Geometric Mediator Structures, Transition Dynamics and Force Constants

The present of mimics gravitational-field models at large radii but departs significantly at small radii to produce a magnetic-field (spin) mimic as well as a weak-field mimic and the Fermi constant) and a strong-field mimic without an infinity at the origin singularity) The structure is constituted by a core-region within which the propagation- velocity, by virtue of the distorted metrics, is greater than c and exhibits a “partial light trapping phenomenon”, a “black hole”. Warping or distorting our spatial-manifold requires energy but with limits as to the degree of distortion thereby predicting and describing fundamental-electromagnetic-particle structures as well as gravitational (dark-matter, black-hole) structures. Abstract It is shown in the present work that the distorted-space model of matter can describe conventional force-constants and transition-mediator structures. We use the verbiage “distorted” to communicate the concept of “energetic warping” to distinguish “spatial warping” from “classical matter warping” , although the concept of “matter” is in fact, in the present context, the “geometric distortion energy” of the spatial manifold itself without a classical “matter stress energy source” . The “ distorted-geometry ” structures exhibit non-Newtonian features wherein the hole or core-region fields of the structures are energetically-repulsive (negative pressure), do not behave functionally in an r -4 manner and terminate at zero at the radial origin (no singularity). Near the core of the distortion the magnetic fields dominate the energy-densities of the structures thereby departing from classical particle-structure descriptions. Black-body and gray-body and structural modeling transition dynamics

region within which the propagation-velocity, by virtue of the distorted metrics, is greater than c and exhibits a "partial light trapping phenomenon", facilitating and duplicating "black hole" behavior. Distorting the geometry in our spatial-manifold requires energy but with limits as to the degree of distortion thereby predicting and describing fundamental-electromagnetic-particle structures as well as gravitational (dark-matter, black-hole) structures. Such a geometric description of localized warping or distorting of the spacetime manifold would seem (?) to constitute a "first-principle" model of the universe. The work also describes geometric "energytransition mediator" structures and the dynamics of the transition process.
A historical quote from Wheeler's work [2][3] published in 1955 reads; "In the 1950's, one of us [4] 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." Additional work in this field continues, some of which is cited in references [5][6][7][8][9][10]. The present treatment departs from these cited "geon constructional methods" in that we do not radiation-emission and structural modeling lead to a description of transition dynamics and photonic entities.

Introduction
Physical transition processes are presently mathematically represented in "quantum-terms" as a manifestation of a "strength-of-interaction coupling-constant" operating on an "initial-state" wave-function particle-descriptor to produce a "final-state" different wave-function particledescriptor; one particle transforms to another particle (a different energy-state) via the forces present at the transformation site. The actual physical description of the structural-changing dynamics is not part of these quantum-mechanical operational-mathematical renderings although an "intermediate" mediator-structure [12] is envisioned.
The intermediate mediator-structure in the beta-decay transition process, the conversion of a neutron into a proton, electron and a neutrino, is a W -BOSON PARTICLE and the "strength-ofinteraction" has been labelled "the Fermi-constant GF" after the physicist who successfully modelled this physical process. Since energetically such a massive particle would seem to pose an energy-conservation problem, an introduction of the Heisenberg principle is incorporated to constrain the "existence time interval" or lifetime of the mediator-structure; Δt ΔU ≡ ħ/2.
We have also successfully and precisely modelled and mimicked this transition process in the "distorted-geometry" model of matter [11] as a product of boson mass-energy and boson physical-volume, a "geometric maximum-curvature condition" and a magnetic-field based (r -6 ) distortion-energy, with structural details which are not forthcoming in present-day quantum mechanics, force-carrier-fields [12] notwithstanding.

