Electro-gravity via geometric chrononfield

In De Sitter / Anti De Sitter space-time and in other geometries, reference sub-manifolds from which proper time is measured along integral curves, are described as events. We introduce here a foliation with the help of a scalar field. The scalar field need not be unique but from the gradient of the scalar field, an intrinsic Reeb vector of the foliations perpendicular to the gradient vector is calculated. The Reeb vector describes the acceleration of a physical particle that moves along the integral curves that are formed by the gradient of the scalar field. The Reeb vector appears as a component of an anti-symmetric matrix which is a part of a rank-2, 2-Form. The 2-form is extended into a non-degenerate 4-form and into rank-4 matrix of a 2-form, which when multiplied by a velocity of a particle, becomes the acceleration of the particle. The matrix has one U(1) degree of freedom and an additional SU(2) degrees of freedom in two vectors that span the plane perpendicular to the gradient of the scalar field and to the Reeb vector. In total, there are U(1) x SU(2) degrees of freedom. SU(3) degrees of freedom arise from three dimensional foliations but require an additional symmetry to exist in order to have a valid covariant meaning. Matter in the Einstein Grossmann equation is replaced by the action of the acceleration field, i.e. by a geometric action which is not anticipated by the metric alone. This idea leads to a new formalism that replaces the conventional stress-energy-momentum-tensor. The formalism will be mainly developed for classical physics but will also be discussed for quantized physics based on events instead of particles. The result is that a positive charge manifests small attracting gravity and a stronger but small repelling acceleration field that repels even uncharged particles that have a rest mass. Negative charge manifests a repelling anti-gravity but also a stronger acceleration field that attracts even uncharged particles that have rest mass. Preliminary version: http://sciencedomain.org/abstract/9858


Introduction
The motivation of this theory is to show that matter can be put into correspondence with an acceleration field. There are two ways that measurement of proper time by a physical clock between events will be shortened: either by gravity in which the clock moves along geodesic curves but in curved space-time or by other interactions that prevent the clock from moving along a geodesic curve. These two approaches have to appear in the equation of gravity in order to describe Nature by a fully geometric model. The latter is not anticipated by the metric tensor alone and therefore it requires a new approach.
In this work, we will study the gradient of a real scalar field P which is i i dx dP P  . If a physical particle moves along the integral curves that are formed by i P then its velocity is where c is the speed of light. For convenience, throughout the paper we will use the notation are the affine connection, also known as second-type Christoffel symbols. We derive,   It is immediately evident that the vector We identify this representation with foliation theory, (Reeb, 1948(Reeb, , 1952 [1] and Godbillon-Vey, 1971, [2], [3] [1] of the foliation that is perpendicular to the 1-Form  .
The representation of the vector  leads to a far simpler term than the one represented by the Reinhart-Wood formula [2]. For this cohomology class, the following holds, where here R denotes the real numbers. This cohomology class is an invariant of the foliation F and is a closed 3-Form. An interesting property of the Reeb vector is that its restriction to the foliation F integrates to 0 on each closed curve on F.
A generalization of (2) to the complex numbers is easily defined    (5) In which the second term is positive because the differentiated Z P i vector has a low index.
The first term becomes, We add (6) and (7) to get (5) and notice that unlike the case of a velocity of a particle 2 c V V i i  . The squared curvature of the integral curves that are generated by i P is expressible, according to differential geometry, by the measurement of how much the unit vector Z P i changes along an arc length parameterization t of the integral curves. Calculation of the second power of trajectory curvature of integral curves along a conserving field, can be left as an exercise to the reader but the author prefers to present its calculation. This calculation is valid for all integral curves that are generated by vector fields that are scalar gradients. In our case, the integral curves should not be geodesic if they pass through material fields.
Caution: The t parameterization may not be the time measured by any physical particle because the scalar field from which the vector field is derived may be the result of an intersection of multiple trajectories along which P is measured. However, a particle that follows the gradient curves will indeed measure t even if its trajectory is not geodesic. Let t be the arc length measured along the curves formed by the vector field  P . By differential geometry, we know that the second power of curvature along these curves is simply Norm , see "APPENDIX -The time field in the Schwarzschild solution"), An arc length parameterization along these curves is equivalent to proper time measured by a particle that moves along the curves, and in the real numbers case, P can be indeed time. Unlike velocity's squared norms, Z is not constant. (11) Non-geodesic motion, as a result of interaction with a field, is not a geodesic motion in a gravitational field, i.e. it is not free fall. Moreover, material fields by this interpretation prohibit geodesic motion curves of particles moving at speeds less than the speed of light and by this, reduce the measurement of proper time. We return to the idea of acceleration by material fields.
We recall the work of Tzvi Scarr and of Yaakov Friedman [5] which used an anti-symmetric matrix to map a 4-velocity vector  V to a 4-acceleration vector  a . Since (2), (12) such that c is the speed of light, where  V is the 4-velocity of a material frame and  A is the Scarr-Friedman matrix [5]. The known relation 0    V a is obvious. The real valued action above (11), will lead to a very different energy momentum tensor than that of a simple real valued scalar Klein Gordon energy momentum tensor, instead of  is the permeability constant of vacuum,  F is the electro-magnetic tensor.

