User:Keshav73

pattern revealed in the fine structure constant digits(where as it is true that these patterns are to a extent base 10 number system specific so that these are numerological apparently but these numerologies are subsequently eliminated in this paper by absolutising (physically only and not mathematically) base 10 number system with significant and relevant justifications. Purely mathematically however base 10 is as good as any other base and are completely equivalent):http://www.facebook.com/jha.dilip.1 on the wall (by DILIP KUMAR JHA,email address: jhadilip73@gmail.com):(derivation of Fine Structure Constant Formula, derivation of Gravitational Coupling Constant Formula and The Origin Of Radiation Reaction Force):      FOR: THE ULTIMATE PHYSICAL EVIDENCE, SEE BELOW:'The predicted value of Fine Structure Constant at zero energy(at Euclidean infinity (

'''Which means that it Q.E.D. calculation enabled value)) is, α=0.0072973525696314079556342228226322....... . The predicted value of Fine Structure Constant at physical infinity and again at zero energy(which means that it is Q.E.D. calculation disabled value'''

)is: .007297352541549875....... . The predicted value of Gravitational Coupling Constant for the case of two electron is:1.751159404………×10^(-45). Here is a formula for fine structure constant that could well be the most probable one on the planet. According to a rough estimate the odds against it are less than one part per zillion(1/10^40,read the paper to see how) and all this backed up by a very elegant derivation, which is likely to be posted soon at this place. It reveals the pattern in the fine structure constant digits. Watch it unfold as you input the formula given below on your computer or calculator. 1)                    4y^3+12y^2+5y-4=0 Y=.3892285591……………. 2)                     (1-x^(2)ln(y+1)^(-2)ln(A)^(2))ln(A)=1-x^(2)y^(-2)      Here ‘ln’ is natural logarithm and ‘A’ is the fine structure constant that is .0072973525…………………. Substitute the latest co-data value of ‘A’ and watch ‘x ‘unfold as .0730730370………………….  . Note that the inverse of fine structure constant is 137.035999……………………..  . Here the under lined(in bold) digits are 073 in reverse order. If this appears to be casual then substitute inverse of muon electron mass ratio (1/206.768282…………).  ‘x’ now will turn out to be .0670370…………… . Note the repetition of underlined digits. This sequence seems to be a hybrid of sequences of fine structure constant and the muon electron mass ratio.  This hybridization is not expected to last beyond five or six significant digits as muon is identical (except for its mass) to an electron only upto this limit as revealed in comparison of their gyro magnetic ratio and thus the muon electron mass ratio can be considered a fundamental dimensionless ratio (though perhaps not as fundamental as the fine structure constant) of electron only in this limit. The pattern is expected since the fact that fine structure constant is unit less means that its value can be predicted mathematically, however it is also at the same time a physical constant(after all its place of origin is physics) which means that it could be determined experimentally. The conflict can be settled if we say that its more accurate value can be deduced mathematically from less accurate value known experimentally which explains the rational periodic character of x. The experiment could be a mathematical trial and error to know the first few digits of 'x' which can be then be continued indefinitely in accordance with the pattern, then solve for fine structure constant in reverse up to unlimited places of decimal.

To conclude then the coincidences are too much to be ignored as plain coincidence. It is proportionate enough to convert the fiction into a fact. To calculate ‘A’ then take x=((73073037037)/(10^(12)-1 ))NOTE:THIS PAGE IS UNDER CONSTRUCTION Bold text. FOR: THE PENULTIMATE EVIDENCE IN FAVOUR OF THE FINE STRUCTURE CONSTANT FORMULA, SEE BELOW:

Here is the promised derivation.  FOR: ANOTHER PENULTIMATE EVIDENCE, SEE BELOW:A  POSSIBLE  MEMORABLE  BREAKTHROUGH:  Because this paragraph constitutes ,what could be a memorable breakthrough with regards to the fine structure constant formula, so I have decided to place it at the top of this paper. However the readers are supposed to go through the entire paper first, particularly the fine structure constant part, before he or she could appreciate the full significance of this paragraph. From the whole complex of equations and numbers we abstract the periodic trend and concertize,generalize it in two numbers r0=73073037037/(10^(12)-1) and r0^2-r2^2=38388181/(10^(12)-10^(4))……..(8). Instead of treating A=α and e the base of natural logarithm as fundamental we treat these two numbers as fundamental. we substitute x for all A and y for all e( in base of all logarithms and external to it also where ever it(e) occurs) in equation (7) and (8)and then solve for x and y. The y is extremely close and slightly different from natural e. This creates a system of mathematical equations and numbers which are internally consistent even though it may overtly appear to be physically, at dis-par with the hypothesis of α as a generalization of natural(that is base of natural logarithm) e. But that is not a point of concern here, since e is a mathematical not a physical constant unlike A=α and so mathematical that is internal consistency alone define it completely. So too should be the case with y. The right most ln(A) in left hand side of equation (7) is now replaced by (ln(A)+ (r/A)(dA/dr)) as was originally the case. Earlier the second term(product) in the bracket was taken to be zero since at finite r corresponding to r’=.3892285591……. , dA/dr was taken to be equal to zero asymptotically. Now however we want to retain this term since we want to compute A=α at arithmetic infinity and find a match for latest CODATA value of fine structure constant there since the state of art conventional QED calculations, I suppose assumes a flat space time background where infinity means just arithmetic infinity which is synonymous with physical infinity there. This product is however undefined at r=infinity since dA/dr is again zero there. This is the way out of the impasse: Formally speaking r(dA/dr)=ΔA ,is the increment to fine structure constant. So we may write the product in the bracket as ΔA/A. We next substitute A=e as this is the case at infinity to get Δe/e since A and e run into each other from opposite directions. Now dA/dr=-de/dr is expected as a consequence of the fact that A is generalization of e. The extra minus sign is indicative of the fact that A and e run into each other from opposite directions. Thus for the product we have -ΔA/e. Thus the product is zero at physical infinity and -ΔA/e at arithmetic infinity. Thus now it is defined at these two places the only place where the theory is designed to be defined, as α is asymptotically convergent and so fundamental only at these two places. At intermediate points, it and the product term exists only as a mathematical remnant. Through two points only a straight line is uniquely defined.That product is therefore definitely linear. Hence for the product the second term in the bracket we may take the arithmetic mean of 0 and –ΔA/e which is –ΔA/2e. we now solve equation (7) ,modified as stated for ΔA With y=2.718281828461898908259130790560522343143 replacing all the e(that is base of all logarithms) in that equation, together with usual values of A=.007297352541549875…… and r0=73073037037/(10^(12)-1) in that equation. The result is ΔA= 0.00000000002808153198871023374609675.We add this increment to usual value of A= 0.007297352541549875966923989076535512184536 at physical infinity to have a incredibly precise match for latest codata value(at arithmetic infinity) that is A'=A+ΔA=0.0072973525696314079556342228226322....... A miracle indeed. Thus, Fine structure constant ,the incomprehensible number has suddenly begun to look like comprehensible.....(to be continued). A GENERAL NOTE ON THE FINE STRUCTURE CONSTANT:Here and below ,some though not all of the amazing numerological coincidences( if one tries to put it that way) would simply disappear if one switches to a number system with a base other than 10. This prompts us to think whether base 10 number systems are in any way fundamental. Mathematicians know that if one allows for the bases to be non-integer as well then e (the base of natural logarithm) is a fundamental base(within the limits of current advances in mathematics) in the sense that mathematics then becomes simple, elegant and even economical(technically speaking in the sense of radix economy). Now the origin of all counting in algebra and arithmetic are integers. Fractions and decimal numbers comes only later on as a late development. So integers are self contained and fundamental and by analogy with e, it could be that number system with integer base 10 is some how fundamental (but only physically and not mathematically, that is the data corresponding to value of any of the fundamental dimensionless constant is skewed such that the symmetry inherent in those data are more discernible in base 10 number system than others. Purely mathematically however base 10 is as good as any other base) in the sense that the fine structure constant is more revealing in the base 10 number systems than others and that it is quieter elsewhere. Philosophically and scientifically the fine structure constant is a mathematical number born physically rather than the converse. That’s a materialist point of view. Hence it is a connection between the material world and its offshoot the consciousness(at least the mathematical consciousness). Existence of this conditional ,absolute base 10 number system(mathematical language)is symptomatic of this material objectivity of consciousness where as at the same time its wider equivalence(as is borne out of previous human experience) with other number system is symptomatic of its ideal subjectivity and of its relative (though not absolute)independence from matter.(to be continued...........)

