^{1}

^{2}

^{3}

The expression of the Maxwell magnetic monopole was employed to correlate the space to space projection that gives rise to the Gell-Mann standard model, and space to time projection which gives the leptons; and how does it correlate to the Perelman mappings from the homogeneous 5D manifold to the Lorentz 4D manifold, together with correlating the physical consequences caused by the breaking of the Diagonal Long Range Order [DLRO] of the monopoles quantum states affected by the motion of massive particles in the Lorentz 4D boundary of the 5D manifold, which leads to gravitons and the gravity field via the General Relativity covariant Riemannian 4D curvatures metric equation.

The homogeneous 5D manifold was presented several years back [

The homogeneous 5D manifold obeys Fermat’s Last Theorem [

[ c t ] 2 = r 2 . (1.1)

Any energy expressed in terms of 2hv, would generate a monopole of strength ±2ec [

± 2 M → μ r . (1.2)

The factor 2 comes from total spin 0 degeneracy of the pair of e , p , s ; − e , − p , − s , where s is up or down spin of the massless spinors, and momentum p is along r such that this M → state is in the DLRO and μ is the magnetic permeability. The potential has the conventional unit of Joule.

In order to understand properly the process of physical generation of mass and charges from the homogeneous 5D space-time manifold, it is essential to first construct this 5th component of the Maxwell magnetic monopole potential.

The 5D homogeneous space-time manifold, together with the uncertainty principle, results in the presence of 5 vector components potential fields. The first 4 vector potentials A(0) to A(3) are those of the Maxwell electro-magnetic vector potentials, while the 5th component A(4) is explicitly derived recently for the representation of the 5D as specified by the massless spinor fields expressible in tU3D × 1D [

Should we be able to add a net linear momentum, from the motion of a massive nuclear in the r = 0 5D frame, with charge 0 or q to the monopole inducing current magnitude M and −M, The currents generating magnetic field will be replaced by q v → + M → N ( + ) and q v → − M → N ( − ) , where v → is the velocity of charge q. Where N ( + ) and N ( − ) are the energy integrated weight factors, then these two opposite magnetic monopoles will give raise to an attractive quantum well in the 5D manifold, as it changes the monopole strength to M ′ = q v → + M → N ( + ) and − M → ″ = q v → − M → N ( − ) Where v → not be along “r”. Unlike M, which is a Boson state, the nuclear charge +q can be either a Fermion or a Boson. Hence the Fermionic and Bosonic statistics of M → ′ , M → ″ is broken and gives us the attracting effect on the massive “q” by a V(5) monopole-monopole potential well:

V ( 5 ) = − M ′ M ″ μ r ≅ − [ 2 e c ] 2 N ( + ) N ( − ) μ r (1.3)

with “r” fixed by ct, irrespective of whether t is fixed or not.

The term ( q v ) 2 can usually be neglected, as it is normally extremely small for the nucleus of q charge as N ( + ) and N ( − ) are canonical ensemble generated, temperature-dependent, large thermal quantities. As such V(5) is valid only for a macro thermal system, not the initial space dimension projections that created the basic leptons and hadrons, similar to the Perelman-entropy mapping. However when q v → exist, the boundary conditions of M → ′ and − M → ″ are altered, and M → ′ as well as − M → ″ are no longer Bosonic and hence in the Bose condensate state.

It should be pointed out that due to the Pauli exclusion on the e-trino, anti-e-trino, the repulsive magnitude M^{2} is given by the products of ( e , p ) × ( − e , − p ) × ( e , p ′ ) × ( − e , − p ′ ) . Where ( e , p ) and ( − e , − p ) represents the e-trino and anti-e-trino spinors with p and -p momentum, that is only radially outward, so sign cannot be changed. Note that e-trino state commutes with anti-e-trino states, while anti-commutes with another e-trino state. Thus M^{2} can be expressed as

1 4 { ( e , p ) × ( e , p ′ ) − ( e , p ′ ) × ( e , p ) } × { ( − e , − p ) × ( − e , − p ′ ) − ( − e , − p ′ ) × ( − e , − p ) } .

The e-trino and anti-e-trino pair states are Boson. But due to its antisymmetry form for arbitrary p and p ′ , its thermal average has 0 net weight.

