Coupling relativity, quantum mechanics and neutrinos for the synthesis of matter

Professor Costas Vayenas and his colleagues Dionysios Tsousis and Dimitrios Grigoriou, explain how the use of special relativity to study the motion of neutrinos at fm distances leads to Bohr-like models with relativistic gravity as the attractive force

For the past fifty years, the Standard Model (SM) of particle physics has provided the basis for describing the structure and composition of matter. According to the SM, protons and neutrons, which belong to the hadron family of composite particles and are the components of atomic nuclei, are made up of elementary particles called quarks which are held together by a force called Strong Force. (1) No quark has ever been isolated and studied independently, and their masses are estimated to be comparable to those of baryons, i.e. of the order of 1 GeV/c2. These masses are 100 billion times (10.11) larger than the neutrino masses (10-1 at 10-3 v/c2) (2) which are by far the lightest particles, as well as the most numerous, in our Universe.

Inertial and gravitational mass increases with speed

Einstein’s theory of relativity (both special (SR) (3.4) and general (GR) (5)) is one of the most remarkable scientific achievements in human history. SR has been experimentally confirmed thousands of times and there have also been many confirmations of GR. Most confirmations refer to macroscopic systems and only recently (6) the astonishing strength of SR and GR has been demonstrated inside hadrons, deep within the femtocosmos of the lightest elementary particles, ie neutrinos.

Space contraction, time dilation, and mass increase with particle velocity are the main features of SR, because the particle velocity relative to an observer, at rest with the center of rotation, approximates of the speed of light c and therefore of the Lorentz factor γ, defined as γ = (1 – v2 / vs2) –1/2approaches infinity.

Thus, considering three particles rotating symmetrically on a cyclic path using their gravitational attraction, FG, as the centripetal force, then FG can become surprisingly strong. Indeed, SR dictates that a particle of rest mass mo has a relativistic mass γmo, (3.4) and a longitudinal mass of inertia γ3mo, equal according to the principle of equivalence (6.7) with its gravitational mass γ3month.(7.8) Therefore, using the definition of gravitational mass in Newton’s law of gravity, it follows:

Fg = Gm2ohγ6 / (√3r2) (1)

where r is the radius of rotation. To find r and γ it is also necessary to use the de Broglie equation of Quantum Mechanics:

γmohvr=nћ (2)

This allows to obtain for n=1 and mo≈43.7 meV/c2valued (6.8) from the Superkamiokande measurements, (2) that r≈0.63 fm and γ≈7.163.109so γ6≈1.35.1059. Therefore, the rotational speed is very close to c and the gravitational force is, surprisingly, according to equation (1), 59 orders of magnitude greater than the normal non-relativistic Newtonian force! (Fig. 1) This force is equal to 8.104 N, equal to the weight of 100 humans on earth.

Relativistic Mass Augmentation: Neutrinos, Quarks and Hadrons

In addition to causing such an amazing γ6~1059 times the increase in gravitational attraction, the special elativity also causes an incredible γ ~ 7.168.109 increase in the mass of the three rotating neutrinos so that the mass of the composite particles increases from 3(43.7 meV/c2) at the neutron mass of 939.565 MeV/c2 (Fig. 1). Conversely, if the mass of the composite particles, 3γmo, is that of a neutron (939.565 MeV/c2) then the rest mass, mo, of each spinning particle is that of the heaviest neutrino proper mass, (9) i.e. 43.7 meV/c2in good agreement with the Superkamiokande measurement of the heaviest neutrino mass. (2) Therefore, special relativity reveals that quarks are relativistic neutrinos and also shows that the gravitational mass of neutrinos, γ3mo, is enormous, i.e. of the order of the Planck mass (ћc/G)1/2 = 21.7 mg per neutrino! This therefore also implies, in conjunction with equation (2), that the gravitational force from equation (1) is equal to the strong force, ћc/r2which is a factor of 137 stronger than the electrostatic force of the positron-electron pair at the same distance. (1)

Minimization of the energy and calculation of the mass of hadrons

The RLM shows that the maximization of the Lorentz factor γ leads to a better stability of the composite particles by minimizing –5γmohvs2which is the potential energy of the rotating neutrino triad (8) and, at the same time maximizing the Lorentz factor γ and therefore also the hadronic mass produced m = 3γmo = 313/12 (mP1mo2)1/3where mP1 is the Planck mass (=(ћc/G)1/2 = 1.22.1028 v/c2). This simple expression surprisingly gives a mass value that differs by less than 1% from the experimental neutron mass of 939.565 MeV/c2.

quantum mechanics, neutrino motion

quantum mechanics, neutrino motion
Figure 1. (Left) Schematic of the Rotating Lepton Model (RLM) of the neutron and its three equations, based on special relativity, quantum mechanics, and conservation of energy, respectively. (Right) Plot of the FG force from equation (1) as a function of the Lorentz factor γ; FN is the Newtonian force FG (γ=1) and FS is the strong force; mo and mn are the rest masses of neutrinos and neutrons respectively. The centripetal force, FG, is equal to 7.98.104 N, comparable to the weight of 100 humans on earth.