ABSTRACT TO, “THE TWO CENTER THEORY OF RADIATION RESISTANCE(REACTION)”:(theory of electron)

The electron, as everybody knows is an enigmatic particle and so is presumably every elementary particle of which it is a prototype. One aspect of this enigma is that the physical origin of radiation reaction force for a single electron is not known to the scientific community yet. In fact this is an unsolved problem of physics and is a place where classical electrodynamics fails which is widely believed and rightly so to be related to its failure to comprehend point charges in correct mathematico logical terms. This paper seeks to explain the physical origin of that force based on a simple postulate that the electron is neither solely a point particle nor solely a extended object, rather ( like other dualistic hypothesis in physics) it is both, what  can preferably be called an extended point. Even though there are several logical justification in the theory for such a hypothesis, it can also be worked out by the process of elimination. There is no satisfactory theory based on electron as a point particle or as an extended object ,hence it is surprising that nobody tried the idea of electron as an extended point throughout such a large chunk of socio-scientific space(whole world) and time(more than hundred years now for the puzzle ). The extended point idea for the electron absorbs the singularity which is so reminiscent of the electron as the electron radius is made to tend to zero. In this theory there is no self force on the model electron yet there is the radiation reaction force a fact that has puzzled eminent physicists of the stature of even R.P. Feynman. How? By stretching and/or squeezing space time interval in the vicinity of electron by the conspiring space time between the observer and the electron in accordance with the space-time equations that follow from the extended point hypothesis, resulting in production of an excess over the Newtonian expression for net external force (F=ma) ,the excess that is identified as the radiation reaction force in the theory.

To lend more credibility to the theory equations modeling extended point electron(extended point model of electron) has been derived, very simplistically and independently from Maxwell’s equations.

Also in the end as an experimental confirmation to the theory a very simple and elegant derivation for the value of fine structure constant at zero energy has been worked out completely within the framework of the theory. This derivation is scientific in essence and is completely opposed to the numerous numerological, mystical and pseudo scientific explanations of the value for the fine structure constant that exists in the market. Finally In solving the electron problem the door to the solution of other elementary particles might have been opened. This could be a fair beginning and a revolutionary one too.

When an electron or for that matter any electrically charged particle accelerates, it radiates electromagnetic waves. The power radiated & transported to infinity is given by the Larmor’s formula which is experimentally verified & theoretically deduced from the laws of classical electrodynamics in a logical manner. This suggests that somehow an extra force acts on the electron opposite to the external force, the work done by which is negative and is radiated as energy in much but loosely the same way as heat is generated by friction (due to negative work done by the force of friction) when a block is moved over a rough surface. This also implies that the force needed to accelerate a charged particle is greater than to accelerate a neutral particle of the same mass and for the same acceleration. From Larmor’s formula and energy conservation consideration the expression for the radiation resistance F,rad can be derived and is already known to the scientific community. But it is one thing to deduce the expression for this force from energy consideration and quite another to explain its physical origin. Many attempts have been made by physicists all over the world which includes physicists of such repute as H.A.Lorentz, P.A.M. Dirac, R.P. Feynman, J. Wheeler etc. but none is complete(and therefore incorrect in totality) because all the related contradiction are not resolved till date. This aspect is the only one where, classical electrodynamics fails and is related to its failure to comprehend point charges in mathematico logical terms.

In this paper it is shown how the electron behaves simultaneously like a point as well as an extended object, what I did prefer to call an extended point. Such a view point leads to the resolution of all the fundamental contradiction related to the previous understanding of the electron and to the present one also. Everything comes out of it with correct expressions and numerical coefficients, like the radiation resistance force and the Newtonian reaction to it. This paper is in three parts. The first part deals with the origin of radiation resistance, the second deals with the derivation of the key equation postulated in the first part from Maxwell’s equation and the third part discusses the fine structure constant as every logical theory of electron should.

But most interestingly the extended point idea of the electron leads automatically to the idea of two centers for the electron. This in itself leads to almost a magical solution to the problem even though the idea is quite contrary to the commonsense. This paper thus gets its name.

NOTE-The symbol meanings used in the paper are provided at the end.

The two center theory of radiation resistance

Part-1.

'''The origin of radiation resistance (reaction) ' The electron is a tricky particle and if we are to comprehend it we must act trickier. ''

We begin with the Einstein’s velocity addition formula, which in a sense is the end point of the Special theory of relativity as also the apparent end point and at the same time a fresh beginning of human cognition since the same can not end abruptly.

So what’s the mathematical point as regards to the velocity addition formula? Let us inspect that formula. U*V=(U+V)/(1+UV/C^2) where ‘*’ can be thought of as some kind of generalized addition since in the special case of ‘U’ and ‘V’ much less than ‘C’ which is physically finite or ‘C’ shooting to infinity in imagination only, we have U*V=U+V. We now have an interesting fact which is almost trivial to check out that, U*C=C*V=C which is just the property of the number infinity. Thus we find that the arithmetic infinity has been modeled by a finite number which is here ‘C’ with respect to the mathematical operation ‘*’. The infinity has been finitized. The plan of science is abstraction and generalization and we too would be indulging in these two processes in what follows. But before we do that let me add that the idea of restricted or limited infinity apparent in the Einstein’s velocity addition formula is not exclusive to this formula only rather there is infinity of other formulas sharing this property for instance the millennium formula- U*V= (U^2+V^2-U^2V^2/C^2)^1/2 as also U*V=U+V-UV/C. However only the Einstein’s velocity addition formula is uniquely preferred by nature whose reasons are not going to bother us at the moment. In a more mathematical language, only Einstein’s formula is allowed by Lorentz transformation equations while all the remaining infinity of A.B. initio formulas are not compatible with those transformation equations. Hence forth whenever we use ‘*’ we understand that we have the Einstein’s formula in mind unless mentioned otherwise. Let us now strip the Einstein’s velocity addition formula of all physical meaning. Then it’s a true modeling of the number which is arithmetic infinity by a finite number. It is a pure mathematical act now. To consolidate the perspective we have taken let us now switch to different symbols and physical meaning as regards to that formula- r1*r2=(r1+r2)/(1+r1r2/r0^2) the generalized sum and r1#r2=(r1-r2)/(1-r1r2/r0^2) the generalized difference. It can be easily checked out that both these formulas model the number infinity by a finite number that is ‘r0’ and that the latter follows automatically from the former by replacing ‘r2’ by ‘–r2’. Further as regards to the summation formula we think of ‘r1’and ‘r2’ as two consecutive segments in the same straight line from a fixed point ‘o’’ in our spatial universe and ‘r0’ as some kind of analogue of what we can loosely call the radius of the ideal and observable universe with constant and uniform density of matter, energy and momentum. We now know from general relativity that such a universe is actually a spherical(constant positive curvature) or a pseudo spherical(constant negative curvature) manifold (depending upon the mean density of matter in the universe) of dimension three disregarding time and dimension four including time. It is also known that such a curved manifold is actually equivalent to a flat space with properly shrinking or stretching space and time scales; at all points except infinity. For a spherical manifold of constant positive curvature it is given by dr’=dr/(1-cr’^2)=dr/(1-r’^2/r0^2) where the last equality easily follows from simple dimensional considerations. The sign of ‘c’ depending on whether the manifold is spherical or pseudo spherical. In this formula ‘dr’ refers to the proper differential length at a distance of ‘r’’ from ‘o’’measured by the own personal scale of ‘P’ and ‘dr’’ the length measurement of the same from the stationary observer at ‘o’’ with his own personal scales.