On the other hand the attractive monopole-monopole magnitude comes from the product between M and −M, where the −M vector is obtained by interchanging e with −e and vice versa in the definition of M giving by the radial current 2ec. This change will no longer lead to the antisymmetry of the e-trino pairs and the thermal weight factor is not 0.

Since the 5D manifold irrespective of “t” is given by Fermat’s sum, and with charge conservation, therefore irrespective of the 4D space as long as there is a finite size of the opening for the 1D subspace that allows for r along that 1D expansion, N ( + ) = N ( − ) = N . In other words, the monopole potential well becomes space inhomogeneous. In fact N would be proportional to the ratio of the solid angle “S” of the opening as compared to the spherical solid angle 4π, and is a linear function of the temperature, as M (having the unit of ampere-meter) represents a boundary-condition-dependent Bosonic state, with degenerate energy E. V(5) is thus modified by s = S / 4π . And for first order approximation we should modify N ( + ) N ( − ) by N ( + ) N ( − ) s . And we can get from the Bose distribution the thermal number of such M states

N = a . (1.4)

where a is derived from integrating the Bose distribution over d E k T from h t ( n k T ) = C , where n is an arbitrary number, such that at t = t o , the time of the Big Bang, we have n h t o = h v ( o ) which must exceed the rest energy of the quark rest mass energy m ( Q ) c 2 , such that at least it is enough to create both electrons and protons and neutrons in the Lorentz boundary domain, while maintaining net charge neutrality. The higher the temperature T, the smaller will be C when h v ( o ) is fixed, leading to a larger N number and thereby more massive particles created in the Lorentz boundary domain which implies in the nucleus more fusion, and in the stars and planets more mass, even implying a larger galaxy. Hence the quantity C corresponds to the chemical potential of the grand canonical ensemble. Should t → 0 , C → ∞ , a → 0 , implying without extra input of starting creation of the monopole potential fields at a finite initial t ( o ) > 0 value, the homogeneous 5D manifold cannot have any M as well as Maxwell EM potentials, irrespective of the uncertainty principle, because there were no sources in the 5D manifold. However with C > 0 and finite

N = 4 π k T h ∑ j = 1 ∞ e − j C j < ∞ (1.5)

where j is a positive integer.

This sum “a” is finite, and proportional to T, the absolute temperature, implying also that the quantum well cannot be completely cancelled by a finite number of + or − massive charges created. On the other hand, if the monopoles M and −M are replaced by M ′ and − M ″ the Bose statistic, and the Bose condensation is broken, similar to a physical feature employed by Higg’s theory. With the quantum well potential completely changed by equal number of relativistic massive charges of opposite signs in the 4D Lorentz boundary domain.

The mass associated with the + q = n e charge is not necessarily equal to that of the sum of n − e massive spinors. Both masses from the 5D field theory are the result from space projection into 4D Lorentz manifolds via the space to time projection P_{0}, and the pure space dimension reduction projection from 4D to 3D through P_{1} enacted during the creation of the 5D universe, at t ( o ) > 0 the instant of the Big Bang, when the monopoles must be created. It was assumed in the 5D book such projections will not happen again in later time [

The difference between M ′ → and − M ″ → comes from the symmetric pairs of e and −e within M → and − M → with different energy values partly converted into q = + e n massive nucleus of mass m ( + ) or n , − e massive electron spinors, with electron mass m ( e ) via P_{0} the 4th space component projection onto the time axis, mapping 5D into Lorentz 4D × SU(2), where SU(n') represents the semi-simple compact Lie Groups of n' dimensions, and from +e massive spinors [the proton] via P_{1} the 4th space component onto the remaining 3 space components, making up the protons and neutrons of the nucleus, such that the proton is defined by the gauge confined u , u , d quarks, and the neutron as u , d , d quarks; where u is the 2 e 3 up quark, and d is the − e 3 down quark. Or breaking 5D into Lorentz 4D × SU(3). Hence m ( e ) n ≪ m ( + ) where m ( e ) is the lowest energy state, the electron rest mass m ( e ) as the nucleus creation takes far more energy than the n electrons creation from the energy of the homogeneous 5D Fermat energy-momentum manifold, breaking the quantum homomorphic M → and − M → distribution and the Bose-Einstein condensation over the 4D homogeneous space, as given by the third phase angle of 0 to 4π in the Fermat’s representation. Such a breaking of the homogeneity of the 5D manifold must occur first before the deformation of the 5D space-time into separated 5D manifolds with separated 4D Lorentz manifolds in the topology of 3D space of doughnut shapes as given by the Perelman-Ricci flow mapping. In fact it is important mathematically to realize the change of M → and − M → along “r” of the 3D homogenous space to M → N + q v → and − M → N + q v → , where the massive q charge velocity v → is an arbitrary 3D vector. This implies the resulting monopole current is no longer necessarily along “r” and will no longer obey Gauss Theorem. It is this feature that implies that the monopole vector potential must be expressed in a covariant representation due to the motion of the massive charge q, similar to the treatment of the Coulomb potential of a charge q. It is this correlation that leads us to the Riemannian curvature for the Lorentz space-time due to the dynamic distribution of masses within, even when these masses can be considered as charge-less on the classical scale, and as the monopole quantum well vanishes when s becomes 0, or by the change in frame to v, when