SU(2) X U(1) symmetries -partially symplectic space-time
is the Levi-Civita tensor (not symbol as the Levi-Civita symbol is a tensor density and not a tensor). It is easily verified that Kronecker delta.In the real numbers case, there are two ways to extend   A to a regular matrix and to keep the norm of the acceleration vector after the extended matrix is multiplied by vectors perpendicular to both Z P    and to  U . These matrices are, (14) and it is easy to see that (14) is the matrix we have been looking for and it also results in an immediate degree of freedom in the representation of the acceleration matrix by two additional vectors to  and  U but not in the matrix itself. (14) is quite similar to Dirac matrices but unlike them, it describes two acceleration planes and not a bi-spinor [6].
In particular, , see [7], that do not affect   A may be applied to   B . These B itself but only in its representation vectors, i.e., the normalized gradient of a scalar and its Reeb vector. As was suggested in (14), the singular

Partially Symplectic space-time
For space-time to be symplectic, it is enough to show that for coordinates s x and the following holds: The result is that the following form is not zero   and a reflection in the complex case. Consider an action operator as the third root of Gram's determinant of 3 Reeb vectors.

SU(3) symmetry
We may want to express the acceleration matrix (here c is not the speed of light and not the previous coefficients but a scalar field) and their gradients that span the foliation's tangent space k h in a covariant formalism but we need some constraint on  P .

Condition:
This condition is not trivial and in general, Consider the following matrix: for some scalar function  whose gradient i  is in the foliation perpendicular to  P etc. and in the same manner replace i  by a normalized unit vector Also, we demand orthogonality, Now comes a little trick: (4), it is obvious that the first two terms constitute minus twice the Reeb vector,

Invariance of the Reeb vector under different functions of P
Here we wish to explore another degree of freedom in the action operator of the "acceleration field" which results from the Reeb vector, as shown by a representative vector field i dx dP which is tangent to a non-geodesic integral curve. We wish to show that P can be replaced with a smooth function . We will prove the invariance of m U where P is real, however, a similar proof is also valid where P is complex and where P is replaced with a smooth function of P . Suppose that we replace P by f(P)such that f is positive and increasing or decreasing, then Which proves the invariance of the Reeb vector