This is shear mathematical diplomacy. We have managed to satisfy all camps. Those who believe that the universe is finite and also those who believe that it is infinite. But diplomacy aside there is a deeper truth hidden in these acts. That the boundary conditions of the universe can not be given it has to be assumed and the assumption needs to be rationalized which means that both possibility must be included since there is no reason to prefer one over another or both must be excluded which is a trivial nonsense. To develop our simple mathematical machinery further let us see if we can model the number zero by a finite number, zero being the other extreme of cognition. As in quantum mechanics at these extremes the subjective boundary conditions fuse with the objective reality in the very process of cognition only to later assert as purely an objective law of nature. I am here referring to Heisenberg’s uncertainty relations. In quantum mechanics which is an objective probabilistic theory the absolutely possible as also the absolutely impossible events are the extremes of subjective cognition of the objective world viewed probabilistically.

So how do we model the number zero by a finite number? In modeling infinity the constant ‘r0’ could in pure mathematical thought be any number very small or very large that is even quasi-infinity. If it is infinity then it portrays a flat Euclidean world. In Einstein’s velocity addition formula the analogue of ‘r0’ that is ‘C’ is the ultimate speed, another property of infinity, and then it must be thought of as a quasi-infinite number. So must be the case with ‘r0’. To convert it to a quasi-zero then it must be reciprocated and then for dimensional consistency r’1r’2 and correspondingly r’^2 must be reciprocated too. Thus the relevant formulas are r’1##r’2=(r’1+r’2)/(1+r0^2/r’1r’2) which modifies to r’1##-r’2= r’1#r’2=(r’1-r’2)/(1-r0^2/r’1r’2) and in differential form to dr’=dr/(1-r0^2/r’^2) by letting r’1 tend to r’2 infinitely closely. It can be easily checked out that in these formulas r0 models the number zero with respect to the operation of ‘##’ or ‘#’ since r’1##r0=r’1 and r0##r’2=r’2 and similarly for ‘#’. In analogy with the Einstein’s relativistic formula for the relative velocity we have r’1, r’2 and r’ the distance of dr that is point ‘P’ from ‘o’’ as measured by the permanently stationary observer at ‘o’’. Here r’1 and r’2 differ by infinitsimallysmall amount and thus r’1~r’2~r’. The sign of approximation may readily be replaced by the sign of equality if one so wishes.

The Larmor’s formula for radiated power is given by P= 2/3 (e^2a’^2)/c^3 .From energy consideration the radiation resistance force is then given by F,rad=2/3(e^2a`)/c^3. However it is one thing to calculate the radiation resistance force by energy considerations and another to know the physical origin of the same. The Abraham Lorentz formula for the radiation resistance force (also called the self force since it was derived on the assumption that the self fields are exerting force on the model electron) is as follows: F,self=αe^2x``/(ac^2)+2/3e^2x```/c^3 +βae^2x````/c^4 +……. . Here the α of the first term also interpreted as the inertial term depends on the assumed charge distribution so too is β, γ etc. of third, fourth terms etc. The terms beyond the second are called the structure terms. While the second term is the radiation resistance term and which does not depend on the assumed structure of the particle. Now this precisely is the feature of a point that is structurelessness but no sooner do we realize this that we put a=0(this particular ‘a’ stands for the classical electron radius) to emphasize that point, we soon find that the term beyond the second all vanish while the first term soars to infinity implying that it is an extended object. This purely theoretical contradiction is backed up by experimental results also. Till date experiments suggests that electron is a point particle without any structure while the theoretical assumption of point particle leads to infinite self energy problem. In quantum electrodynamics the problem of infinite self energy is handled with the process known as renormalization which is considered adhoc and dubious even by its originators even though it gives excellent agreement with experiments. Let us then claim that the electron is at once a point as well as an extended object, it is what I call an extended point. This duality is not an isolated example in physics; in quantum physics too we come face to face with the wave- particle duality for matter as well as radiation. Thus nature has hinted us of the direction of the truth as regards to radiation resistance. But how did she do it? Actually the hint that is the problem and the solution is manmade through Lorentz and we have been really late in catching up with it. More than hundred years now! The means for calculating the force in macroscopic situation has been extended to evaluate the force on an indivisible particle like the electron which in a sense is both right and wrong, right because the electron is an extended object and wrong because it is at the same time a point object without any internal structure and without any sign of divisibility. Our task then is to set the wrong part also right.

R.P. Feynman talks about throwing away the first term by denying self interaction however he then asserts that then the second term is also automatically thrown away and it is like throwing away the baby with the bathwater. That precisely is the idea! Why don’t we throw away the baby and the bath water simultaneously and then recover the baby alone later on. It will be shown later that how this can be done.

At the macroscopic level the force on a discrete or differential charge element is calculated by the field due to all the remaining charges of the system, the self force is never considered and it gives rise to results that are in full agreement with the relevant experiments. How can then the self fields be supposed to be acting on an indivisible point particle which is only a special case of macroscopic distribution of electric charges. Let us then for the moment deny the idea of self interaction and consequent self force for an accelerating electron which arises due to finite speed of signal propagation and the resultant time delays involved due to which the force from the back to the front is lesser than the force from the front to the back.

When an electron accelerates then the Newtonian ‘ma’  is always there  wheather or not(may be not, for some exotic and as yet unknown reason) an external force acts on it. The problem is to construct the external as well as the extra radiation resistance force out of it. How can this be done? A little thought will make anybody guess that this can be done by stretching and / or squeezing space time interval in an appropriate manner. Now since our model electron is an extended point we can use r1##r2= (r1+r2)/ (1+r0^2/r1r2) or else r1#r2= (r1-r2)/(1-r0^2/r’^2) or the mathematically more useful differential expression dr’=dr/(1-r0^2/r’^2) to model the space around the electron. I take this as the first postulate. As a result we shall find that time also gets modeled in a certain manner when the constraints of special relativity are applied on to it.

In the last expression ‘dr’ refers to the local or the proper or the ‘on the site’ space differential element measured by an observer stationed there itself that is at point P while ‘dr’’ refers to the measurement of the same by an fixed and stationary observer at ‘o’’ and ‘r’’ is the distance of ‘P’ from ‘o’’ as observed by the observer at ‘o’’. ‘dr’’ is called the generalized space differential element. Now since the speed of light must be a constant for all observers and reference frames we have the following relations dr=cdt, dr’=cdt’, r=ct, r’=ct’ and also r0=ct0 where t0 is some time constant corresponding to the space constant r0=ct0.

Integrating by separation of variable the expression dr’=dr/ (1-r0^2/r’^2) we have r’+r0^2/r’-r0=r where we have used the boundary condition that r’=r0 implies r=r0. This plausible boundary condition is our second postulate. It is plausible like the first postulate since r0 models zero which must be formally identical in ‘r’ as well as the ‘r’’ coordinates.

We are now in a position to moot the idea of two centers for the electron. One given by r’=0 and another given by r=0. The later splits into two centers as r’=r0 (1+3^1/2i)/2 and another is its complex conjugate. We can choose either of them to be dynamically active centers in a sense that will become clear shortly. This is because no extra information is carried by the either of the two complex roots that is not carried by the other. In fact knowing one, one automatically knows the other. So any one of the two can be taken to be dynamically active and the other as redundant. Even the entire quadratic with real coefficients can be constructed out of the knowledge of one of the two complex roots only.

The magnitude of these complex roots is also ‘r0’. This means that in the real world projection of the electron it has two active centers. One, that is r’=0 (that is ‘o’’) defines the conventional center itself and relative to this the other r’=r0 (that is ‘o’) is on the surface of the spherical electron as we assume. Thus the patterns of field lines outside and inside the electrons are different. The external field lines seem to meet at r’=0 where as the internal field lines actually intersect at r’=r0.