s [ M N ] 2 = [ q v ] 2 = [ n e v ] 2 (1.6)

representing the charge neutral mass case when the simultaneous condition on both the positive q and negative − n e massive charges having the same Fermat’s amplitude value r, which is the quantum well “r”. Since N is linear in T, hence the right hand side of (1.6) which includes v 2 must also varies as s T 2 . When that happens, the charge neutral moving mass in the Lorentz 4D domain observes no monopole well effect, and the nuclear fusion ceased, and the remaining binding is replaced by the forming of atoms by Coulomb attraction between the nucleus and its orbiting electrons. But most important it gives us a relationship between how s → 0 due to v 2 of the nucleons in the nucleus mass shell, forming the Poincare sphere, as evolved through the Perelman-entropy mapping. Using the expression for N as an infinite series sum as given by (1.5) the relationship between s and N, can be expressed as a double sum series equation

∑ j = 1 ∞ ∑ j ′ = 1 ∞ s ( j , j ′ ) j j ′ e − ( j + j ′ ) C = K ， (1.7)

where K is a constant.

As M → and − M → are by uncertainty principle required to be along the same vector r → of the 5D manifold so that V ( 5 ) = M A ( 4 ) is also a solution of the 5D metric operator, hence s ( j , j ′ ) must be confined by a gauge restriction of h c M = h 2 e , which defines a gauge loop and thereby the Boson field M can be decoupled by a gauge transformation. This loop integration requires defining a z vector direction, and coupled to the conversion of M → and − M → into 2 closed current loops of J or −J, where the magnitude J is still 2ec, but is no longer of DLRO, and made of the pair e , p x r ′ , s ; and − e , − p x r ′ , s ′ , and separated along z, in a 2D × 1D space representation, where the spins s , s ′ are not necessarily opposite, and the gauge loop radius that decouples M from A(4) for r ′ < r , the Fermat’s amplitude. Such a single closed J loop also leads to e , − e annihilation, unless J is split into 2 parallel but separated ec closed loops, which would generate a magnetic dipole field, and obeys Chern-Simons gauge, leading thus to a Perelman-Ricci Flow mapping.

Hence s ( j , j ′ ) has a solution depending on

e − ( j + j ′ ) C , (1.8)

where the chemical potential C is a positive number. In fact C must be related to the q and n e k T the open 5D core of 2D radius “ r ′ ”. This form of s > 0 corresponds to the Bose condensation of N being broken by the presence of q v → . Therefore the higher the 3D × 1D core temperature the more nuclear and electron matter will have to be created on the final closed shell.

Its relationship to N is the physics that basically gives us the Perelman-entropy mapping from the Perelman-Ricci Flow open doughnut Lorentz manifold into the closed Poincare sphere, when the isolated monopole no longer exist. This triple relation between q = n e , v c and N, a function of increasing temperature will result in a phase diagram for elements formed at T. Hence the complete periodic table is reached in cool systems. Thereby, the pure gravity field observed by the observer must be derived from the covariant Riemannian 4D space-time, chosen fixed by the rest frame of the observer fixed on the moving masses formed by elements that gives us the Newtonian gravitation potential of a stationary mass as derived from the curvature metric equation according to General Relativity.

As we mentioned without defining a t ( o ) > 0 for the Big Bang, the 5D homogeneous manifold contains no fields of any kind. Hence in the process of creation of the monopoles at t ( o ) , there must be the total energy conservation given by

〈 E 〉 + V ( 5 ) = 0 . (1.9)

The quantity 〈 E 〉 represents the total ensemble energy of creating the monopoles M → and − M → . It is easy to see that both terms in (1.9) are proportion to T 2 . Hence the temperature of the manifold cannot be determined, by C and s.

