The proton consists of fundamental particles called quarks and gluons. Gluons carry the force that binds quarks together. Quarks are always confined in the composite particles in which they are located. The origin of quark confinement is still a subject of intensive study in modern physics. Recently, for the first time, the pressure distribution inside the proton was measured via deeply virtual Compton scattering [1, 2, 3, 4]. Strong repulsive pressure up to 10^{35} pascals, the highest so far measured in our universe, was obtained near the center of the proton up to 0.6 fm, combined with strong binding energy at larger distances. This recent pioneering experimental [1, 4] and theoretical [1, 2, 3] work on the deeply virtual Compton scattering has opened a new area of research on the fundamental gravitational properties of protons, neutrons and nuclei and has underlined that gravity plays an important, perhaps dominant, role inside protons and other hadrons [1–3].

The dominant role of gravity inside hadrons has also emerged in recent years from the rotating lepton model of composite particles (RLM) [5, 6], in which gravity causes confinement of highly energetic neutrinos in bound rotational states and thus leads to formation of quarks, hadrons and bosons [5–8]. This model follows exactly the steps of the Bohr treatment of the H atom and contains no adjustable parameters (Figure 1). It synthesizes Newton’s gravitational law, Einstein’s special relativity [9], and de Broglie’s wavelength expression, thereby conforming with quantum mechanics. The model does not require any new theory. Furthermore, it fits extremely well with current experimental evidence for the masses and other properties of hadrons. The simple structure of the RLM for the proton is also suggested by the earlier proposed [10] bagel-shaped proton geometry (Figure 1). Without any adjustable parameters, the RLM predicts the masses of hadrons [5, 6], but also of bosons [7, 8] with an accuracy of one percent [5, 6, 7, 8, 11, 12].

Figure 1 Rotating lepton model (RLM) of the proton. Top: Schematic comparison and synthesis of the bagel shape model of protons computed via model wave functions, constructed with Poincar´*e* invariance [10], and of the three-rotating neutrino RLM model of protons [5, 6, 7]. Particle size is dictated by the quark Compton wavelength $\lambda \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}{\phantom{\rule{negativethinmathspace}{0ex}}}^{-}q=\hslash /{m}_{q}c$. The central particle is a positron of negligible speed, thus negligible gravitational mass [7]. Bottom: Confining and repulsive forces in a proton according to [1] and to the RLM. Point O is the center of rotation and point D is the midpoint of A and B.

Two important suggestions have emerged from the RLM analysis [5, 6]: First, quarks and hadrons consist primarily of rotating neutrinos, and their mass is due to the kinetic energy of the rotating neutrinos. Second, the Strong Force can be viewed as the relativistic gravitational force. Here the RLM is used to describe quantitatively the measured pressure distribution in protons and to derive simple analytical formulae for the proton radius and for the maximum measured pressure of 10^{35} pascals. We also show that a proton structure comprising a ring of three rotating ultrarelativistic quarks with radius 0.63 fm describes the proton’s measured pressure profile semi-quantitatively without any adjustable parameters.

Similar to the Bohr model of the H atom, but utilizing gravitational instead of electrostatic forces, the RLM for the proton comprises only two equations, *i.e.*

$$F=\gamma {m}_{o}{\text{v}}^{2}/{r}_{e}=\frac{G{m}_{g}^{2}}{\sqrt{3}{r}_{e}^{2}}$$(1)where *r*_{e} is the radius of the circle defined by the centers of the three rotating particles, which form an equilateral triangle, *m*_{o} is the neutrino rest mass, and *m*_{g} is the neutrino gravitational mass, together with the corresponding de Broglie wavelength equation

$$\gamma {m}_{o}{\text{vr}}_{e}=n\hslash $$(2)which is the historical basis of quantum mechanics and introduces quantization via the integer number *n*. Accounting for the equivalence principle, *m*_{g} equals the inertial mass, *m*_{i}, which as Einstein showed in his pioneering special relativity paper in 1905 [9], is equal, for linear motion, to the longitudinal mass, ${\gamma}^{3}{m}_{o},$[9, 13]. Originally derived for linear motion [9], this useful result has been shown [5, 6], via the use of instantaneous reference frames [13], to remain valid for arbitrary motion including circular motion. Thus it follows

$$\begin{array}{r}{m}_{g}={m}_{i}={\gamma}^{3}{m}_{o}.\end{array}$$(3)Solution of equations (1), (2) and (3) for *n* = 1, the ground state, gives

$$\begin{array}{rl}{\gamma}_{q}& ={3}^{1/12}({m}_{Pl}/{m}_{o}{)}^{1/3};\\ {m}_{p}& =3{\gamma}_{q}{m}_{o}=3{m}_{q}={3}^{13/12}({m}_{Pl}{m}_{o}^{2}{)}^{1/3}\end{array}$$(4)where ${m}_{Pl}(=(\hslash c/G{)}^{1/2})$is the Planck mass and the subscript “q” denotes quark.

Setting *m*_{p}=938.272 MeV/c^{2}, the proton mass, in equation (4), one computes *m*_{o} = 0.0437 eV/c^{2}, which remarkably is within the experimental limits of the mass of the heaviest electron neutrinos (0.048±0.01 eV/c^{2}) [14, 15]. Consequently, the rest mass of quarks appears to be that of electron neutrinos. Also the radius, *r*_{e}, of the proton computed as the quark de Broglie wavelength from equation (2) for *n* = 1 and _{v} ≈ *c*, is given by

$${r}_{e}={\u019b}_{q}=\hslash /{\gamma}_{q}{m}_{o}c=\hslash /{m}_{q}c=3\hslash /{m}_{p}c=0.63fm,$$(5)in very good agreement with the experimental value [1, 16]. It is worth noting that the first equation (4), together with (3), dictate that the gravitational mass, ${\gamma}_{q}^{3}{m}_{o},$of the relativistic rotational quarks is very close to the Planck mass, *i.e.*

$$\begin{array}{r}{m}_{g}={\gamma}_{q}^{3}{m}_{o}={3}^{1/4}{m}_{Pl}.\end{array}$$(6)Consequently the relativistic gravitational force and potential energy computed from eq (1) is

$$\begin{array}{r}F=\frac{G{m}_{g}^{2}}{\sqrt{3}{r}_{e}^{2}}=\frac{G(\hslash c/G)}{{r}_{e}^{2}}=\frac{\hslash c}{{r}_{e}^{2}};\phantom{\rule{1em}{0ex}}U=-\frac{\hslash c}{{r}_{e}}\end{array}$$(7)which are the values anticipated for the strong force between two quarks [16]. These values are a factor of *α*^{−1} = 137.035 stronger than the Coulombic attraction and potential energy of an *e*^{+}*e*^{−} pair at the same distance.

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