It is well known that neutrinos come in three different “flavors”, namely electron neutrinos, muon neutrinos and tau neutrinos. These flavors are obtained by mixing neutrinos of the three mass types (or mass eigenstates), i.e. m3 mass neutrinos (heaviest), m2 mass neutrinos and type m1 neutrinos (the lightest) for the Normal Hierarchy. Using equation (1) and the experimental masses of the hadrons, we calculated the mass values ​​of the composite particles plotted in Figure 2. The agreement with the experimental composite mass values ​​is better than 2%. Conversely, the experimental mass values ​​of hadrons or bosons can be used to calculate the three neutrino masses. Concordance with the experimental values ​​measured at Superkamiokande (2) is less than 5%.

Uniting forces

The fact that the Newton-Einstein gravitational equation (1) matches the experimental mass values ​​of hadrons so well shows that, given special relativity, gravity is sufficient to describe the strong force. The equally good fit to the experimental mass values ​​of W±, Z0 and H bosons show that relativistic gravity is also sufficient to describe the weak force. Indeed, in both cases at the limit of large γ we obtain Fg = Gm2 P1/r2 =G(ћc/G)/r2 = ћc/r2 which is the value of the strong force. (1) Similarly, for the weak force we also obtain Fg = ћc/r2. It can therefore be concluded that the strong and weak forces have been unified with Newtonian gravity (γ = 1) in the RLM via equation (1). (10.12)

In summary, the RLM reveals that our known universe is the product of the combination of neutrinos, electrons, positrons, Einstein’s relativity and the wave-particle dual nature of matter, as described by the de Broglie equation of quantum mechanics. (12.13)

quantum mechanics, neutrino motion
Figure 2. Comparison of the calculated RLM masses of the composite particles with the experimental values. (10) Agreement is better than 2% without any adjustable parameters. The three approximate mass expressions presented in the figure provide the order of magnitude of the masses of hadrons and bosons. (ten)

The references

  • Griffiths, Introduction to elementary particles. 2nd ed. Wiley-VCH Verlag GmbH & Co. KgaA, Weinheim, 2008.
  • Takaaki Kajita, Nobel Lecture: Discovery of atmospheric neutrino oscillations, (2016). Program. Physical. 691607 – 1635 (2006)
  • Einstein (1905) Zür Electrodynamik bewegter Körper. Ann. of physics., bd. XVII, S. 17:891-921; English translation of Electrodynamics of Moving Bodies ( by GB Jeffery and W. Perrett (1923).
  • P. French, Relativity (WW Norton and Co., New York, 1968).
  • W. Misner, KS Thorne and JA Wheeler: Gravitation. WH Freeman, San Francisco, (1973).
  • G. Vayenas & S. Souentie, Gravity, special relativity and strong force: a Bohr-Einstein-de-Broglie model for the formation of hadrons. Springer, New York (2012).
  • G. Roll, R. Krotkov, RG Dicke (1964) The equivalence of inertial and passive gravitational mass. Annals of Physics 26(3):442-517.
  • G. Vayenas, S. Souentie & A. Fokas (2014) A Bohr-type model with gravity as the force of attraction. Physics A, 405:360-379.
  • G. Vayenas, D. Tsousis and D. Grigoriou, Calculation of neutrino masses from Hadron and Boson masses via the Rotating Lepton model of elementary particles. J Phys. : Conf. Ser. 1730012134 (2021).
  • G. Vayenas, D. Tsousis and D. Grigoriou, Calculation of the masses, energies and internal pressures of hadrons, mesons and bosons via the Rotating Lepton Model. Physics A, 545123679 (2020).
  • G. Vayenas, AS Fokas, D. Grigoriou, “Catalysis and autocatalysis of chemical synthesis and hadronization”. Appl. catal. B, 203582-590 (2017).
  • G. Vayenas, D. Tsousis, D. Grigoriou, K. Parisis and E. Aifantis “Hadronization via gravitational confinement of fast neutrinos: Mechanics at fm distances” Zeitschrift für Angewandte Mathematik und Mechanik (2022).
  • Halliday, R. Resnick, J. Walker, Fundamentals of physics11th edition, Wiley (2018).

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