric, magnetic, and gravitational flux distribution of the quantum on a similar basis to static charged rest mass electron-proton flux distributions in their own inertial frame. Eddington's detection of the deflection of light by the sun, and Tolman's detection of a gravitational component to electromagnetic radiation, previously attributed to spacetime curvature, are re-interpreted.  The quantum's spacecurve flux distribution is shown in pair production to gain a spin moment, an added degree of freedom, through the interaction with the field of a nuclei, enabling the spherical expansion of the quantum into the electron and positron flux configurations.   

I. Unified energy vector 

Electromagnetic waves are introduced in physics textbooks as described by the Poynting-Heaviside energy flux vector S of equation (1), and depicted as a linearly polarized wave, as shown in the animation of Fig.1.   

S = `.`(`*`(c, `*`(`/`(`+`(`*`(4, `*`(Pi)))), `*`(E, `*`(x, `*`(H))))), 1) 


 Fig. 1. Conventional electromagnetic wave animation 

Where the electric field component E, and the magnetic field component H, can also be expressed in terms of flux.  The terms field and flux are related in that, field refers to a volume of space where each point is assigned an intensity vector, while the term flux refers to the lines of force the field vectors make as they originate on positive charges and terminate on negative charges.  The quantity of flux indicates the quantity of charge, and in turn the quantity of energy.  Of course, as is the method in quantum mechanics, any graphical depiction of flux, or lines of force, must be interpreted as expectation, or average values of a measurement probabilty distribution, arising from the uncertainty principle. 


In this sense, the closest convergence between quantum mechanics and general relativity lies in their constructs of flux curvature and the geodesic curvature of a spacetime metric.  Accordingly in the present unified flux hypothesis, the gravitational interaction between "a thin pencil of light" and a nearby massive particle, first detected by Tolman [1][7] in 1934, and explained on the basis of the spacetime curvature generated by the light, will be re-interpreted as the result of the symmetric addition of a gravitational cross product to the electromagnetic cross product of equation (1).  This is presented as an alternative to the gravitational cross product first proposed by Heaviside [2] soon after the original energy flux vector was introduced.  An experiment is proposed in section IV to test the contrast between the present hypothesis and the general theory. 

The basis for the Heaviside gravitational cross product, written `+`(`-`(`*`(e, `*`(x, `*`(h)))))in the original notation, was a "convective current" of matter from which gravitational flux arose analogous to the electric current generation of magnetic flux.  The present alternative basis for the addition of a gravitational cross product has the advantage of further applying the postulates of relativity and the Minkowski spacetime metric, which came several decades later, to the transverse expansion of the flux and not just to its longitudinal motion at the constant velocity c.   

First, a quantum of electromagnetic energy E = `/`(`*`(hc), `*`(lambda)), is postulated to be a wavetrain, or wavepulse, having a coherence length `Δs` = `*`(N, `*`(lambda)) equal to its wavelength with N = 1.  An extrapolation is then made from the reference circle of one dimensional simple harmonic motion, so that the transverse electric flux density displacement D = `*`(`ε`[o], `*`(E, `*`(sin(`*`(omega, `*`(t)))))) of a linearly polarized plane electromagnetic wave in its own inertial frame indicates that, just as in any other form of one dimensional harmonic motion = the return force per unit displacement per unit inertia [3].  This would be to place the transverse expansion/restoring force of the flux on an even par with that same flux's force of external interaction.  To object then that a transverse expansion/restoring force is at work in the mutual expansion of the cross products, or even in the original single cross product, would be tantamount to maintaining the position that nothing physical is occurring in the wave's transverse displacement.  Consequently, the first requirement for every force can be met, that for every force there must be an equal and opposite force, to symmetrically complete the cross of S with a gravitational cross product 




where `∝`(G[E], `+`(`-`(E))) and so that the gravitational cross product of course has the same sign and direction as the original electromagnetic cross product.  The maximum radials of the consequent gravitoelectromagnetic flux distribution, which distribution is to be described in more detail later, are then as depicted in the animation of Fig. 2. 