Since imaginary numbers have cropped up in our problem a question naturally arises as to how real are imaginary numbers, the answer to which is that it is as real as the real numbers since historically all numbers were invented to allow certain mathematical operations to be performed which could not be performed with the previously known variety of numbers. Imaginary numbers too arose under similar circumstances. However despite this fact there are no positions available on the real number line to represent it geometrically and in this sense it is truly imaginary. Leibniz has put it well even though idealistically, that, “Imaginary numbers are a wonderful flight of God’s spirit; it is so to say almost an amphibian between being and non-being.” In a more scientific language it can be said that it is a connection between what is known and what is as yet unknown but knowable in due course.

Generalizing our equation r’+r0^2/r’-r0=r so that r’ and consequently r can take on complex values as well we find that the resulting equation is analytic since it satisfies the Cauchy-Riemann condition for analyticity. Also the relevant derivatives are continuous. This has the consequence that dz/dz’ exists meaning that it has the same value no matter from which direction ‘z’’ approaches ‘z0’’ in the complex ‘z’’ plane. Now let the imaginary part of these complex number approach zero than in the limit dr/dr’ equals dz/dz’ so that analyticity now means that dr/dr’ has the same value no matter in which direction ‘dr’ or equivalently ‘dr’’ is oriented. This is valid along any infinitesimally thin strip along the radial direction whose length represents the real direction and the thin transverse length the imaginary direction.

 Velocity and acceleration transformation formulas:

We already have dr’=dr/ (1-r0^2/r’^2). The corresponding time transformation formula can be easily found from the constraint of special relativity. It is dt’=dt/ (1-t0^2/t’^2).combining these and the special relativity constraint of the constancy of the velocity of light we find that v’=v and a’=a (1-r0^2/r’^2) as velocity and acceleration transformation formulas respectively.

Derivation of the expression for radiation resistance and its reaction from energy considerations:

It can be shown that the electric field inside as well as out side the electron satisfies the Poisson’s equation the fundamental equation of electrostatics despite the weirdness of the electric field inside. No exception to the law summarized by this equation has been found, even to the tiniest distance probed till date.

Since we are dealing with point even though it is an extended point it is obvious that the charge should be uniformly distributed inside the electron since a non- uniform distribution can be used to distinguish one part of the electron from the other and this is not possible since a point electron has no parts. The other possibility is that there is no charge in the interior and that all the charges are on the surface that is in the form of the shell. Now this is impossible since then there would be no electric field in the interior apart from having no charge and this would be the trivial case. Also if there is no electric field then there is no energy and mass equivalent and consequently no ‘ma’ and thus no F, rad as some kind of surplus over ‘ma’ as we expect it to be. It should be noted now that the center r’=r0 is skewed to the periphery relative to the center r’=0 only and that the charge distribution is normal hence the electric field is given by E=ρR/3€0 instead of E=ρr’/3€0 where ‘r’’ is equal to the distance from r’=0 to ‘P’ and ‘R’ is equal to the distance from r’=r0 to the point P. Such an electric field does satisfy the Poisson’s equation in the interior of the electron. The external electric field is given by E=q/4П€0r’^2 and this too satisfies the Poisson’s equation as is already known.

Let the electron be given an acceleration of “a”. By this is meant that the center rather the two centers of the electron has this instantaneous acceleration. Let ar’,t’ refer to the local acceleration of any point in the interior of the electron at a point ‘r’’ and at a time t’=r’/c in ‘o’’ frame of reference which is instantaneously at rest and also not accelerating. Also we assume that the electron is instantaneously at rest in this frame but accelerating so that we are actually dealing with the non relativistic case. Then a0,0=ar0,0=a since both the centers given by r’=0 and r’=r0 are of equal physical status, there being no reason to prefer one over the other. Thus by means of Maclaurin’s infinite series expansion we have ar0,t0=ar0,0+t0a`r0,0+t0^2/2!a``r0,0+t0^3/3!a```r0,0+…….. . ‘a’ here stands for the instantaneous acceleration of the two centers. Since ‘t0’ is tiny the terms beyond the second may be neglected if the multiplied time gradients are not too large.

Now ar’,t’ can be thought of as an event and if 0<t<t0 then it ought to explain ar0,t0 and at the same time be explained by ar0,0. that is the present must be explained as a consequence of the past where as the future must be explained as a consequence of the present. Again a mechanical impulse moving from present to future is equivalent to negative of the same impulse moving from the future to the past. We thus have ar’,t’=a0,0-ar0,t0=a-(a+t0a`+t0^2a``/2+….  )=-t0a`-t0^2/2a``+….. = s (say)..........(A1)

The fields which were invented or rather discovered by Faraday to explain action at a distance has its own independent existence independent of the charge that generates it. It can harbor energy, momentum, angular momentum etc, in short every thing that a material particle can carry. It thus has a life of its own. That’s the conclusion from the field theory. The acceleration impulses thus are like mechanical impulses traveling over a fixed stationary field background spanning the entire space, at the speed of light. If this were not rigorously correct then also it would be correct as a first approximation.

Now let ar’,t’ be measured as a’ by ‘o’’. Then as shown earlier a’=ar’,t’(1-r0^2/r’^2). We take as our relevant differential volume element dv’=(2Пr’sinө)(r’dө)(dr’) ,du=1/2(€0E^2)dv’ where E=ρR/3€0, also since E=mc^2(here ‘E’ stands for energy equivalent of the mass and ‘du’ stands for differential energy) therefore we have dF’=dma’=dmar’,t’(1-r0^2/r’^2) thus now F’=double integral of (a’dm) between the limits of ‘r’’ as ‘0’ to ‘r0’ and of ‘ө’ as ‘0’ to ‘П’. while integrating we write ‘a’’ as ar’,t’+(a’-ar’,t’) that is a base integral plus a surplus integral. Also we use elementary trigonometry and thus R^2=r’^2+r0^2-2r0r’cosө. Here ‘ө’ is the angle between the rays ‘r’’ and ‘r0’ emanating from the center r’=0. The base integral evaluates to 4/15(e^2s/c^2r0)~ -4/15(e^2a`/c^3)=F,ext. at the same time the surplus evaluates to -2/3(e^2s/c^2r0)~2/3(e^2a`/c^3)=F,rad, ignoring the square and higher order terms in Maclaurin’s infinite series expansion. Thus the radiation reaction force has come from some kind of stretching and squeezing of the space time intervals due to change of scales.

Now for the physical origin and interpretation of the same we proceed as follows. Let ‘dr’ at ‘P’ be observed as ‘dr’’’ by ‘o1’ where ‘o1’ is some point between ‘o’ and ‘o’’. Then dr’=dr/(1-r0^2/r’^2) and dr’=dr’’/(1-r0^2/r’’^2) where r’’ is the distance of ‘o1’ as observed by ‘o’’, thus we have dr/(1-r0^2/r’^2)=dr’’(1-r0^2/r’’^2). Now if ‘r’’’ >> ‘r0’we have r0^2/r’’^2~0 and hence dr’’=dr (1-r0^2/r’^2) that is dr’’=dr’. Thus ‘dr’ is viewed by a faraway external observer ‘o’’’ (‘o1’ externalized) in the same way as by the internal absolute observer ‘o’’. By faraway observer is meant a theoretically infinite distance but practically a very large distance compared with r0 from the electron say from a distance of 100000r0 or so, such that the error then would be tiny and could be neglected.

So has the nature conspired by self dilating and contracting space time differential interval to produce the radiation reaction force? Well to this I would say that it doesn’t matter so long as the conspiracy cancels itself out. This will be clear shortly to the readers. The cancellation will be in accord with the Newton’s third law of motion.

For now, since all the physics mentioned above is the same for ‘o’’’ as for ‘o’’ as shown earlier, it implies that the radiation reaction force or the bootstrap force is due to stretching and squeezing of the conspiring space-time intervals between ‘o’’’ and the electron or more precisely between the measuring apparatus of ‘o’’’ and the electron.

