The Conservation of Energy and Momentum in the Positron-Electron Annihilation According to Complete Relativity

According to Complete Relativity (an improvement of Special Relativity where a new postulate was introduced: an electron cannot move at average speeds less than u0 = αc / π where α is the constant of fine structure and c is the speed of light), and with the hypothesis that the electron and positron are two three-dimensional electromagnetic spherical standing waves, the electromagnetic standing waves of an electron-positron pair can overlap in the low energy conditions and generate a single γ ray of E = 1.022 MeV. Instead the actual physical theories consider a model of positron-electron pair annihilation, where conservation of momentum requires the creation of two 511 keV photons moving toward opposite directions. But in the paper we do not consider any photons: there is only a progressive electromagnetic wave compound by two electromagnetic standing waves. Only a thorough test will be able to decide the right model.


INTRODUCTION
In 1981 I studied computer hard disc vibrations using the mathematic bi-Laplacian operator. The mathematical solution of this activity was published on TI Journals in the paper: Vibrational Modes of Thin Flat Circular Plates Calculated Using the Dynamic bi-Laplacian Equation [1]. During my research I asked myself: Can I simulate the atomic nucleus with a solid sphere and study it with the bi-Laplacian operator in polar coordinates? Hence, a new idea in Fundamental Physics, where particles are considered as electromagnetic spherical standing waves, was developed in five articles.
The first considers the Dynamic bi-Laplacian Equation, in order to calculate the Magic Numbers of atomic nucleus 1 . On July 2009 the paper entitled: The dynamic bi-Laplacian Equation in polar coordinates and the magic numbers of atomic nucleus [2] was published online on the Physics Essays website. The focal point of considering electrons, atoms and atomic nuclei as a superimposition of e.m. standing waves, is a modification of Einstein's Special Relativity. In accordance with Appendix B of [2], I introduced a new postulate: an electron cannot move at average speeds less than u0 = αc / π where α is the constant of fine structure and c is the speed of light. The postulate characterizes the Complete Relativity Theory.
The hydrogen atomic model founded on the electromagnetic standing waves [3]. This paper proposes a new threedimensional electron model. The electron model [ψ(r,t)=(sin(kr)/r)cos(ωt)] criticizes the electric charge notion. In reality the innermost spherical nodal surface of the electron (that corresponds to the first root of sin(kr)/r) is surrounded by other concentric nodal surfaces (that correspond to the other roots of sin(kr)/r). According to Einstein's formula E = mc 2 , the energy of the e.m. standing wave (averaged over time) in the innermost sphere, has an equivalent in mass. Moreover, the averaged energy of the e.m. standing wave of the electron outside this sphere can be considered to be generated by an equivalent electric charge stored on the innermost spherical nodal surface. So the innermost sphere can be considered as a spherical particle: the electrically charged electron. Then the electric charge would only be an ideal concept (averaged over the time period of the e.m. wave). The best physical result in [3] is the calculation of the fundamental spectral frequency of the molecular Hydrogen. Another important result can be achieved by considering the energy function of the electron: We can interpret E0 ( r ) as the total energy of the e.m. standing wave of the electron inside a sphere of radius r concentric with the electron. Then E0 ( r ) can represent the potential energy of the electron at a distance r from the centre. So then its derivative represents the field of forces in accordance with the value of the vector radius r. Instead deriving the function E0 ( r ) of the positron, we can conclude that at the centre of the positron there is an impulse function of force. Then between the electron and the positron, both their structures are completely asymmetric. Nevertheless at great distance they appear symmetric. [4]. This paper shows a new Periodic Classification of the Elements which differs from the current Classification for the heavy elements of VI and VII Period (see Table 1). His organization is according to the fractal form of the atoms. In the new Classification, Ytterbium can be seen as a useful element for the cold fusion tests. Furthermore, a new phenomenon appears: a new set of e.m. theoretical (albeit unsteady) wavelengths of Hydrogen spectrum. Should this phenomenon be revealed, then it would confirm the proposed Fundamental Physics Theory that was named Electromagnetic Atomic Theory (EAT). The Doppler Effect according to complete relativity [5]. This paper highlights the zone of the e.m. spectrum (from the γrays to X-rays) covered by the Doppler effect applied to the γ-ray of 1.022 MeV produced by a positron-electron superimposition. We analyze the Doppler effect in Complete Relativity when the wave front moves away from the observer. When the p/e pair moves in relation to the relativistic observer with a ratio β > 0.5774 we observe an inversion of the normal behaviour: instead of an increase in the wavelength of the e.m. radiation, we can observe a decrease in the wavelength. In addition, when the p/e pair moves away at round the speed of light, we can observe a blue shift due to the relativistic addition of the speeds. Furthermore, the paper shows a new way of considering the photoelectric effect.

The Hydrogen Atomic Model Based on the Electromagnetic Standing Waves and the Periodic Classification of the Elements
Finally in The ideas behind the Electromagnetic Atomic Theory [6] some ideas are indicated in order to develop the Electromagnetic Atomic Theory. In the paper a new vision of the Cosmos is discussed.

THE POSITRON -ELECTRON ANNIHILATION
We consider the electromagnetic (e.m.) spherical standing wave of a positron [ψ(r,t)=(cos(kr)/r)sin(ωt)]. Into a sphere of radius r (the origin O of r is the centre of positron) total energy E0 + ( r ) of positron is worthy: where: α = constant of fine structure , re = electron radius ; re is a value obtained by the following Eq.
We study the case whose conditions are: γ1 = γ2 = 1 and ( a , b ) → 0. Then Eq. (9) is converted into: In the same conditions of the energy calculation [ γ1 = γ2 = 1 and ( a , b ) → 0 ] , we can write the differential of momentum dQ as: where: When the p/e pair tends to overlap, then b → 0 and δ ( rb ) → δ ( r ) Since εt → 0 and the last member of Eq. (15) is a finite real number , then the force F tends to infinity . But this is the condition for a collision between the positron and the electron. Since the force F tends to infinity , also the acceleration of the positron-electron pair tends to infinity when εt → 0 ; then the speed u1 tends to the maximum value possible: the speed of light c. Then β1 → 1 , and during the collision of the e/p pair, Eq. (15) becomes: being: Then the two e.m. standing waves of the electron and positron modify suddenly their status and transform themselves in a γ ray of 1.022 MeV. Energy and momentum of the e.m. progressive wave created by the annihilation of the p/e pair are worthy: On the other hand, since: the γ ray direction is in agreement with positron velocity. 2 In accordance with Classic Mechanics and with Figure 1, during an elastic collision between two bodies, momentum is espressed by the equation: = 1 1 − 2 2 where: m1 , m2 ˃ 0 . In our case, in accordance with Einstein formula E = mc 2 the following Eq.(12) and Eq.(13) show two opposite mass densities equivalent to the correspondent energy densities: where Eq.(13) is negative (positive), the analogous part in Eq.(12) is positive (negative). This reflection gives explanation for the plus sign in Eq.(11).