The Proton and Neutron as Distortional Structures in the Geometric Manifold

— It is shown in the present work that the distorted-space model of matter as extended to mimics of the proton and neutron can be described as muonic-based structures. These distorted-geometry structures exhibit non-Newtonian features wherein the hole or core-region fields of the structure (negative pressure core for a positive-charge positive-pressure large-r structure) do not behave functionally in a r -4 manner and terminate at zero at the radial origin (no singularity). Of particular interest is that of r -6 energy-density behavior at structural radial distances near the core of the distortion, a region also displaying potential-well behavior.


INTRODUCTION
For this distorted-geometry (DG) modeling, we maintain and expand the geometrical perspectives inherent in the "Curved empty space as the building material of the physical world" supposition of Clifford [1]; "RIEMANN has shewn that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. Similarly, he says, although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very small portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space.
I wish here to indicate a manner in which these speculations may be applied to the investigation of physical phenomena. I hold in fact (1) That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them. (2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave.
(3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial. (4) That in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity.
A quote from Wheeler's work [2] published in 1955 reads; "In the 1950's, one of us [3] found an interesting way to treat the concept of body in general relativity. An object can in principle be constructed out of gravitational radiation or electromagnetic radiation, or a mixture of the two, and may hold itself together by its own gravitational attraction…A collection of radiation held together in this way is called a geon (a gravitational electromagnetic entity) and is a purely classical object….In brief, a geon is a collection of gravitational or electromagnetic energy, or a mixture of the two, held together by its own gravitational attraction, that describes mass without mass." Subsequently at The International Congress for Logic, Methodology, and Philosophy of Science in 1960, he [4] began by quoting William Kingdon Clifford's [1] "Space-Theory of Matter" of 1870 and stated "The vision of Clifford and Einstein can be summarized in a single phrase, 'a geometrodynamical universe': a world whose properties are described by geometry, and a geometry whose curvature changes with time -a dynamical geometry." A comprehensive and general treatment of the historical, geometric and physical foundations of modern geometrodynamics is found in the already cited publications of Wheeler [2][3][4]. Additional work in this field continues, some of which is cited in references [5][6][7][8][9][10]. The present distorted-geometry treatment departs from these cited "geon constructional methods" in that we do not constrain the distortional descriptions to only gravitational couplingconstant produced structures.
The classical Riemannian four-dimensional curvature equations have been applied to describe the "localized geometrical distortions and associated energy distributions" at both quantum-and galactic-level magnitudes and radialextensions. By requiring that the geometric distortions mimic the physical characteristics of the elementary particles, a coupling constant between energy and geometry is produced. The theoretical modeling and calculational procedure is limited to those geometric-distortional families satisfying an equation-of-state, which also expresses static, sphericallysymmetric Maxwellian tensor behavior.
Functional solutions to Riemann's geometric, non-linear, coupled, partial-differential equations have classically not been forthcoming. A solution however has been found for the Riemannian equation-set by describing the "distorted" space as satisfying the "equation of state", Eq.(1). The Riemannian field equations, produced by the present "metric-solution", Eq. (2), are of such a character that, over portions of the radial extensions of the distortion, the geometrical tensor elements exhibit negative, as well as positive, curvature-magnitudes and energy-densities. The field-observable in the negative energydensity (negative pressure) spatial region (the core region) is non-Coulombic and non-infinite at the radial origin. Mass, electric charge and magnetic moments have been simulated for the down-quark, up-quark, electron, tauon, muon and neutrino as well as for a hypothetical beta-decay transitionmediating distortion-particle; this process also generates a geometric expression for the Fermi constant [11] and conversely predicts a mass-value for the W-boson calculated to the precision of the electron g-factor (gyromagnetic ratio). Figure 1 displays the field energy-density quantities for three of these particle structures.
The dependence of the field quantities on mass-energy and electric-charge is apparent in the Figure. The field energydensity magnitudes should be compared with the experimental data in reference [17].
By incorporating gravitational-field structures into the geometric modeling and simulation process, a refined coupling-constant is engendered. This process recovers the gravitational coupling-constant of general relativity and leads to a structural description of gravitational-like geometricdistortions.
We showed in [11] that the propagation velocity in the core region (negative pressure core for a positive-charge, positivepressure (at large-r) structure) of these distorted-geometry (DG) structures was approximately 1.5 times that external to the core; see Fig.2. This feature, which is a fundamental attribute of such structures, is equivalent to a "partial light trapping" phenomenon and for gravitational structures would be classically associated with "black holes".
A "geometric maximum-energy-density" feature was successfully exploited to geometrically explain and quantify the Fermi constant [11]; in addition a "stability-based minimum-energy-density" condition was fundamental to describing the structure of the "stable DG electron" feature. Energy-densities are equivalent to pressures and the core and shell regions of the structure hold the system in stable equilibrium. Fig.3 graphically displays such structures. In the perspective of [11], the DG model is a departure from the classical geometry model where the Einstein Curvaturetensor is the stress-energy-tensor describing the "material contents" of the energy distribution. The DG model is rather viewed with the energy-content residing in the warping or distorting of the manifold and therefore in its geometrictensors, and the "curved empty space" [11] referred to above is a "localized curved or distorted space" devoid of an "external or foreign" causative matter-entity. 2.