If we plot a graph between ‘r’ and ‘r’’ with ‘r’ as the ordinate and ‘r’’ as the abscissa then we have almost a upward opening parabola since for r’=0,r0,α` respectively we have r =α`,r0,α` respectively. Thus for a given value of ‘r’(>r0) we have two real values of ‘r’’ say r’1 and r’2. The graph between ‘t’ and ‘t’’ would be similar but only scaled down by a factor of ‘c’,  since t=r/c and t’=r’/c. For the acceleration impulses traveling at the speed of light ‘c’, r1 corresponds to t1,r2 corresponds to t2 and r’1 and r’2 corresponds to t’1 and t’2 respectively on the corresponding time graphs. The minimum or the apex of this parabolic graph corresponds to r’=r0 and consequently r=r0 as well. Since the acceleration impulses are traveling at the speed of light and radially outwards even though the acceleration vector itself could have any direction, we the integrator are also assumed to be traveling with the same speed, always remaining by the side of the local acceleration in the fabric of the electron’s space-time. Thus when we are at r’1 at a time t’1 then in r coordinates we are at r1 which means that we are also at r’2 at that time potentially and which is actualized only at time t’=t’2 when we would also have visited that same point in the same time classically that is without any tunneling effect through ‘r’ coordinates which in the above mentioned sense seems to be shorting the universe of ‘r’’, coordinates. Thus to calculate the Newtonian reaction to the radiation reaction action the relevant integral which would be worked out shortly has to be integrated twice within the limits from r’=r0to r’=ά

The relevant integral and the reaction to the radiation resistance is evaluated as follows. We take as our relevant volume integral dv’=4Пr’^3dr’ (where r’>r0) which is equal to the volume of a thin spherical shell of radius ‘r’’ and thickness ‘dr’’. Also du=1/2(€oE^2 dv’),E=q/(4П€or^2) ,dm=du/c^2. and dFr as the differential force which is a reaction to the radiation reaction force implies that dFr=dma’. Integrating this expression within the limits of r’=r0 to r’=ά we have Fr=2/3(e^2s/c^2r0)~(-2/3e^2a`/c^3), ignoring the higher order terms due to smallness of ‘t0’. Thus the conspiracy at last has cancelled out there by validating Newton’s third law and in a way that’s truly fantastic of nature.

''Notes:1)since the self force is zero on the model electron so the usual Lorentz force calculated is zero and so do not make any contributions to radiation reaction force calculated in this theory.2)Poincare stresses are not needed to stabilize the electron as there is no self force. so there is no correction to mass and hence the radiation reaction force as calculated in this theory.In any case non electrical contributions are immaterial in a completely electromagnetic theory of electron like this one.''

Part-2

Derivation of the integral of the 'Two center theory' equation from Maxwell’s equation

Electrostatics is summed up in the Poisson’s equation del^2(ф)=(Ρ/€0) where ф= electrostatic potential, Ρ= local charge density and €0=electric permittivity of free space. We wish to apply this equation in the space outside the electron, derive the extended point law there and extrapolate it to generalize the law to the entire space whether inside or outside the material electron. One should note at this point that the Poisson’s equation is something like a quasi absolute truth if not an absolute truth since any other alternative to describe electrostatics is not only inconsistent experimentally but also theoretically and internally. Outside the electron Ρ=0 so Poisson’s equation becomes the Laplace’s equation del^2(ф)=0 which in spherical coordinates (r’) becomes -     (1/r’)d^2(r’ф)/dr’^2   =0. Consider now the identity (1/r’)d^2(r’ф’)/dr’^2=(1/r’)d^2(r’ф)/dr’^2=0………..(A) where ф’ is the potential in r’ coordinates and ф is the potential in r coordinates which are different mathematically but same physically. One should note that we have impregnated the above identity with the possibility that a point in space outside the electron has two coordinates r’ and r. let us now see as to what the equation returns. We know that the electric field outside the electron at large distance from the electron is radial and it is given by E=-dф’/dr’ or by E=-dф/dr so too are the radial coordinates r’ and r, we also know that on the surfaces r’=constant or r=constant implies E= constant. Since we are tracking things physically so we let E=c1=constant in the above two equations. We then have c1=-dф’/dr’=-dф/dr. solving these that is by integrating these we have ф’=-c1r’+c2 and ф=-c1r+c3, substituting these in equation (A) and solving the resulting differential equation we have r’^2-rr’=c3r’+c4 where c3 and c4 are constants of integration. We now use r=r0 implies r’=r0 and vice versa in this equation. We then have c4=-c3r0^2, substituting this into the same equation and applying the condition for uniqueness or single valuedness of r’ at r=r0 we equate the discriminant of this quadratic equation in r’ with r=r0 to zero we have c3=r0 and thus also c4=-c3r0=-r0^2. Thus outcomes the key equation: r’+r0^2/r’-r0=r.

Part-3

The fine structure constant as a generalization of the number e, the base of Natural or Naperian logarithm No theory of electron is complete without a word or two about the fine structure constant.Here, there is more on offer than just that. .	In part one of the two center theory we introduced an equation connecting the local and the general space differential as dr’=dr/(1-r0^2/r’^2) where dr=r’1-r’2 and consequently the label r in dr was rather  improper. It should have been dr’ instead but then it would mix up with the other dr’ in the left hand side of the equation. The way out is to define a generalized differential Dr’ in the same variable r’ but in a different operator D in place of d and replace the ordinary looking differential dr’ in the left hand side of the equation. Thus we have Dr’=dr’/(1-r0^2/r’^2) similarly we can define Dr=dr/(1-r0^2/r^2). Now we can take the ratio of the two equations to form what I call the generalized deriviative as follows- Dr’/ Dr  =(r’^2/r^2)((r^2-r0^2)/(r’^2-r0^2))(dr’/dr) …………(1). thus connecting the generalized deriviative Dr’/Dr with the ordinary Newtonian deriviative dr’/dr. This deriviative Dr’/Dr is the hall mark of  the mathematics  of the two center theory just as dr’/dr is the hall mark of the ordinary calculus. Now we make an interesting observation that e^x is the only function in all of mathematics whose deriviative is the function itself. Consequently the  number e sums up the peculiarities of the ordinary Newtonian calculus. This observation permits us to calculate the value of fine structure constant α by defining it analogously. If we let dr’/dr =r’ then it’s solution is r’=ce^r where c is the constant of integration and e^x is the primitive integral. Similarly if we let Dr’/Dr =r’ then r’=c’α^r should be its solution where c’ is a constant and α is the fine structure constant(or may be a fine structure variable in which case we would like to get its value at infinity or equivalently at zero energies), a generalization of the number e, the base of Naperian or natural logarithm.