Energy density by an acceleration field -Reeb vector at the classical non-covariant limit
We now show, how much energy density does this term represent.
For a clock that moves along the integral curves, formed by Z P  , we have from (2) and (11) In special relativity, the squared curvature of a trajectory of a particle is expressible by its 4acceleration, divided by the squared speed of light, The classical limit of a gravitational field is not covariant and that even worse, the classical field is intrinsic to the body of mass M that generates gravity; however, it is valid tool for the assessment of a physical model. We consider small mass at rest in a Newtonian (obviously not covariant) gravitational field. By the principle of equivalence, this mass is accelerated, otherwise it would freely fall. So, if a force field can keep small mass from falling, the field's classical limit of energy, is the same as the energy of the classical non-covariant gravitational field. This results hints at the energy of an acceleration field that opposes weak gravity. Summation of the squared norm of non-covariant 3acceleration, 2 a of clocks that are kept from falling in the weak gravity generated by the mass M is Where K is Newton's gravity constant. Now we calculate the non-relativistic and non-covariant negative potential energy So from (20) and (21) implies the following relation between energy and the non-gravitational acceleration field that prohibits geodesic motion, where 2 c  is the energy density and  is the mass density.
Note that unlike (24), (23) is not a covariant expression. What does it mean in the non-covariant classical limit of the electro-static field E . Since an electric field is also a form of an energy density,  is the permittivity of vacuum and from (24) we can infer the following non-covariant classical limit, where 2 E and 2 a are square norms of the 3-vectors E  and a  . We can infer, The acceleration in (25) is dauntingly small and very difficult to measure. It requires an immense field of 1 million volts over 1 millimetre to expose an acceleration of uncharged clocks, which is about 8.61cm/sec^2, less than 0.01 g, providing that there are no other fields that cancel out this acceleration. In fact, we will see below that charge also generates gravity and that for the choice (23), the acceleration will be about 4.305 cm/sec^2. By the principle of parsimony, the fact that this acceleration field stores energy, i.e. K a Density Energy  8 2   means that this acceleration is aligned with the electric charge, electro-static field curves because this can explain the electric charge attraction and repulsion by simply, increasing or decreasing the energy stored in such a weak acceleration field. As we shall see, if instead of K 8 we develop this theory such that K 4 divides the square norm of acceleration, no acceleration of neutral particles will be measured within a homogeneous electrostatic field. This is because, we will develop the Euler Lagrange equations of the Ricci scalar plus (11) and see that charge also generates gravity and not only inertial mass does.
We now use a covariant terminology of 4-acceleration  a and   a a a  2 .
As a more general theory, we can write Another important remark is that in the classical non-covariant limit, the divergence of the electric field can be written as, Such that  denotes charge density and not the previously defined energy density.
By experiments done by Hector Serrano, for NASA, the author believes that the acceleration field of even uncharged clocks in an electric field is towards the electron and out of the proton. A relation between charge and gravity can be developed, leading to unprecedented repercussions on the feasibility of Alcubierre Warp Drive, reference is given where it is discussed later. There is a remark of Serrano [9], about a moving capacitor in vacuum, in a reply to Peter Liddicoat: "Actually by the generally accepted definition of what constitutes high vacuum 10^-6 Torr is about in the middle. This pressure is about equal to low Earth orbit. More importantly at this pressure the 'Mean Free Path' of the molecules in the chamber is far too great to support Corona/Ion wind effects. We've tested from atmosphere to 10^-7 Torr with no change in performance either. However, I'm glad the results have you thinking. It looks simple, but trust me it's not".. Hector Serrano has mentioned in a patent [10], that a capacitor manifests weak thrust also in vacuum. Another indirect evidence is the Flyby Anomaly [11] which is possibly caused by ionosphere charge. For further evidence, see Timir Datta et. al. work as an elegant way to focus field lines by metal cone and plane and to observe an effect [12]. The author believes the acceleration of charge-less particles in an electric field is from positive to negative. In section 9 it is shown that there is an electrogravitational effect opposite in direction to the acceleration of an uncharged particle in an electro-static field. There is at least informal evidence that the elecro-gravitational effect shows thrust of the entire dipole towards the positive direction [9] and the author does not imply asymmetrical capacitors of 1 -0.1 Pico-Farad with 45000 Volts. It is shown that such capacitors -according to the calculations in section 9, assuming a roughly approximated acceleration proportional to the gravitational field -will not manifest any measurable effect of at least 1 micro Newton thrust. Most likely is that any measurable thrust, using such small capacitors, will be solely based on ionic wind.