THEORETICAL FOUNDATION
A "constitutive relation" or "equation-of-state" was used as descriptive of the distorted-space volume (the physical spatial volume encompassing the distortion-produced energy ), that is, where the metric quantities g 11 ≡ -e µ and g 44 ≡ e ν ; ν ' = 0-2+ (3) Riemann's geometric equations (Tolman [13]), presented here as Eqs. (4), are expressed in the metric-variables # and # and the manifestation of the composite coupling-constant appears in the geometric quantities γ and the geometric "transformation radius" R0, both determined from the "distorted spatial volume" with electromagnetic and/or gravitational energy-density components.
While the "energy-density Equation-of-state" Eq. (1) is equivalent to Maxwell's "energy-density" form for static, spherically-symmetric Maxwellian tensor behavior [11,13], the field equations, in both the EM realm and the gravitational realm (Q = 0), exhibit r -6 geometric, static, spherically-symmetric Maxwellian tensor behavior which we have interpreted as constituting a "magnetic monopole" mimic (what is a "magnetic monopole"?).
Building on the earlier work, [11,12], we apply the geometric concepts to produce a proton-and neutronconglomerate of distortional-geometric muon-mimics.
This section is a brief summary of the development in [11] and is included here as a less cumbersome introduction or reference to the theoretical modeling concepts.
Symbols [11] are R0 is the geometric transformation radius (R0electron is calculated from the fundamental-particle magnetic-field component) and ro is the radial value at which the energydensity distribution-function transitions from the core-value to its negative in the shell (r → ∞) region; ro ≡ R0 uo follows from solution for the uo roots of the field equations (Eqs. (6)) We also define γ ≡ 2 R0

Rs
Rs ≡ Schwarzschild radius = 2 κ Mc 2 (7) For the electron we write (me ≡ mass electron, Qelec = 3), The core-radii for the EM structures are inversely proportional to the mass-energy of the distorted-geometrystructures and therefore it follows that the ro radius for the muon, or anti-muon, is ro(muon) = me c 2 m muon c 2 ro(electron) = 1.61(10) -16 meters. (10) In the earlier development [11] for the geometric Fermiconstant, GF is GF geo ≝ 0fe The uo root values (uo ≡ R0 ro ), for determining ro values, from Eqs. (6), are plotted in Fig.4