The number α sums  up the peculiarities of the two center theory(actually its mathematics) in the same manner as e does the same for the ordinary Newtonian calculus. Since we do not know in the beginning wheather α is a constant or a variable the safe way would be to think of it as a variable as a constant is a special case of a variable. If r’=c’α^r then  dr’/dr=r’( lnα +(r/α)(dα/dr)) ………(2) Now we have a observation that if r0=0 or if r’=α’,r=α’ in Dr’/Dr =r’ we then have dr’/dr =r’. that under these circumstances the generalized deriviative reduces to the ordinary deriviative with the solution r’= ce^r instead of r’=c’α^r. this is expected as this logic is inbuilt into the equations right from the beginning. Thus in these limits α collapses to e. Also in these equations if r=r0 then so must be r’=r0 so as to avoid infinite divergences in equation -1. In these limits the value of c’ can be calculated in the following manner. If r=r0 then r’=r0 so we have r0=c’α^r0 now letting r0 tends to zero so as to attain boundary condition, for finite α we have c’=0 as in these limit α=e which is finite. So the characteristic function r’=c’α^r collapses to r’=0 resulting in its de characterizing in terms of α. The likely physical reason for this is the singularity at the origin in the classical electron theory at r’=r=r0=0. To avoid this singularity we try shifting the origin to r=c1 and r’=c2. Then we have r’-c2=c’α^(r-c1). if we now substitute r’=r=r0=0 we have -c2=c’α^-c1 so that we have r’-c2=-c2α^r, if we now set r’=r0=r tends to zero as well as simultaneously α tends to e then carrying the proper limits we have c2=-1,( as lim (e^x-1)/x with x tending to 0 equals 1). We thus have r’+1=α^r or r=ln(r’+1)/ln(α)………(3). Before proceeding further we have to overcome two problems-  the first problem is that if the electron is considered a rigid sphere in one frame (moving or at rest) then from special relativity it would become an ellipsoid in some other moving frame of reference. So the spherical symmetry of the two center theory and the extended point model(which is a function of a single parameter r0) will surely break down which is actually absolutely necessary for its survival. The only way out is to assume that the electron circumference in three dimensions is either exploding or imploding at the speed of light. Then once a sphere it will always be a sphere in every inertial frame of reference. Thus we must relativize our actual physical equation in the above mentioned sense from which generalized differentials were created. That equation was dr/dr’=1-r0^2/r’^2 from which we have r0=r’(1-dr/dr’)^(1/2)……….(4) Also we have dr0/dr=dr0/cdt=c/c=1 where the r0 has been assumed to be expanding at the speed of light c in the local coordinates. Differentiating both sides of equation 4 with respect to r’ and setting dr0/dr=1 in the left hand side we have -     dr/dr’=(1-dr/dr’)^(1/2)-(r’/2)(d(dr/dr’)/dr’)(1-dr/dr’)^(-1/2)………..(5) The second problem is that we want to find α at infinity but the very way we have set up our equations and definations it turns out to be e which is different from the experimental value of α (at infinity that is r’=α’ and also r=α’ from equation 3 for finite α). The contradiction can only be resolved by claiming that the theoretical  value of α=.007297352541549875……..as will be derived in this paper shortly is the value at physical infinity that is at the boundary of observable universe or the likes but not at mathematical or arithmetic infinity which exist only in imagination as a mathematical abstraction. The challenge thus now is to find the finite value of r’ that models infinity that is r’=α’. For this purpose the trick is to tag α with e in accordance with the logic and symmetry that at infinity α=e. substituting this in equation 3 we have r=ln(r’+1). Substituting this again in equation 5 and solving the resulting cubic 4r’^3+12r’^2+5r’-4=0 we have r’=.3892285591……. in some properly rationalized system of units.( there are two other roots which seems to be physically irrelevant). This point then should represent the physical infinity. We now eliminate dr’/dr from equations (1)and(2) and solve for Dr’/Dr we then write the equation Dr’/Dr =r’………..(6) we then eliminate r from equation (3) and (6) to get an equation connecting r’,r0,and α. In this equation we substitute dα/dr =0 together with the finite value for r=ln(r’+1)/lnα in the substitute expression for dr/dr'. Thus this term in dα/dr vanishes. The fact that dα/dr has been equated to zero at that finite point is logical since it defines the finite point r’=.3892285591…….. modeling infinity and it is obviously zero at the arithmetic infinity. We then substitute the derived value of r’=.3892285591………… and the latest codata experimental value of α=.0072973525698 to get a periodic number for r0=.0730730370………..,the zeroes representing the turning point between anticlockwise(positive) and clockwise (negative) on a circle with number 0,7,3 written in say anticlockwise order, thus those numbers are simultaneously positive and negative and hence can not be non zero. Thus the extended pattern for r0 seems to be (73073037037)/(10^(12)-1). Assuming this to be a true guess one can substitute this in the same equation and solve for α. This periodicity in my opinion is expected(there is no intention of being numerological here on my part ), since the fact that α is dimensionless means that its value can be predicted mathematically and the fact that it is at the same time a physical constant means that its value can be known experimentally. This contradiction can be resolved if we declare that its more accurate value can be generated from its less accurate value known experimentally. So r0 instead of α is expected to be a periodic rational number as it would be too trivial to expect α directly to be rational periodic. Thus we have the following formula for fine structure constant- (1-r0^(2)ln(r’+1)^(-2)ln(A)^(2))ln(A)=1-r0^(2)r’^(-2).......(7), where ln is the natural logarithm and A the fine structure constant.To calculate A then substitute r0=(73073037037)/(10^(12)-1) in the above formula. 'Here is an update on the Fine Structure Constant: Here I wish to explain the periodic number .0730730370 and .0670370. Since the FINE STRUCTURE CONSTANT is origin wise a physical constant therefore it is ultimately a statement of the physical world even though we are here seeking to derive a mathematical formula for it. This however does not undermine its physical nature. After all this formula can be derived as in this paper only by a proper comprehension of the physical laws. A purely mathematical deduction of the same without due regard to physical laws is in my opinion an utopia. That these numbers are rational periodic follows from its dual status of it being a physical(origin wise) as well as mathematical(unitlessness wise) constant as already briefed in the paper. There are however other reasons, for instance: the fine structure constant α has been shown to be a generalization of the number e the base of natural or Naperian logarithm. Now one of the most beautiful as well as so far mysterious relations in all of mathematics is the Euler’s identity e^(ix)=cos(x)+isin((x). It connects algebra ,something abstractedly mathematical to geometry,something concretely physical. The i=(-1)^1/2 is a periodic thing as a function of its exponent n and it is this periodicity in the ultimate analysis that is responsible for the periodicity of sine and cosine of geometry and perhaps all of mathematical world and again perhaps all of physical world since complex periodicities there can be broken into elementary sine and cosine by means of Fourier Analysis as is already known. Even the value of the geometric constant pi( π) can be worked out from the approximate version e^(ix)=1+ix for small x. see The Feynman Lectures On Physics, vol-1, chapter: Algebra. Thus periodicities are expected in α formally but in r0 actually since periodicities in α directly would mean that it value can be worked out to any degree of precession by merely extending that pattern from its initial seed value(first few digits) obtained experimentally and that would undermine the physical laws leading to its derivation as in this paper. In other words it would be a compromise with the physical status of that constant. Also periodicities of a (fundamental)constant number can only mean periodicities in its decimal expression since its value is not amenable to change(at least to first order at infinity that is equivalently at zero energies). Now y=e^(ix) is the equation of a unit circle. And the periodicities in the sign of i^n(--++--++….. as n=2,3,4……,for n=2 the defining quadratic i^2+1=0 is obtained) we take as fundamental as it is completely divorced from numeral values and can be associated with direction of travel of a unit circle, very naturally. A minimum of three numbers are needed on a unit circle or a closed loop for that matter to distinguish clock wise(negative) motion from anticlockwise(positive) motions. Now from where do we procure these three numbers? Well the first number can be taken as zero since it marks the turning points of motion and so is simultaneously positive and negative. As discussed earlier the value corresponding to  the ordered set of these numbers must match the  seed value(first few digits since their chances of stability is most in the event of α being a running coupling constant) of α(if the formula derived is for α^-1=137.035999…..) or α^-1(if the formula derived is for α=.0072973525….), since both are dimensionless and therefore both fundamental formally. Which is which can be established by mathematical hit and trial with the formula on a calculator or computer. We can in fact reverse this argument and claim that α^-1 should be a 3 digit large number(integer part only). One of these digits is 0 where as the other two happen to be 3,7. Physical experiments are not necessary to establish this as it can be done with a mathematical experiment( hit and trial ) to establish this relations between α and r0 using the formula relating these two as derived in the paper. Actually zero cannot occupy the first place in integer part of α^-1 as then it would become insignificant and a two digit number. Analogous schemes apply for muon- electron mass ration. But how do we know if we had worked out a formula for α or α^-1 if both are fundamental? Well it must be α=.0072973525….,as can be known by hit and trial mathematical experiment and not its inverse as we(actually nature through us) require some kind of relative or formal independence from physical laws leading to the derivation of α, in order to push through the mathematical point(that is the correlation between the first few digits of r0 and α^-1 that we talked about in the previously). Then α^-1 would be relatively independent of r0, the electron radius in some properly rationalized system of units. The contradiction of two fundamental constant can be trivially be resolved if we say that α=α^-1 that is α=1. Hence in the hit and trial mathematical experiment the first trial value should be α=1 and then upwards for α^-1 upto 138 in case of α and 207 in case of electron muon mass ratio. Only the integer values needs to be checked. Also therefore α or α^-1 should be as close to 1 as possible. One last point, an important one and a clinching one too: if from the above equations for fine structure constant we choose to eliminate r0 instead of r to get (ln(α)^(3)ln(r’+1)^(-2)-r’^(-2)=ln(α)r^(-2)-(α^(r)-1)^-2) and treat r as a free mathematical parameter as we did previously for r0 then then substitute α=.0072973525…… then for r we get .067000……,the code for muon electron mass ratio talked about in the introduction of this paper. The three zeroes after 067 perhaps signifying the vacancy to be filled in by 037 procured from the decimal expression for r0. This coincidence is fantastic and and takes the odds against the theory to minuscule level, less than 1 part per 10^20. I leave then upto the peoples(both lay man and specialists) to decide whether the search for a fine structure constant formula be continued or ended abruptly. NOTE THAT HERE AND BELOW, THE SYMBOLS (A) AND (α) ARE USED SYNONYMOUSLY FOR THE FINE STRUCTURE CONSTANT. THE PENULTIMATE  EVIDENCE