Experimental problems -electron mobility
The down side of the non-geodesic acceleration is that it is about 10 orders of magnitude smaller than the accepted and known electric field interaction. For example, negative charge suspended above the Earth will cause charge to move in the ground. This charge will have a much stronger effect than the interaction with the acceleration field as is, and will cause a shielding effect i.e. the fields will cancel out within the Earth. Even the almost ideal insulator, i.e. diamond crystals, have impurities such as Nitrogen Vacancies [13] that allow charge carriers to move in the lattice i.e. high electron mobility. In the purest diamonds, the NV impurities are about

Vaknin's theory
We quote here one of the four models of Vaknin [14] as follows: This work contains a possible realization of space-time as an ideal geometric object that becomes physically accessible only where a wave function which is called "chronon" collapses. The physical model is therefore of events and not of particles. This paper offers the idea that matter occurs where the Reeb vector is not zero. Showing consistency of this model with Quantum Mechanics is a very difficult task although it is possible to show that the energy of an electric field is stored in an acceleration field by replacement of the electromagnetic tensor with the anti-symmetric acceleration field.
Vaknin's description of the realization of event is as follows:"Time as a wave function with observermediated collapse. Entanglement of all Chronons at the exact "moment" of the Big Bang. A relativistic QFT with Chronons as Field Quanta (excited states.) The integration is achieved via quantum superpositions".
The main difference between Vaknin's approach and the author's approach is that Vaknin's approach is algebraic where the author's approach is geometric. Thus, the outcome is two different theories that discuss a similar idea. We now show the simplest implementation of Vaknin's model as a quantization idea of time by collapsible events, as an additional constraint to the action  is the root of the negative metric tensor determinant for the volume element, such that The term 4 c is the speed of light to the power of 4 and K is Newton's gravity constant. 2 Curv is a generalization of the square norm of the Reeb vector to the complex numbers field, is the Levi Civita tensor (not symbol) and We generalize the acceleration field energy density from As we saw in (18), P does not have to be the proper time measured along curves. Instead, it can be a function of such proper time. (18) motivates the decomposition of P into wave functions because P does not have to be a monotonically increasing function of the proper time measured along integral curves formed by k P . The problem is that P is not any wave function of a particle. The simplest physical interpretation of the wave function is that it describes events in space-time and not particles. Therefore, P becomes a sum of wave functions and ) ( ...
is a decomposition of the function P as a sum of wave functions.
As quantum states, these event wave functions ,… must be normalized to probability 1 on the space-time manifold and they should be independent of each other as was written in two integral constraints. The best motivation for the constraints is given by Vaknin [14] and Storkin [15] where they emphasize that physical events are discrete. Storkin considers decreasing probability functions of the number of events in a given -volume, e.g. Poisson distribution within 2+1 dimensions Minkowsky space-time. Causal sets are partially ordered graphs of events along paths in space-time. The approach of this paper is more robust than that of Storkin because causal sets are the result of the order of events along the integral curves that are naturally formed by k P along with the mentioned constraints that induce a countable set of wave functions.

Auto-rotation:
In this section, we will study the field of a particle, whose rest mass energy is presumably stored in an acceleration field. The equation can be reduced to a probability density. k  is not a vector in the usual sense of General Relativity because it transforms between different coordinate systems with the help of spin connections and not with the help of affine connections. The philosophy of the Dirac equation, stems from the motivation to represent spatial spin axes as three orthogonal quantum states and to predict their probabilities. Any measurement of a spin is either +spin or -spin, no matter from which angle the physical measurement is performed. This property of the spin is very different than that of the classical mechanics spin. The philosophy of this paper is very different than that of Dirac, because it is based on a fully geometric interpretation of matter as acceleration fields. We begin with a quest for k  that will be an ordinary vector, unlike in Dirac's equation. We know that if k k *   is a probability density, then by (24), if all the energy of the particle is in its acceleration field, then we must have