Fig. 4
Radial-zero functions u0 (from Eqs. (6)) relative to electric-charge values; uE0 for the electrical function, uM0 for the mass-energy-density function and uB0 for the magnetic-energy-density + function. Abscissa charge-values are displayed as a function of the index Qq. uo(Qq) is graphically used for the zero-radial-function uo. Also indicated is the E_field quantity, u0_core = uE0(Qq = 0), and a gravitational "black-hole u0_bh" value.
Therefore, we see that the static-spherically-symmetric Maxwellian tensors exhibit the same stress and energy relationship as the geometric tensors. Finally, it is of interest to examine the ratio of the 1/r 6 tensor-component to the 1/r 4 tensor-component in the construction of the geometric fields. The geometric-energydensity or field equations [13] are (the "d" symbolism denotes the "distorted space" modeling) In discussions of the negative energy-density (pressure) core-regions of this universal (EM as well as gravitational) distorted-geometry structure, it should be emphasized that a negative energy-density gravitational feature (a repulsive gravitational force or negative pressure) is non-Newtonian. The hole or core region-fields of the structure are repulsive thereby stabilizing the structure, do not behave functionally in an r -4 manner and terminate at zero at the radial origin (no singularity). This field behavior is a fundamental feature of these "distorted-space" structures; the field exhibits r -4 , r -6 and r -n dependences in both the core and shell regions and is thereby able to account for "Newtonian"-, "weak"-and "strong-fields"(see Eqs. (14,15)).
To further illustrate the structural character of the "distortional-geometry mimics", we compare at "near-core radial regions" the geometrostatic field quantities Fd 14 2 and Fd mag 2 which are also the geometric mimics of the Maxwellian tensors. For both gravitational and electromagnetic distortions, the magnetic field component, Fd mag 2 , is non-zero at the "radial field zero", Fd 14 2 = , or "core radius (ro)" (in the gravitational realm, this field feature would seem responsible for accretion-disk and galaxy-matter rotational-distribution behavior). Figures 5 and 6 are constructed for the "distorted-space electron-mimic" although the energy-density distribution functions are qualitatively similar in the normalized radial variable u =  Actually, the Fd 14 2 fields contain r -6 elements of a magnitude comparable to the magnetic-field strengths Fd mag 2 , resulting in a significant departure from the classical Newtonian r -4 (or r -6 ) behavior. The fields also exhibit potential-well behavior as they radially (shell values) transition to repulsion (if attractive in the shell) at the holecore radius ro. 3.

PROTON AND NEUTRON MODELING
We use the following experimental data [14][15][16]  Calculating the binding energy of the proton, modeled as a structure of 4 muons and 5 anti-muons, we get BEp (binding energy of proton) ≡ 9 Mµ -Mp = 12.6504 MeV. Similarly, the binding energy of the neutron ≡ Ben = 9 Mµ + Me -Mn = 11.358 MeV. The muons and anti-muons are posited to coexist within the immediate field environment of the composite structure as, for example, neutrons existing within elemental structures, or, in other field-induced [15,16] modifications. For comparison to other nuclear structures, a binding energy per nucleon, where a "nucleon" definition here incorporates the muon, is BEp_µ = 1.40555 MeV/nucleon and Ben_µ = 1.318666 MeV/nucleon.
For the DG_mimicked-proton, the electric-field is that of an anti-muon (the other 4 muon/anti-muon pairs exhibiting zerocharge or self-neutralization [16] and zero g-factors). The gfactor value of the 4th muon is field "flipped" and a resultant sub-structure of two anti-muons and the "flipped" muonproduces a "field induced" composite g-factor, reduced from the free-field value of (2.002331) per anti-muon to (1.861898) per anti-muon, to create a proton g-factor of (5.585694). The proton (stacked-muon) radius, assuming a 3-layer, 3 muon/layer, @ 4 ro-muon separation (center to center) between muons, in a stacked-geometry-configuration, would be approximately 10 ro_muon or 1.0625(10) -15 meters.
For this DG_mimicked-neutron, the electric-field is that of the DG_mimicked core-proton plus the field of the shroud DG_mimicked electron registered on the same center as the protonic-core (see Fig.7). In other words, a zero-charge field (at large radii) but a "DG-field" near and within the ro(electron) radius. The neutron ro radius is that of the shroudelectron-mimic or ro(neutron) = ro(shroud-electron) = 2.197(10) -14 meters.
The energy-density features of these structures are illustrated in Fig. 7. The field-energy-density distribution functions are recast as, for example, U_pro(r ) ≡ (Fd 14 ) 2 = -g 11 g 44 jTd 4 4 + Td 1 1 k = g 11 g 44 j 2 Td 2 2 k and the neutron distribution is defined as