In this theory the fine structure constant α has been worked out as a generalization of the number e, the base of natural logarithm. Now every number is its own generalization too. So too must be the case with e. Therefore α=e the trivial solution should be included in the general solution to equation (7) that is along with α=.0072973525415498………….. . Indeed this cubic equation in lnα has two independent solution for lnα and hence also for α. However substituting α=e in this equation (7) leads to contradictory result r’=0(besides it can not be solved for r0 as it cancels out), while we know that actually r’=.3892285591……….. . From a purely mathematical point of view this is because the base of logarithms in that equation is also e which prevents α=e. so we choose a unknown base x for all the logarithms in that equation (7) and solve for it with α=e substituted in that equation  simultaneously( and with the usual values of r0=(73073037037/(10^(12)-1)) and r’=.3892285591.....). The result is x=2.6786728……….. . we next fix the base of all the logarithms in the equation to this constant value( with the usual values for r0 and r’) and solve for A(that is α), the fine structure constant proper but a dummy one( the true one being .0072973525.....). The result is α=.007337450477………. . Now when we changed the base of logarithms we are likely to have changed the physics. To restore the original physics we return to the original base e of the logarithms, substitute α=.007337450477………. and solve for r0( with the original value for r’). The result is r0=.073160670438……… .Now this is amazing( it is nothing less than a very real miracle): 073 is the code for fine structure constant α, 16, the code for tauon-muon mass ratio( there being no 0 here as its integer part is a two digit number so clock wise and anticlockwise movement can not be distinguished and hence there are no turning points implying that there are no numbers which are simultaneously positive and negative that is zero),067, the code for muon-electron mass ratio, 04 is a part of the code 0477 for tauon –electron mass ratio(3477). Later on we shall see that how in a more refined treatment r0=.073160670477……… .Even though this does not provide a direct check on the true value of fine structure constant, it does however check the usual values of r0 and r' as well as the process by which the fine structure constant is obtained from r0 and vice versa, that is the main formula (7)itself. Hence in this manner it indirectly checks the fine structure constant at zero energies. ( To be continued……….) In the mean time can any body of us imagine a unity that is more grand than this possibility. ANOTHER PENULTIMATE  EVIDENCE

Every method has its own advantages and disadvantages.The disadvantage with the previous one was that we had to change the base of logarithms and restore the same which was tantamount to changing the physics and then restoring the same. From another perspective the solution to the equation (7) was α=e and α=.007337450477………. but in the changed base and not α=.0072973525415498……… along with α=e. Though this is not erroneous within the limits of relevant approximation it may appear to be dissatisfying to many( how ever I consider this to be purely a subjective effect). For such persons and for others as well here is another equivalent alternative which is not an iota less thrilling, perhaps even more. Equation (7) malfunctions for α=e directly as pointed out earlier. The indirect alternative is following: Not all of the three roots of equation (7) are independent in α that is A. This is because the coefficient of (lnA)^2 in that cubic equation is zero. In fact the product of the roots is an unity. That is XYZ=1 where X,Y,Z, are the three roots that is the three possible values of A in equation (7). What we wish is that X=e,Y=α=.0072973525415498…….. and Z=X^(-1)Y^(-1)=e^(-1)α^(-1). Since X=e is a problem directly, there fore we take Y and Z as the independent roots which indirectly implies 	X=e but we avoid taking it head on. This is the trick. This way we manage to couple e the base of natural logarithm with α the fine structure constant at zero energies explicitly in equation (7), earlier it was only implicit. We next substitute A=Y in equation (7) and solve for r0^(2)=r1^(2)(say)with the usual value of r'=.3892285591.... and then substitute A=Z in the same equation and solve for r0^(2)=r2^(2)(say)again with r'=.3892285591..... We wish, though not expect (that is we are wishing the unexpected here and it is a wish that would tie up e and α in a perfect union without having to change the base of logarithms in equation (7)) that: r1^(2)-r2^(2)=0, (other forms are possible but this is the simplest equation obtained by eliminating the two r0( treating them superfluously to be the same for the moment) from those two equations obtained by the two substitutions in equation (7) and truth is often the simplest and the most beautiful  ). The calculated number in not zero as expected but it is a number that mirrors the number r0=(73073037037/(10^(12)-1)) which in turn models the number zero in this theory(that is it models the number zero indirectly), in the same kind of cyclical setting as the number 073 and 037 in the number r0. Only thing is that the corresponding numbers are 38 and 81 respectively perhaps procured from the values of r’=.3892285591..... and e=2.718281828...... respectively. One can check on the calculator or computer that in fact r1^(2)-r2^(2)=.000038388181383…………, a fantastic result indeed. I am calling these two evidences (that is this one and the previous one ) as penultimate because even though they are very strong they don’t constitute the ultimate one. For instance we know that the value of π (pi) these days are calculated up to millions and millions of digits on super duper computers, yet no body divides the measured circumference of a circle by the measured diameter of the same up to that precession(that is no body measures the circumference and diameter of a circle to that precession and divide them to calculate the value of π (pi)) to verify the result. One is simply confident in the mathematical sense. This thing is yet to to come up with regards to α: the value of fine structure constant at zero energies, whence it would be called the ultimate evidence,if at all such a happy moment exists. '(THE GRAVITATIONAL COUPLING CONSTANT):

(The ultimate physical evidence.)

(THE SEEDS OF UNITY BETWEEN GRAVITATION AND ELECTRICITY)'' '''

Let us now adapt the electron theory above to the case of universal Gravitation in a manner that is completely analogus to the electron case. As shown in the very beginning of this paper for spherical manifold of constant positive curvature(the ideal universe having uniform distribution of matter energy and momentum) the characterizing space differential relation is dr’=dr/(1-r’^2/R0^2)………(9). Here R0 models the number infinity, finitely and may be called loosely the radius of the physical or the observable universe. In the electron theory case that number was r’=.3892285591………. To suggest consistency we equate the two numbers that is we take R0=.3892285591…….. . Now in a manner completely analogous to the electron case we define a generalized derivative as Dr’/Dr =((R0^2-r^2)/(R0^2-r’^2))dr’/dr………(10). We then let Dr’/Dr =r’………(11) and declare that r’=cα^r………(12) is its solution where the α now is the Gravitational coupling constant(and c is a arbitrary constant) as we know from general relativity that gravity determines the space time relations of the large scale universe. These are also completely analogous to the electron case. From (11) and (12) we have ((R0^2-r^2)/(R0^2-r’^2))(ln(α)+(x/α)(dα/dx))=1 …………(13). Again analogously in equation (12) we shift the origin and apply the boundary condition r=r’ which tends to zero to have r’+1=α^r ……….(14). The term αdx/dα=(α-0)dx/dα=-(Δα)dx/dα=-Δx in the new scale and =Δx in the old electron scale as these scales are oppositely directed, since 0 should be the value of gravitational coupling constant at mathematical infinity which is beyond physical infinity and our physical universe can not gravitationally interact with that point in principle. Thus we have (x/α)(dα/dx)=x/Δx=m/(m-c’)=1 ………..(15) ,where c’ is a constant number modeling 0 and m tends to infinity. Also the term Δα is increment to α between the points c’ and infinity the only points where α is convergent and defined. At the intermediate points it is not defined and the equations mentioned above exists only as a mathematical remnant, so the variation of α with x may be modeled by a linear relationship and so only the first two terms of Taylor expansion has been used as in the electron case(that is fine structure constant case).