The presents of V(5) creates an energy sink in the 3D × 1D space of the 5D manifold, hence any matter in the 4D Lorentz boundary domain would be attracted towards the center of the 5D manifold, the 2D r ′ = 0 , similar to the presence of a black hole in 4D General Relativity theory. But in 5D theory, matter cannot exist inside the 1D monopole subspaces. This requirement can be achieved if the matter in the Lorentz 4D boundary domain has an angular momentum L, such that as its distance R > r ′ to the void deceases towards r ′ , of the 1D M domain, its energy increases so that its momentum ρ increases, leading to a centrifugal outward force sufficient to counter the monopole quantum well attractive strength, by converting part of the energy carried by the M → , − M → states along r into coherent rotating e , − e massless counter L state around r ′ to the revolving mass L, via the reducing of the covering solid angle ratio s, eventually s becomes 0, enacting the Perelman-entropy mapping in the maintaining of balance of net L = 0 as the monopole M’s energy distribution in the homogeneous 5D is reduced to . This feature was discussed earlier by us [

As discussed in the introduction a single isolated m ( + ) massive nucleus with charge “+q” within the non-homomorphic 5D manifold is subjected to the quantum approximate well given by V(5), thus at a given time t, it must be governed by the force balanced equation, equivalent to the introducing of the Lagrangian for the Ricci-Flow mapping given by Perelmann [

q H v c − m ( + ) v 2 r ′ + d V ( 5 ) d r ′ = 0 (2.1)

The first term is the Lorentz force, due to a H field generated by all the moving charges within the 4D Lorentz manifold on q, and including the dipolar field generated by the non DLRO closed J or −J loops. The dipole H field obtained from J or −J is then given by the loop currents thermal averaged over product of two Fermi distributions of e , p x r ′ and − e , − p x r ′ ; [see Fung and Wong [

r ′ = ( m ( + ) c 2 ) ( v c ) 2 q H { 1 + [ 1 − 4 M 2 μ s N 2 q H ( m ( + ) c 2 ) 2 ( v c ) 3 ] } . (2.2)

There are 3 variables, namely qv, r and H. Hence there is no unique solution to “ r ′ ”.

But since r ′ must be real and positive, it means

m ( + ) c 2 ( v c ) 1.5 > 2 M N s μ q H (2.3)

As q = n e and m ( + ) > m ( p ) n where m ( p ) is the proton rest mass, due to neutrons within the nucleus. Thus we have

H < ( m ( + ) c 2 ) 2 ( v / c ) 3 4 μ q M 2 s N 2 (2.4)

As v c according to special relativity is bounded by 1, the nucleus mass m ( + ) and the nucleus r then dictates the upper bound value of the magnetic field H.

Thus since the nucleus r ′ > 0 , and small, it has an upper bound value H = m ( + ) c 2 [ v / c ] q N . As N → ∞ as C → 0 and M is in the Bose condensate state, H = 0 . Even not in the condensed state H value is relatively low because of large but not infinite N when the Bose condensation is broken making nuclear fusion from single proton to nucleus with n > 1 possible. While there is in general no lower bound on H limitation which is also valid in stars and planets as r increases to astronomical value.

For the electron force equation solution we replace with − e ( − v ) replacing qv and r ′ changes from (2.2) to

r ′ = ( m ( e ) c 2 ) ( v c ) 2 e H { 1 + [ 1 − 4 M 2 μ s N 2 e H ( m ( e ) c 2 ) 2 ( v c ) 3 ] } . (2.5)

It is obvious “ r ′ ” for electron is greater than “ r ′ ” for the nucleus with +q charges, even if v c is the same. Condition on H with N given by (2.4) is modified to

H < ( m ( e ) c 2 ) 2 ( v / c ) 3 4 e μ s M 2 N 2 . (2.6)