Theoretical Foundations
Fundamental theoretical and mathematical foundations for this undertaking are presented in the Supplementary Information section.
The "distorted-geometry" mathematical symbols are The distorted-geometry radial descriptor R0 is where S = spin quantity (S = 1 for the boson), ge = gyromagnetic ratio and Q = electric charge quantity (Q = 3 for the boson), then, ℏc (m w c 2 ) -1 and = 2 m w c 2 .
The energy-density structural nature of the 'distorted geometry" solution [11] (Eq. (SI-4)), gives rise inherently and comprehensively to the fundamental force quantities ( , , and ), heretofore characterized as independent entities; a weak-magneticforce, an electric-force and a strong-force at the nuclear core. These force characterizations are here manifested as r -6 , r -4 and complex repulsive-core r -n components of the "one geometric structure". The structure is a balanced internal/external high-energy-density configuration, the difference in internal-pressure vs external-pressure manifested as particle mass-energy. The magnitude of the structural energy-density descriptor function is determined by the mass-energy or geometric-curvature with a geometry-to-energy coupling constant (meters/Joule) also dependent on these physical characteristics; a constant coupling-constant component describes gravitational structures. The "distorted-geometry-solution (≡ DG)", is generated from Riemann's geometric description of a 4-dimensional spacetime manifold applied at localized warped-or distorted-space energy centers.
With the geometric success of mimicking the Fermi-constant as a particle-structure descriptor (the Wboson), which is a "mass-energy*R0 3 "product and which is a magnetic energy-density weak-force maximum and a geometric-curvature maximum (inverse dependence) [11], we posit gravitational, electromagnetic and strong (core)-force "strength-of-interaction-DG" constants as energy-density coefficients of the various r-dependent components of a DG W-boson structure.
However since such tensor-force ( (Fd 14 ) , (Fd mag ) 2 (Fd core ) 2 ) entities are geometrically coupled entities, the classical "independently separable" model (weak plus EM plus strong) is not applicable. We instead use the energy-density maxima in the core region and in the extra-core region to establish the physical strengths of the classical-differentiated force functions. We use the "BLACK-HOLE DISTORTIONAL EXTREMUM [13] (a minimum hole mass) mass-energy" for calculating the "gravitational-interaction-strength" constant GG. Note that the "gravitational coupling-constant, G*c -4 = 0.826 10 -44 meters/Joule, is ~32-42 orders of magnitude smaller than the "EM (electromagnetic) coupling-constants" 8πκ W or 8πκ electron .

Calculational Method for the W-boson mediator
The positive-pressure (positive energy-density) quantity, (Fd 14 ) ( , ≠ ), evaluated at the core-radius functional-extremum, for the Wboson, is The actual DG functional value at the energy-density maximum is 7.64 10 47 Joules meter 3 @ = 2.37 10 −19 while the classical r -4 value is illustrating the magnitude of the contribution to Fd 14 2 from the r -6 and other r -n components.
Similarly, the negative-pressure (negative energy-density) core-maximum (for a W-boson structure) is and EM_mag for Fd mag 2 ) for the BOSONIC-mediator structure, illustrating the 'Strongrepulsive-force (is this gluon behavior?), Weak-force and Electric-force" components. The ordinate in Joules/meter 3 is displayed in logarithmic form and the abscissa, the structural-radii r2, in meters in logarithmic values.
From the "energy-emission dynamics" model in ref. [14], and using the "corrected form" of equations 11-15 (ref. [14] corrected in ref. [15]), we can model and calculate the "lifetime" of this boson-mediator structure as where U0 is the mass-energy of the "energy-emitting" body with a constant density ρ.
Environmental fields [16,17] not included in the structural modeling would influence this "lifetime" as, for example, the stability behavior of a neutron in or out of the presence of nuclear fields. This "energy emission" model is elaborated in the following for the "electromagneticradiation-emission mediator".