General Relativity for the deterministic limit
By General Relativity, we have to add the Hilbert-Einstein action [16][17] [18] to the negative sign of the square curvature of the gradient of the scalar field in order to replace the energy-momentum tensor in the Einstein's field equations. Negative means that the curvature operator is mostly negative. As before, we assume g  is a scalar density of the volume element, R is the Ricci curvature [16] and g  is the determinant of the metric tensor used for the 4-volume element as in tensor densities [17].
The variation of the Ricci scalar is well known. It uses the Platini identity and Stokes theorem to calculate the variation of the Ricci curvature and reaches the Einstein tensor [18], as follows, which will be later added to the variation of The following Euler Lagrange equations have to hold, which we obtain from the minimum Euler Lagrange equation because . In order to calculate the minimum action Euler-Lagrange equations, we will separately treat the Lagrangians,    We subtract (28) from (29) g ) Z can be transformed to a real matrix due to the SU(2) x U(1) degrees of freedom and also be imaginary. From (31), (32) we have,

electro-gravity -unexpected gravity induced by electric charge
We return to (34) and see a startling property of the term This renders Alcubierre Warp Drive [19] a feasible technology by charge separation. Capacitors of several pico-Farads will not yield any measurable thrust [20] because they do not separate enough charge. However, separation of virtual charge that appears during transition states of electrons as they interact with photons, and by a short lived vacuum charge, is not ruled out because they can explain the effect known as EMDrive [21] by Warp Drive [19] caused by (37).

Total acceleration around electric charge
In the classical non-relativistic limit, acceleration a around a charge Q at radius r will be the result of (37) and by the acceleration field that prohibits geodesic motion, see (25)

Proof:
From the zero variation by the scalar time field (36) and we are done.

Conclusion
An upper limit on measurable time from each event backwards to the "big bang" singularity as a limit or from a manifold of events as in de Sitter or anti -de Sitter, may exist only as a limit and is not a practical physical observable because it can only be theoretically measured. Since more than one curve on which such time can be virtually measured intersects the same event -as is the case in material fields which prohibit inertial motion, i.e. prohibit free fall -such a time can't be realized as a coordinate. Nevertheless, using such time as a scalar field, enables to describe matter as acceleration fields by using the gradient of the scalar field and it allows new physics to emerge by a replacement of the stress-energy-momentum tensor. One arrives at electro-gravity as a neat explanation of the Dark Matter effect and the advent of Sciama's Inertial Induction, which becomes realizable by separation of high electric charge. This paper totally rules out any measurable Biefeld Brown effect in vacuum on Pico-Farad or less, Ionocrafts due to insufficient amount of electric charge [20]. The electrogravitational effect is due to field divergence and not directly due to intensity or gradient of the square norm. Inertial motion prohibition by material fields, e.g. intense electrostatic field, can be measured as a very small mass dependent force on neutral particles that have rest mass and thus can measure proper time. The non-gravitational acceleration should be from the positive to the negative charge. The electro-gravitational effect which is opposite in direction and ¾ in intensity, requires large amounts of separated charge carriers and acts on the entire negative to positive dipole.
Thanks to Mr. Raviv Yatom for his financial help and to Dr. Shomir Banerjee for his moral support.
Also, in the Book of Principals by the philosopher Rabbi Joseph Albo, essay 18 appears to be the first known historical account of what Measurable Time -In Hebrew "Zman Meshoar" and Immeasurable Time "Zman Bilti Meshoar" are. His Idea of the immeasurable time as a limit [22], is the very reason for 11 years of research and for this paper.