U_neu(r) ≡ U_pro n (r) -U_elec n (r)
where the "n" subscript denotes the impact of the gyromagnetic factor ge. The DG_proton g-factor produces U_pro(r) + U_elec(r) and the DG_neutron g-factor produces U_pron(r) + U_elecn(r). Absolute values are displayed because of the energy-density-value transition at the core radii. The symbols ron and rop are the DG_neutron and DG_proton core radii.
Some experimental data have been measured for the proton and is detailed in reference- [17] as; "The proton, one of the components of atomic nuclei, is composed of fundamental particles called quarks and gluons.
Gluons are the carriers of the force that binds quarks together, and free quarks are never found in isolation-that is, they are confined within the composite particles in which they reside. The origin of quark confinement is one of the most important questions in modern particle and nuclear physics because confinement is at the core of what makes the proton a stable particle and thus provides stability to the Universe. The internal quark structure of the proton is revealed by deeply virtual Compton scattering, a process in which electrons are scattered off quarks inside the protons, which subsequently emit high-energy photons, which are detected in coincidence with the scattered electrons and recoil protons. Here we report a measurement of the pressure distribution experienced by the quarks in the proton. We find a strong repulsive pressure near the centre of the proton (up to 0.6 femtometres) and a binding pressure at greater distances. The average peak pressure near the centre is about 10 35 Pascals (Joule/m 3 ), which exceeds the pressure estimated for the most densely packed known objects in the Universe, neutron stars. This work opens up a new area of research on the fundamental gravitational properties of protons, neutrons and nuclei, which can provide access to their physical radii, the internal shear forces acting on the quarks and their pressure distributions." In the DG_neutron, the core values of the DG energy density distribution for the co-centered electron are of the same sign as the energy-density shell values of the protonicmuon structures but of an almost infinitesimal value compared to the muonic values. At the electron-core radius the two energy density distributions are of opposite magnitudes and produce a composite neutral charge character (Qneu = 0)( for radii greater than ro(electron)) and the modified g-factors discussed above.

CONCLUSIONS
It has been shown in the present work that the distorted-space, or distorted geometry (DG), model of matter, as applied to fundamental-particle (muon/anti-muon) constructs, can mimic the composite neutron and proton structures. The distortedgeometry structures exhibit non-Newtonian features wherein the core-region and shell-region fields of the structures do not behave functionally in an r -4 manner and terminate at zero at the radial origin (no singularity). This field behavior is a fundamental feature of these "distorted-space" structures; the field behavior exhibits r -4 , r -6 and other r -n dependence in both the core and shell regions and is thereby able to account for, explain and mathematically elucidate "Newtonian"-, "weak"and "strong"-fields." interests for this paper Author Contribution Statement The work and representation was provided by the sole author Correspondence and requests for materials should be addressed to drkoehler.koehler@gmail.com      (6)) relative to electric-charge values; uE0 for the electrical function, uM0 for the mass-energy-density function and uB0 for the magnetic-energy-density + function. Abscissa charge-values are displayed as a function of the index Qq. uo(Qq) is graphically used for the zero-radial-function uo. Also indicated is the E_field quantity, u0_core = uE0(Qq = 0), and a gravitational "black-hole u0_bh" value.