Calculational Method for the Neutrino structure
The "extremum" equation (1) can be rearranged to accentuate the geometric-structural elements giving rise to the "strength of interaction" quantity GF as follows; We see then that the magnetic descriptors theoretically and mathematically allow for solutions describing extremum-structures other than the boson. If ascribed to a neutrino structure, the extremum-structure would provide a magnetic characterization where, for example, using a neutrino mass [18][19][20][21] at 0.02 eV, an electric charge (≠0) is calculated at Q (neutrino) = 1.49(10) -12 (compared to Q =3 for the electron or boson (see equation (2a)) or Q (neutrino) = 7.46(10) -11 for a neutrino mass at 1.0 eV; such a charged neutrino structural description is presently not part of conventional modeling, (Q = 0), which however is experimentally not verifiable.
A structural configuration describing a "stable energy-density mimic of the electron" is described as or Q = 4.24(10) -7 for a neutrino mass of 1.0 eV.
Calculational Method for energy-transition dynamics, the muon mediator and the photon structure In reference [14], we modelled "energy emitting structures" via a "black body construct" realized at the mass-level of a "fundamental particle" with a mass-energy = Universe-mass-energy. Here we posit such a "radiation-energy emitting" structure to describe photon emission dynamics, an energy-transition process. The "Planckian (Stefan-Boltzmann emitting body) power and energy distribution function" is integrated over the infinite energy spectrum and modelled as a spherical entity with radius R; Using the radial-zero value, u0B, of the Fd mag 2 function, converts the normalization radius R0 W to its geometric value, r = ro since u0B ≡ R0 W /ro.
A final extinction time, wherein all of the structural energy has been depleted and converted to photon-energy, is reached at thereby producing a propagating directional photon (multi-particle production allowed) with a time-width tf and inherited blackbody and DG features; we assume a photon with velocity = c and exhibiting the "thermodynamic" body descriptors; "thermodynamic radiation" being understood as "EM radiation" at velocity c. The use of an "explosive" adjective to describe this dynamic feature is better appreciated when examining the enormous energy-densities (10 48 Joules/ meter 3 ) or pressures (Pascals) within these "DG particle structures" (compare to a "stick of dynamite" at ~ 10 9 Pascals).
The extinction-time result can be interpreted as a "photonic-structural-descriptor" where tf ≡ 1/ν and R ≡ ƛ; λ ν = c ; the thermodynamic variable c has an electromagnetic "velocity of propagation" meaning.
Electric charge features are not inherent to this development since "black bodies" have been modelled from thermodynamics and statistical mechanics theory although the charged bosonbody characteristics in the form of the mass-energy density ρ, the total mass-energy U and the geometric radius feature R have been utilized; a simple conceptual model wherein "explosion-or energy-transition information" propagates physically throughout the exploding entity. The maximum-curvature DG-concept, from weak-force beta-decay modelling, produces a maximum energy limit at R min = R0 W , a charge-induced, magnetic-field-(Td 1 1 + Td 2 2 ), r -6 , induced limit and therefore probably not the same limit as for (Td 1 1 ), r -4 , forces. In fact, the ratio of r -6 azimuthally-directed energy-densities to r -4 radially-directed energy-densities is The "material properties" of the "distorted-space" are sufficiently significant in the azimuthal directions as to be responsible for the phenomenon of beta-decay, at least if the "mediator structure" is that of a Wboson.
If one rather considers the muon-structure (an excited electron-structure and a lesser-energy structure than the boson) as the black-body mediator-structure for "classical-radiation-emission", then U max _ ℎ = hc R min = hc R0 = 1.59 10 −10 Joules or 1.23 10 −2 × mass energy , where R0 muon has been calculated from Eqn. (2a).
The DG muon-"photon producing"-mediator fields are displayed in Fig.2;  Fig.2. Distorted-Geometry Energy-Density (field) functions (Eμ_e for Fd 14 2 and Eμ_mag for Fd mag 2 ), for the MUONIC-mediator structure, illustrating the 'Strong-repulsive-force, Weakforce and Electric-force" components. The ordinate in Joules/meter 3 is displayed in logarithmic form and the abscissa, the structural radii rμ, in meters in logarithmic values. Note the energydensity reduction and the increase in radial extent compared to the W -BOSON-character.
Although these distortional structures have been characterized at the outset as stable distortions, we have subsequently exploited the distortional form as the mediating entities in distortional transition processes, suggesting that the structural stability can be of a transient nature and sensitive to environmental "fields". As a supplementary visualizing addition to the geometric modeling we include as a Supplementary Video an animated video (simulating the muon to electron beta-decay, the higher-energy nuclear process).
A black-body emitted, propagating, DG photonic structure is simulated and mathematically detailed, as an example, for the Lyman-alpha line @ ƛ = 121.567 nanometers (labelled R0ν), in Fig.3; the simulation is also displayed in Fig.4 to better communicate the structure of the timevarying "energy-density fields".