AppendixA-The time field in the Schwarzschild solution Motivation:
To make the reader familiar with the idea of maximal proper time from a sub-manifold and to calculate the background scalar time field of the Schwarzschild solution from that submanifold. We choose as a sub-manifold, a small 3 dimensional 3-spehere around the "Big Bang" singularity and therefore this example is limited to a "Big Bang" manifold. So, we want to connect each event in a Schwarzschild solution to a primordial sub-manifold a fraction of second after the presumed "Big Bang", with the longest possible curve under the assumption that no closed time-like curves occur. In this limited case, the scalar field is uninteresting as it does not represent interactions with any charged particle or with other force fields and therefore, the Reeb vector is zero.
We would like to calculate in Schwarzschild coordinates for a freely falling particle. This theory predicts that where there is no matter, the result must be zero. The speed of a falling particle from very far away, as measured by an observer in the gravitational field is Where is the Schwarzschild radius. If speed is normalized in relation to the speed of light then c U V  . For a far observer, the deltas are denoted by and, and .
Here is not a tensor index and it denotes derivative by ! On the other hand Here, is not a tensor index and it denotes derivative by ! For the square norms of gradients, we use the inverse of the metric tensor, And we can calculate Note that here is not a tensor index and it denotes derivative by ! t t r r Please note, here is not a tensor index and it denotes derivative by !
which shows that indeed the gradient of time measured, by a falling particle until it hits an event in the gravitational field, has zero curvature as expected.

Appendix B -Planck Area Gravity -Based on a lecture by professor Seth Lloyd of the M.I.T combined with the Geometric Chronon model and its correlation with sub-atomic particles
This appendix, reduces (34) and (35) from a 4 dimensional Minkowsky manifold equation to a 3 dimensional problem in ordinary Riemannian geometry. We will reach two third order polynomial equations form which guessing the relation between the radius of an infinitesimally small three dimensional ball and the norm of a Reeb vector will yield known energy ratios from the particles world. It is a whole system but it requires further honing in order to become rigid mathematics. We will consider the ratio between an area added or subtracted to a sphere due to curvature and the area of a flat sphere. The non-trivial step in this calculation is an assumption of a scaling factor which later on, yields very interesting results.
Suppose we have an atomic length L , The speed of light is c so the maximal acceleration will be L c c L c 2  . By the real case (33), If the right-hand side if multiplied by 2 1 and then by 4 12 L  then it yields the missing or added area to the sphere perpendicular to the unit vector Z P  [23], [24]. When this sum of sectional curvatures is multiplied by , it yields the area that is subtracted or added due to gravity or antigravity in relation to a sphere. If we consider that we have an "arrow" which is the Reeb vector alone with an infinitesimally small support and it is within a ball of radius and we attribute the divergence of the field to this "arrow" only, then we should not consider the increase or decrease of area around the ball due to the Ricci curvature as influencing due to Gauss law as we should if we consider the source of the field as a charge -like phenomenon. Then we can say that the field appears along a distance and has induced a geodesic motion prohibition as an acceleration, and and . We can play with the added portion area as a portion of energy around a particle. Let us consider, for instance, the portion of the Muon energy, 105.6583745(24) MeV This value is about 0.550304033979167 MeV and the electron mass is about 0.5109989461 MeV. It is a nice thought experiment but we need much more than that. We can't solve (34) or (35) yet but we can at least have a better idea of the field behavior in the Planck scale.
We didn't take into account that the geodesic motion prohibition field i.e. acceleration field changes its density on the sphere in accordance with increased or decreased area ratio, . We consider the Gauss law around an electric charge. So here we present a second approach to area addition and subtraction around an electrically charged particle. (B.4) can be rewritten as a more enlightening term, Such that  is either bigger than 1 or smaller than 1 and denotes the increase or decrease in area.
Note that the term  measures how much the square acceleration field changes as the area grows or dwindles.
The resulting equation is a cubic equation: (B.7) locally reduces the Minkowsky geometry of (34) to a Riemannian geometry in three dimensions. The area is increased or decreased by  and the portion of the area that changes is around a positive charge. The problem is that there is no stable charged particle without spin and therefore our discussion could mean a temporary decomposition of electrically neutral Bosons into two energy states, one temporarily behaving like a negative charge and one like a positive one. The reasoning behind such a claim is that if matter is expressible by a weak acceleration field and the weak acceleration field energy is the energy of an electric field, then elementary neutral particles, even with zero magnetic momentum and with zero electric dipole, should have an internal electric field. The question is how to infer such a structure. The idea is that area changes are relative to energy ratios even if they are changes due to charge electro-gravity and not due to inertial mass. It is a manifestation of a holographic principle [23], [24]. Our modest test will be to divide the Higgs energy by 2 and then either by