Since v c < 1 , H must be inversely proportional to N 2 . But for the nucleus charge q, it is inversely proportional to n N 2 . Thereby the dipolar magnetic H 1D field has to be weak in heavy nuclei. Furthermore, since H is the same for both, it follows that m ( + ) must be proportional to n m ( e ) . It should be noted that the r ′ solution for the electron is outside the nucleus “ r ′ ” by orders of magnitude, which further reduces the mass ratio m ( e ) m ( + ) . This extra ratio comes from the SU(3) Lie Group generators. Therefore for general “r” solutions exceeding that for the m ( e ) solution given by (2.6) as N and n become very large, would result in the 4D Lorentz domain with charge neutral matter and then becomes the pure classical topological Perelman mappings derived from the 5D homogeneous manifold. In conclusion, the nucleons, proton and neutron masses are mainly due to quantum gauge confinement of the repulsive gluon potential generated by the intermediate quark currents [

Astronomical matters, such as galaxies, stars and planets the above force balanced equations is not strictly applicable not because s becomes 0, but rather that we are in a classical gravity domain, where some physicists believe in the graviton theory, where all matters are formed by charge neutral atoms, as the time frozen void 3D × 1D way exceeds the dimension of atoms, and with N being an integer, gravity is quantized, in a way similar to photons. Where the Lorentz force and as the monopole potentials no longer apply for the charge neutral matter. None the less, we believe it is the replacement for the monopole attractive quantum well by the Newtonian gravitation potential in the rest frame of the neutral mass m ( + ) + m ( − ) , with the curving of the Maxwell 4 vector potentials acting on a combined massive large number of charges q, and -ne forming atoms and molecules in a pure 4D Lorentz domain as explained in section 1, as the open surface area S of the Gauss surface shrinks due to the closing by the mass shell around the 3D × 1D space void it must be considered multiplied together with the infinite N 2 magnitude of the monopole well, [see eq. 1.5] due to the lower limit in its energy spectrum when in the specific rest mass 4D Lorentz frame such that the monopole quantum well strength in the mathematical limit s → 0 , as N → ∞ when C → 0 , such that the quantum well s N 2 M 2 strength is replaced by the attractive Newtonian gravity potential with strength G M ( 0 ) where G is the Newtonian constant and M ( 0 ) the mass of the Poincare sphere. Hence with relative moving masses, as given geometrically as Poincare spheres, the gravity potential field from multiple masses must be derived from the curvature arising from the covariant mass frames creating the warping of the 4D Lorentz-Riemannian covariant space-time caused by the dynamic distribution of all these masses in relative motions within the 5D universe, a result discussed in Einstein’s General Relativity. Such a non-linear gravity potential equation without the masses having 3D × 1D voids must then possess a singularity independent to the mass corresponding to the N divergence. In fact it can be easily seen physically from the exact q v → + N M → and q v → − N M → moving charge currents given in (1.3) as q v → , which is on the Lorentz boundary domain becomes comparable to NM in magnitude, that the approximated V(5) is no longer valid, and the shrinking of the open Gauss surface because of the finite sizes of the matter composed of atoms such that the Perelman entropy mapping takes over and closes the mass into a Poincare sphere, the monopole strength will decrease to zero, breaking the Bose-Einstein condensate state. In another word, N is a decreasing function of s and the final Poincare mass sphere should enclose a time frozen 3D × 1D void, such as that suggested by Wheeler’s worm hole [

To mathematically understand how the eigen solutions of revolving charge within a dimension reduced space can be equivalent to the revolving charge’s mass will give us further understanding on the importance of the monopole quantum well effect on the 4D covariant Lorentz space time Riemannian curvature.

It was proposed by Perelman that the Ricci Flow topological mapping of the 5D homogeneous space-time would result in a non-homogeneous 4D Lorentz manifold, with the 3D space in the geometrical shape of a doughnut, with a center 5D open core, such that the 4th space coordinate 1’D is the space coordinate through the vector addition from open z axis through the doughnut core center and the closed changing r as a function of z, of the doughnut tube variable around the core. It is such a 4D Lorentz manifold that properly represents the 4D space-time of a galaxy, where all the stars, planets and meteors are contained. In fact such a mapping can be quantized during the absolute beginning of the 5D universe matrix, by employing the P_{0} and P_{1} projections from this 4th 1’D entangled space representation dimension onto the t and 2D coordinates via P_{0} and P_{1} mapping. By using a 2D × 1D × 1’D space symmetry representation, the 3D Lorentz gauge invariance must now be replaced by the 2D Chern-Simons gauge [