Calculational Method for the Gravitational mediator
The positive-pressure (positive energy-density) quantity, (Fd 14 ) , evaluated at the radius(r_max) of (Fd 14 ) (max) , for the "HOLE_MIN" [13], GRAVITATIONAL STRUCTURE, due to a maximum curvature, is To consider imperfect-black-body radiating-structures (a reduced radiating efficacy by factor ε), the energy-transition process can be written in "DG-model" terms to allow for a "gray-body emission-character" as (equation (9) where R is the radius of the spherical radiating surface and ε, a variable between 0 and 1, is the emissivity of that surface. The transition time for the "DG" W-boson provides an explicit form for energy-transitions of "distorted-geometry" structures and supports the "constant-density" "mathematical-structure" of equation (9). The spectral emissivity ε for the "DG" Wboson structure would have to be ε = 0.0663 to agree with the experimental W-boson lifetime = 0.316(10) -24 seconds [22]. The spectral emissivity ε for the "DG" muon structure would have to be ε = 9.52(10) -6 to agree with the experimental muon lifetime = 2.196(10) -6 seconds [23].
Examination of the DG-structure's energy-density profile in equations Fd mag 2 and Fd 14 2 in the SI, and in Figures 1 and 2, reveals the marked departure from a structural "constant density" feature and the need to use a wavelength-dependent "spectral emissivity εƛ". However the "lowemissivity" of these presently-modeled DG-structures can be understood as the reason for their stability, even though they exhibit rather short-lifetimes; finally, it is energetically admissible that the DG "Wboson mimic" but not the DG "muon mimic" be a candidate for the "mediator structure for low-energy classical radiation-emission". to Fd 14 2 from the r -6 and other r -n components (see Fig.6). Similarly, the negative-pressure (negative energy-density) core-maximum (for this black-hole gravitational structure) is (Fd ) 2 (max @ r = 9.08 10 9 meter) = -1.92 10 23 Joule/meter 3 .

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 (boson, muon and gravitational) constructs, can produce structures satisfying "particle mass-energy-transition" or "mediator" dynamics.
Earlier successful mimicking [11] of "Fermi-described beta decay" has been extended to a mediator description of "classical radiation-emission" and a "gravitational energy-transition mediator" entity.
Therefore, we see that the static-spherically-symmetric Maxwellian tensors exhibit the same stress and energy relationship as the geometric tensors [SI-1], The present geometric-modeling endeavor, with its Maxwellian-tensor-form mimickingcomponent, has produced the fundamental and limiting agent for the currently-studied distorted geometry, namely a particular constraining functional relationship between the geometrydefining tensors (for an empty-space geometry, all of the components of the energy-momentum tensor are zero). In using this simple equation-of-state, equation (SI-2), as a restricting distortional-model tensor relationship, we thereby elicit the metric-defining differential equations for such a family of geometric distortions. The geometric-energy-density or field equations, after using solution Eq. (SI-4), are repeated here (from [SI_2]); also see [SI_1];

STRUCTURAL EQUATIONS
The calculational treatment employs the isotropic coordinate description of equation (SI-1) and utilized by Tolman , where the system of equations represented by equation (SI-1), is shown more explicitly in equation (SI-3) in mixed tensor form; 3 3 , Metric coupling, that is terms such as μ'ν', are apparent in the fundamental curvature equations. The usual notation, where primes denote differentiation with respect to the radial coordinate r and dots denote differentiation with respect to the time coordinate t, is employed. We are considering the static case (where total differentiation replaces partial differentiation) as was also used for Schwarzschild's (gravitational) interior and exterior solutions for the model of an incompressible perfect-fluid sphere of constant density surrounded by empty space . In that work a zero-pressure surface-condition and matching and normalization of the interior and exterior metrics at the sphere radius were used as boundary conditions.
Tolman  has shown that the energy of a "quasi-static isolated system" can be expressed as "an integral extending only over the occupied space", which we will allow to extend to infinity, and where the total energy of such a sphere is therefore expressed as This mass-energy representation will be used throughout in calculating the distortional massenergies. The distortional-tensor energy-density amplitudes manifested in these presently calculated geometric representations are both negative and positive, that is, there are both negative energy-density [SI-2] and positive energy-density regions internal to the distortions. However, the modeled distortions for the mimicked elementary particles all exhibit positive mass-energies. Since geometric distortional fields arise from the same energy-density tensors, the negative energy-density geometric regions are also sources of negative energy-density field quantities.
(R0 is the normalizing radius after mimicking EM and gravitational forces) ) S ℏ and g e = 2.0023193043 6 (for the electron) .
The Td and Fd symbolism is used for the "distorted geometry" tensor quantities. The field equations, in both the EM realm and the gravitational realm (Q = 0), exhibit r -6 geometric behavior which we have interpreted as constituting a "magnetic monopole" mimic (what is a "magnetic monopole" ?).
A 2-dimensional plot of this structure is included here to help visualize the "distorted-geometry" model.