MeV
This energy is the model dependent vacuum constituent Quark energy according to Zhao Zhang et. al. [25].
According to this paper, no neutral particle can avoid having an internal structure, otherwise, the particle would not be able to manifest an acceleration field as energy. This leads to the possible model of BS Meson, Z Boson and Higgs Boson as either oscillating + and -charge such that both the magnetic and electric dipoles are zero, or as spinning + and -charge such that both magnetic and electric dipoles vanish. The problem is the Z and the Higgs bosons which are considered elementary particles. The Z boson mass is . If we split this mass into two charges, then 1/ 192.00515087160028 of area around the negative charge will be added, which is considered as proportional to mass [23]. But that portion is of half of the mass that splits to two charges, so we seek1/384.00258393161619 of the mass of the Z boson as having a physical meaning.  Here is a summary of the electro-gravity energy in the Planck scale around a positive and a negative charge that split an elementary boson and by this, these energies are beyond the Standard Model. Table 1. Presumed Beta and Alpha energy ratios due to a splitting of an Elementary charge-less boson into positive and negative charge -a supposed process which is beyond the Standard Model. A second assumption is that the magnetic moments and the electric dipoles of such bosons are zero, therefore split charge should be fluctuating and so is the area around positive and negative charge.
Area contraction ration around e+ r1=0. The mass ratio between Muon and the Electron (B.7) took into account the change of a unit field through a unit area as that area expands due to nonzero Einstein Tensor. The field however should be concentrated along an "equator" of a ball and zero at its poles due to spin around a negative charge. In this calculation, we only take into account the uneven distribution of the field but not its motion due to spin. The integration of the acceleration field should be of a field, Where is the angle from the "equator". A ball integration on two hemispheres is then, If the field is uniform then the integration would be And the ratio between (B.11) and (B.10) is which means that the acceleration field has to grow by in order to sum up as in (B.11). We can imagine as a vector that points towards or outwards -of an integral curve in space-time but that the Minkowsky norm of the field is always the same, only the probability that this vector points towards a certain direction in space-time changes. This idea leads to the compensating scaling value in (B.12). The ratio , which is very close to the ratio between the mass of the Muon to the mass of the Electron, 206.768277. The difference is expected because we did not take into account a spinning field. We followed the M.I.T professor Seth Lloyd offer that addition or subtraction of quantum area, means addition or subtraction of energy and reached . For the solution of (B.14) We get about 44.63955018. These ratios ~1/45 and ~1/207 could mean a decay path for charged leptons where the numerical stability of 1/45 is worse than that of 1/207. That energy is small but beyond the energy of any Neutrino mass. It is an unknown energy. Should it be a particle, this particle is beyond the Standard Model and its existence should manifest itself through a g-2 Muon anomaly. The ratio between the electron's energy and this energy is 0.5109989461 / 0.00004187500790 which is approximately, 12202.95760492718728 almost 12203. We can get this value if we return to the (B.7) roots and see that their multiplications a bit lower than 12203, but (B.7) assumes a gravitational field caused by an acceleration field around a negative and around a positive charge with no spin. The roots of (B.7) yield 1+ and 1-area ratios. These area ratios multiply to 12027.11454948692699 and not to 12027 < 12203. The exact number is obtained if we look at the following polynomials: (B.16) Which is 1 + or 1-the portion of area added around a negative charge or subtracted around a positive charge such that the acceleration field is smaller by a factor of . We did not use, however, the other root of (B.14) for which This ratio is very close to the mass ratio between the W boson and the Tau particle.