Without considering M → ′ and − M → ″ on the curving portion, it is interesting to investigate the Hamiltonian for the electron eigenstates just around the core at z = 0. This Hamiltonian is in 2D, M → ′ and − M → ″ are reduced to only M → and − M → along up and down z and making the monopole quantum well V(5) resembles that of the relativistic 2D hydrogen, with Coulomb like potential 2D monopole-monopole attractive potential − 4 e 2 c 2 μ r = − 4 e 2 ε r due to ε μ = 1 c 2 . Although the monopole-monopole quantum well is expression wise equivalent to the Coulomb attractive potential of − 4 e 2 , topologically it is totally different. Nonetheless the eigen solution of a revolving massive electron would satisfy the Chern-Simons gauge. In the 2D Coulomb potential the charges are massive and the positive charge +e is the proton at the origin r = 0 , while the much lighter electron is in a relativistic ground state orbit, with the binding energy precisely equal to the electron reduced mass energy. In the case of the 2D quantum well produced by M and −M along z, the massive electron ground state binding energy must then be precisely 4 times the electron rest mass [_{1} choices 4 × 4 × 4, a result that also generate the gluon field via gauge confinement that gives the major portion of the proton mass. Hence the monopole quantum well strength should be weighted by 64 for the quarks. Note that all M monopole states are charge neutral. Therefore if a −e massless spinor is projected onto t, and becomes an electron, then one +e must also be projected by P_{1} to form a proton. Hence because of the 1D having a 4 fold choice, in terms of the quark rest mass derived from m(e), must be 4 × 4 × 4 m(e), which is roughly 32 MeV. This value for the quark rest mass agrees well with data fitting for the mesons and baryons to within error due to the relativistic modification on the quark constituent masses within each hadron [

Hence here the 2D relativistic ground state of the monopole quantum well is − 4 m ( e ) c 2 . Thereby the total 2D hydrogen like quantum well after creating the electrons has no more energy [_{0} projection of creating electrons in the presence of Perelman-Ricci Flow mapping. In short, the electron mass m(e) is created by the 2D monopole-monopole attractive quantum well under Chern-Simons gauge transformation via the Perelman-Ricci flow mapping same as the P_{0} projection from the 4th space component onto time, resulting in an electron placed within a 4D Lorentz manifold. This interesting approximate result implies that the M → , − M → 1’D like currents would be continuously ejected from the doughnut 5D core of a galaxy, according to the 5D homogeneous energy-momentum metric, at t = 0 , E and p are infinite, hence despite the creation of numerous galaxies from the Big Bang, the galactic center of the galaxy will still have infinite amount of energy to create M → and − M → magnetic pole currents radiating outward along the 1’D, although these M → and − M → energetic bosons will not be projected to form new SU(2) × L and SU(3) × L manifolds they can excite the leptons and nucleons causing ionization as well as nuclear fission, and any star system that spirally revolves outward on the Lorentz doughnut surface, would periodically receive these M → , − M → high energy like monopole beams. Since, M → and − M → with high energies above the electron and quark mass must preserve the equal numbers of electrons and protons, causing atomic ionization and nuclear fission, which upon thermal cooling afterward can lead to new nuclear fusion and the changes in the element composition in its matter shell, it means a planet, like earth, which star the sun revolves on the 4D Lorentz doughnut manifold would necessarily encounters intense periodic cosmic storms from the galactic core, the period length is equal to the revolving period of the star around the galactic center, which normally would be of the scale of millions of years. Such intense ec loop currents around the galactic core creating H z ′ tilted to the galactic plane [_{2} within the planet’s atmosphere. Due to this cosmic effect on our earth’s climate exists, recent climate changes on earth as solely due to human created CO_{2} effect cannot be considered as conclusive. In fact we can make a crude model on this effect. Let us say the galactic core system orbital radius around the Milky Way core is R, and its revolving speed is “w”. Then the approximate period T ( o ) = 2 π R w . The sun is however on the surface of the galactic doughnut tube of radius “r”, and inclined to the galactic plane with angle theta.

Then the position of the sun to the galactic core is

R ′ = R + r sin 2 θ + ( 1 − cos θ ) 2 = R + r 2 ( 1 − cos θ ) (3.1)

There are therefore periodic effects based on the J currents loops emitted from the core’s curvature creation of a tilted radiation belt energetic particle hitting planets like earth as it enters and leaves the belt.

First because the solar system spirals outward from the galactic core R is an increasing function of time. Thus for each orbital period, a change of the period

of D T = 2 π w d R d t occurs.