 Fig. 2 Quantum gravitoelectromagnetic wavepulse maximum radials 

Recently Tajmar and de Matos [4] have derived gravitational permittivity `ε`[g]and gravitational permeability mu[g]constant analogs from Newton's gravitational constant G and Einstein's field equations so that 


`and`(`ε`[g] = `/`(`+`(`*`(4, `*`(Pi, `*`(G))))), `/`(`+`(`*`(4, `*`(Pi, `*`(G))))) = `+`(`*`(1.19, `*`(x, `*`(`^`(10, 9), `*`(Unit(`/`(`*`(kg, `*`(`^`(s, 2))), `*`(`^`(m, 3)))))))))) ,                                                       (3) 






A necessary result, of course not only for now so that both cross products travel at the same speed in the same direction, but for later when certain tests have to be met.  Consequently, wherein the entity which provides the restoring force for a wave displacement is defined to be the medium for the wave, in that the expansion of the gravitational cross product provides the restoring force for the electromagnetic cross product, the gravitational cross product is then the medium for the electromagnetic cross product, and the electromagnetic the medium for the gravitational, wherein there is no string.  This mutual relative medium then becomes the relativistic medium when observed from external reference frames.  Consequently this relativistic medium hypothesis presents an alternative to string/M-theory, which seeks to describe the vibrations of a hidden curled-up 10 dimensional string medium as the common medium of all the electromagnetic and gravitational particle-wave modes of the Standard Model. 

The interpretation remains then that gravitoelectromagnetic waves propagate of their own through space, however to take the matter further, it is necessary to recognize that for something to qualify as a physical entity it must have a flux distribution in all three spatial dimensions.  As a physical entity, the quantum must occupy a volume.  Meaning in particular that each photon, in its own inertial frame, must also have a flux distribution along its direction of propagation, along the wavelength of the wavetrain or wavepulse, even though from an external frame that dimension is Lorentz contracted to the flux plane of equation (1). Consequently, starting with the transverse or radial distribution, the lemniscate of Bernoulli provides the form to begin to quantify the spatial flux distribution of a photon, as like flux tends to spread out.  

The radius of the lemniscate curve `*`(`^`(r, 2)) = `*`(`^`(a, 2), `*`(cos(`+`(`*`(2, `*`(theta)))))), the transverse displacement, is calibrated in units of so that r = `+`(`*`(`/`(1, 8), `*`(lambda, `*`(sqrt(cos(`+`(`*`(2, `*`(theta))))))))), where the maximum amplitude, the maximum distance of transverse expansion of opposing flux is then `+`(`*`(`/`(1, 4), `*`(lambda))), as shown in Fig.3.  Meaning that as the node of a quantum passes through a plane, and then travels a distance the opposing flux expand radially overall by the same maximum distance. Thereby bringing the transverse displacement of the wave in accord with the postulates of relativity, so that the maximum transverse displacement is equal to the longitudinal over that interval.  Where generally has an area of here the area calibrated to the quantum is then Of course, once again, one is confronted with the same problem of a singularity at r = 0, as has always been the problem with the electron rest mass flux configuration.  However, here the Planck length l[P] = `+`(`*`(1.616097442, `*`(`^`(10, -35), `*`(Unit(m)))))is taken to be the shortest length from which a physical quantum entity, meaning the root of its flux, can emerge so that one can write of a Planck radius `*`(r, `*`(P)) = `+`(`*`(`/`(1, 2), `*`(l[P])))from which the opposing flux of the cross products expand radially. 