MeV
The energy of the Tau particle is 1776.82 MeV so the delta is 23.93739300965255 MeV. Dividing this energy by the Tau energy yields 0.01347204163035791470154545761529 = To understand where such a value can come from, we will return to (B.1) but this time with when in the real numbers format.
(B.7) transforms into a simpler equation instead of because which means .
In the very same manner (B.13) and (B.14) turn into which yields the following second order polynomial root equations, . The roots are easy to calculate especially that we know that one has to be slightly above 1 and one root is below 1. The average of area ratios is a bit surprising if we take out the sign, which means that like in (  The fine structure constant can be viewed as related to an energy portion of an emitted photon to the energy of an emitting charge. This is the subject of this section. Until now, all we did is to guess polynomial roots and see how they are related to energy portions via added or subtracted area portions. Now we are going to be more speculative because we will try to find out the average distance on a sphere when this sphere has non zero Riemannian curvature. We will try to have an educated guess of its curvature and to use this value in order to derive the Fine Structure constant. We will continue with the idea of an acceleration field where here C is the speed of light and in the real numbers case we relate square acceleration to the Reeb vector square norm . The fine structure constant is related to an interaction with a charged particle without spin transfer. If we think of a charge in such an interaction as a random point on a sphere then it is easy to see that the average distance between two random points on a sphere S2 is such that is the radius of the sphere. So if we ignore the addition of area around a negative charge or the subtraction of area around a positive charge, we have to multiply by and we have and (B.7) transforms into: (B.27) or and we find an emergent inaccurate approximation to the inverse of the fine structure constant: Close to 137.035999173 which is the inverse of the fine structure constant but not enough. We need to take into account the effect of dilation and contraction of areas around the electric charge in order to get a better value. In flat geometry, the distance between two points on the sphere that have an angle with the center is Integrating on the sphere and dividing by the area, we get the average distance, (B.30) Nice but why to see an interacting charge as a random point on an infinitesimally small sphere, pointed to by and why to assume that the geometry on the sphere is equivalent to a field that comes from the center, regardless of the direction of the field ? That idea has a lot to do with a popular idea, the Holographic Principle. We continue with the approximated calculation in which due to (33) there is a systematic error. The software involves basic understanding in numerical analysis and was written in the C language and not with the default Python that is used in the academic world simply because Python is too slow for 100000 refinement steps that resulted in 1000,000,000 integrand summations. for (i = 0; i < FINE_STRUCTURE_ITERATIONS; i++) { r_f1 = FUNCTION_average_distance(r_root1); r_root1 = (192 * r_root1 * r_root1 + 2 * r_root1 / r_f1 -1.0/(r_f1 * r_f1))/192; r_root1 = cbrt(r_root1); r_f2 = FUNCTION_average_distance(r_root2); r_root2 = (192 * r_root2 * r_root2 -2 * r_root2 / r_f2 -1.0 / (r_f2 * r_f2)) / 192; r_root2 = cbrt(r_root2); r_result = sqrt(1 / ((r_root1 -1)*(1 -r_root2))); printf("%.9lf, x1=%.9lf, x2=%.9lf2, 1/f1=%.9lf, 1/f2=%.9lf2\n", r_result,r_root1,r_root2,1/r_f1,1/r_f2); } } int main() { while (_kbhit()) _getch(); // Clear keyboard input.
puts("Press Enter to exit the console."); getchar(); return 0; } C -Towards a unified force theory -1,2,3 types of accelerated clocks The following action can be extended to U(1) x SU(2) and to SU(3) symmetries by considering more than one Reeb vector.
The physical meaning of  ) 2 ( U is another acceleration field in another plane. We will consider (C.2) as the "Electro-Weak Geometric Chronon Action".
In the three dimensional space, Minkowsky perpendicular to  P we can view three holonomic vectors fields that span the foliation tangent space as required by the Froben ius theorem . These can be locally described by three gradients,

The sign of time
This idea is quite similar to Dr. Sam Vaknin's idea of Quarks of time, see [14].
Consider the Levi -Civita tensor (and not the Levi -Civita symbol which is a tensor density and not a tensor) we can establish a vector which is parallel to the flow  P but can have either a positive or negative sign when contracted with  P , we have