Second, the effect of the direct free charge radiation trapped in the galactic radiation belt created by the J or −J closed loops around the core, which must not be aligned with Lz, the galactic angular momentum due to annihilation forbidden path overlap of the e-trino states. Hence solar activity occurs only if the galactic belt hits on the solar system, and in turn causes the solar radiation periods, which as compared to the spiral period T ( o ) is

T ′ ( o ) = T ( o ) [ r R ] 2 ( 1 − cos θ ) (3.2)

For the Milky Way r R ≫ 1 , however for the sun, θ is a small angle [

In the previous sections, we studied the e-trino, anti-e-trino opposite momentum 1’D monopole currents which are created from energy in the 5D manifold. For any totally matter enclosed sphere, the time frozen spherical boundary imposed on these pairs states would necessarily convert the open M → , − M → like magnetic currents into perpetual ec closed loops. The perpendicular axis z ′ to the closed ec loops cannot be in exact alignment to the in phase e-trino, anti-e-trino rotating perpetual pairs as we had mentioned in section 3, that would generate a self rotation of angular momentum Lz, as it will cause e , − e annihilation due to t > 0 only in the 5D metric. This is clearly observed in stars and planets. In fact this same restriction must also happen with the ec closed loop states around the doughnut shaped galactic core. Take the Milky Way, as example, the closed ec loops, would then generate a dipolar magnetic field H z ′ , that is tilted to the galactic plane. Hence, all charges trapped within this H z ′ field created by the closed ec loops around the galactic core will give raise to a tilted radiation belt as is well known in the Milky Way to astronomers [

Nuclear shell model [_{2} green house gas, plus changes in oxygen and nitrogen composition due to decomposition of oxides and nitrates from the crest. And with water vapor if the surface is above freezing, after cooling creates surface water pools, forming seas and lakes and streams. To achieve a man make planet core temperature change via nuclear fission and fusion for example on Mars, [see recent news article on Mars.] require solving many technical problems. For example, how to send a carefully controlled nuclear fission device into the planet’s core intact and then trigger it. If we can achieve and solve these technical problems, then perhaps, it is the best humanitarian reward we earned from our previous technological development on creating nuclear weapons?

It is best to summarize the important mathematical theorems that are employed in the 5D theory we present in this paper. First and foremost, all homogeneous manifolds satisfy the Fermat’s Last Theorem. So does the 5D. Second, all space and time measurements must obey the uncertainty principle, thus the Fermat’s sum gives the same dimensional homogeneous quadratic operator, with only plane wave solutions. The boundary on such a homogeneous space-time manifold is a manifold that is one space dimension lower. Thus for the 5D homogeneous space-time manifold, it is the 4D homogeneous Maxwell manifold. It is this theorem that provides us with the 5th component vector potential, the magnetic monopoles. The Perelman-Ricci Flow Theorem allows for the connecting of 4D Maxwell domains into a Lorentz 4D doughnut domain, thus chopping the single 5D universe into interconnected tU3D × 1D space-time topology with the initial absolute zero magnitude point of the Fermat’s sum into a line of zero dimension, similarly, with the energy-momentum metric, the initial single infinite energy point from the original 5D into disconnected multiple zero spread points of infinite energy. These mappings are mathematically profound, because only 0 and ∞ can be divided arbitrarily into, no matter how many and remain exactly 0 and infinite. Lastly, if in each 5D manifold, t is fixed, then it follows the 4D space that is arbitrarily divided and separated into Poincare spheres with time independent voids of 3D × 1D. Such topological subspaces are obtained via the Perelman-entropy mapping. Such mapping must follow from the closing up of the connected doughnut 4D manifolds. However, none of these mathematical theorems can give us the space to time and space to space dimension reduction projections, as projection transformation has no inverse, and must be enacted, meaning that it can only happen by command, an implication similar to the creation of the 5D metric itself, coming from the universe creator.

We thank Ms. Winnie So, Ms. Elize Yeung and Dr. Anthony Cheng for their help in typing this manuscript.

The authors declare no conflicts of interest regarding the publication of this paper.

Wong, K.W., Fung, P.C.W. and Chow, W.K. (2019) A Quantum Representation of the Homogeneous 5D Manifold and the Perelman Mappings of 5D onto Non-Homogeneous Lorentz 4D Manifolds. Journal of Modern Physics, 10, 557-575. https://doi.org/10.4236/jmp.2019.105039