Fig.3 E x H + GE x GH transverse flux distribution 


The volume occupied by the quantum's flux is not only necessary to qualify it as a physical object, but of course necessary in the quantification of energy.  In electrostatics, where work equals force times distance, the potential energy associated with a static charge distribution, a static flux configuration,  is equal to the work required to bring a number of charges into a volume, and can be calculated from the field or flux density itself by assigning an amount of energy to each volume element and then integrating over all the space where there is electric field.  The flux expansion volume for each of the four radial fields of the quantum is then found by integrating down the z axis sinusoidally over one λ as in equation (6), and as depicted in Fig.4 corresponding to the electric field lobe with the unit cubes.   


`and`(lambda[V] = `+`(`*`(32, `*`(Int(Int(Int(`*`(sin(z), `*`(r)), r = r[P] .. `+`(`*`(`/`(1, 8), `*`(lambda, `*`(sqrt(cos(`+`(`*`(2, `*`(theta)))))))))), theta = 0 .. `+`(`*`(`/`(1, 4), `*`(Pi)))), z... 


 Fig. 4.  Quantum flux expansion lobe with unit cubes 


For example, the flux of a quantum of the fourth harmonic of a Nd:YAG laser, lambda = `+`(`*`(266, `*`(nm))), expands into a wavelength volume lambda[V] of , which works out to .   Consequently in the rendering of the quantum expansion volume of Fig.5 the two plot3d are parameterized in the Minkowski metric by equation (7), so that they geometrically cross the cross products in spacetime 

[ct, `+`(`/`(`*`(`/`(1, 8), `*`(lambda, `*`(cos(phi), `*`(sin(t))))), `*`(`+`(1, (`*`(`^`(sin, 2)))(phi))))), t, `+`(`/`(`*`(`/`(1, 8), `*`(lambda, `*`(sin(phi), `*`(cos(phi), `*`(sin(t)))))), `*`(`+`... 



Fig. 5. Double ∞ quantum flux expansion volume 

 where t = `*`(T, `*`(`/`(`+`(`*`(2, `*`(Pi))))))as  The flux extend radially out from the surface of the flux tube of radius r[P], again completely radial as is the electron, but not of uniform field intensity or flux density.  The maximum being for the first lobe at r = r[P], theta = 0, z = `+`(`*`(`/`(1, 4), `*`(lambda))), so that the flux density decays radially throughout the volume by `+`(`*`(`/`(1, 8), `*`(lambda, `*`(sqrt(cos(`+`(`*`(2, `*`(theta)))))))), `-`(r))as r goes from r[P].  Which is as depicted in Fig. 6, an animated rendering of the quantum flux distribution, as a matrix of spacecurves passing through the flux plane of equation (2) fixed at the origin.  






 Fig. 6.  Quantum flux distribution animation 

 The magnitudes M of the quantum's field intensity vectors, having spatial components only in the xz flux plane, extend then from a flux tube of length λ and radius r[P] to 

`*`(`^`(M, 2)) = `+`(`*`(`^`(c, 2), `*`(`^`(t, 2))), `-`(`^`(`+`(`/`(`*`(`/`(1, 8), `*`(lambda, `*`(sin(t), `*`(cos(phi))))), `*`(`+`(1, `*`(`^`(sin(phi), 2)))))), 2)), `-`(`.`(`^`(`+`(`/`(`*`(`/`(1, ... 


  Similar then to the potential energy content of a flux distribution volume of charged rest masses in their own frame, the energy of the unified flux distribution of the quantum increases as the volume of its flux distribution decreases.  However, since in free space the relative flux distribution cannot or does not change in form from one wavelength to another, it is in the coupling to the fourth dimension of time where a higher energy content, the force times distance of wavelength contraction is generated.  In the most common system of units, the fundamental quantity of energy of the quantum's flux distribution is then of course 6.626 multiplied by the fourth dimension of timethen multiplied by frequency, yielding the energy relationship E = hfof the quantum as is so well known. 


continued in Relativistic medium from unified energy vector Part 2: 

II.  Light deflection 

III. Electron-positron pair production 

IV. Experimental test 

V.  Conclusion