Extending the analogy further we have from (9): R0=r’/(1-dr/dr’)^(1/2). Relativizing(as in the electron case) this expression, that is differentiating this expression with respect to r’ and setting dR0/dr =1(this is also consistent with the physical observation that the boundary of the observable universe ought to be receding at the speed of light, the horizon effect, due to expansion of this universe) and substituting r=ln(r’+1) and then solving for r’ we get r’=-1.93756489………. This point should them model the number zero here just as r’=.3892285591……. modeled the number infinity in the electron theory. The minus sign is due to the opposite direction of the new scale as compared to the scale in the electron theory.Let us now imagine two coaxial circles(also coaxial with the poles) on two dimensional spherical manifold modeling the electron and the universe. The first one modeling zero(r0) and the other one modeling infinity(R0). Therefore the distance r’=+1.93756489……. locates the origin of the universe centered at one pole of this sphere with respect to r0 the point that models the origin of the electron theory which(origin) is at the other pole(diametrically opposite) of this same sphere. Thus the effective distance of r0 from this origin that is consistent with the two scales is r’=r0+1.93756489…..(forward movement)+1.93756489…..(then the backward movement bringing us back to the point r0 or which is the same as the further forward movement by the same amount on the closed spherical manifold bringing us to the same circle r0 from the other side of the manifold)=3.94820283….. ,where r0=(73073037037)/(10^(12)-1) as in the electron theory.We now substitute this value of r’=3.94820283……. together with r=ln(r’+1)/ln(α) and equation (15)in equation (13) and solve for α the gravitational coupling constant on the electron boundary. The result is α=1.751159404………×10^(-45). The measured value being 1.7518….. ×10^(-45) which is experimentally known only to four significant digits the fifth digit being uncertain. Thus the digits come right, the order come right and it is nothing less than an absolutely fabulous result. And even more fabulous is the fact that r0 can be eliminated to express this Gravitational Coupling Constant as a function of Fine Structure Constant there by shunning numerology(base 10 specific) of any kind here, whatsoever. An important note for the formulas for gravitational and electric coupling constant in this theory:.Even though these formulas are derived physically so that the parameters like r0,r’,r etc. had a physical meaning then, but once derived they cease to be anything physical as I think, they are just mathematical parameters in the abstract. That is why their numerical values are weird but they do not compromise the inner consistency of the theory.

THE EQUATION OF MOTION OF THE ELECTRON(THE POINT PARTICLE LIMIT):

We are going to work out the equation of motion of electron firstly in the fake space-time coordinates r,t and not in the true ones r’,t’. It should be recalled that the former is fake because it is the cumulative total of the corresponding differentials in a succession of reference frames each at a different point in space where as the later is for a fixed stationary observer at the center of the electron. We do so because even though it is fake it helps us to compare our results with those of classical electrodynamics in a straight forward manner since this is the coordinate that we use there unmindfully. The fact that these coordinates are fake in this sense however does not imply that they are useless. They mirror the other useful coordinates r',t' through a well defined mathematical relation connecting them and therefore can not be so. It only means that they should be interpreted with care. In fact it can also do things that the true ones can not do. For instance the true coordinates are not well defined in the interior of the electron or equivalently in these coordinates the equation of motion is not well defined for time less than t0. Therefore, here the fake coordinates r,t can be our only guide. From equation (A1), for small t0(in the point electron limit) we have a=-t0(da/dt) where “a “ is the local acceleration. Also it can be easily shown that “a” and “da/dt” refer to the same point and same time. Separating the variables and integrating(taking the limits of “a” from “a0” to “a” and that of time from 0 to t), we have a=a0e^(-t/t0) where e is the base of natural logarithm. Now comes a key point that this derivation is purely kinematic not dynamic as external force has not been considered in any real sense.Therefore this formula describes the motion of electron from that time onwards (t=0) when the external force acting on the particle has suddenly been withdrawn. The corresponding equation in classical electrodynamics as can be checked from any standard text book on classical electrodynamics is a=a’0e^(t/T) where a’0 and T is some constant. Contrary to our case this is an exponential runaway which is physically unacceptable. Thus here our equation of motion of electron is physically consistent and its all due to change in sign of time t, mathematically. Thus for us this is small(mathematically) yet also a big (physically)change for ours is an exponentially decaying solution in the absence of an external force. However the changes are not abrupt for abrupt removal of that force. This is understandable as the radiation reaction force would damp out any rapid change in the acceleration vector. In classical electrodynamics this exponential runaway can be eliminated by setting a’0=0 but then there pops up an even more unpleasant consequence: the electron starts accelerating even before force begins to act by a small time interval T. However in our case there is no such need and therefore there is neither the defect of a runaway nor an accausal preacceleration. But the point is, is this a hint for time flowing backwards for a time T or equivalently t0 in our case in the very beginning of the motion of electron? Perhaps yes!(but only within the framework of pure classical electrodynamics) But why? Because : Our time transformation formula is dt’=dt/(1-t0^2/t’^2) which follows from the space transformation formula dr’=dr/(1-r0^2/r’^2). The way we define dr as local or “on the site” space differential it follows that its sign can not change no matter where we measure them since r can only increase. So too must be the case with dt. But then it follows from a simple inspection of time transformation formula that dt’ has opposite sign for t’ less and greater than t0. Thus time seems to flow backwards in the interior of the electron if it flows forward in the exterior of the electron which we all know it does. However the duration t0 is very small, smaller than at least 6*10^(-24) seconds very loosely speaking if we interpret r0 as the classical electron radius and t0=r0/c where c (3*10^8 m/s)is the speed of light. How smaller than this time, it actually is we don’t know at this moment but nevertheless a formal possibility of this ultra microscopic time travel exists in principle. A strong evidence of the reversal of the arrow of time: If we integrate the equation of motion of electron, deduced above within the limits of velocity v from 0 to v and time t from 0 to t we have v=a0t0-a0t0e^(-t/t0). Now for infinite times the second term goes to zero in the limit if a0 and t0 are finite ,as we all believe and thus we have v=a0t0 in that limit. If in conditions of very large force or whatever if a0 is sufficiently large (not necessarily infinity) then no matter how small t0 is but not zero, it is quite possible for v to exceed c ,the speed of light the ultimate speed in the universe. Similar is the case for displacement of the electron. Thus the motion of the isolated free electron becomes unbounded both spatially and kinematic ally. The problem is more serious in classical electrodynamics where even the acceleration is unbounded apart from the displacement and speed due to the exponential runaway mentioned earlier. Consider now the motion of a particular electron moving randomly in a material conductor in the absence of any external electric field. The motion of the electron is bounded due to random collisions with other atoms or electrons giving an average velocity and average displacement of zero. Due to randomness there are on an average as much forward movement as backwards. If now,we abstract the material conductor away leaving only that particular electron behind how can the electron be in a state of random motion so as to make its spatial and kinematic parameters bounded when there is nothing material left to collide against randomly. The only feasible solution is the reversal of the arrow of time at the moment t=t0. Due to reversal of the arrow of time the motion must reverse at these moments producing an average of zero. The electron undergoes mathematically an abstract random dance not due to collisions with any thing material but rather, as it is due to collisions with its own temporal destiny(t=t0) and this mechanism is quite possible and acceptable within the frame work of this theory. (TO BE CONTINUED………………………).

' Epilogue.

I have worked out this theory of electron in an attempt to resolve the contradictions associated with the previous understanding of the electron. The theory is aesthetically pleasing and internally consistent in the scientific sense. However judgment on a theory can not be passed on the basis of aesthetics and internal consistency alone rather a theory must be substantiated by experimental evidence even though a correct theory automatically becomes aesthetically pleasing since it simplifies the so far complex piece of reality so much and in some cases to the point of almost trivializing it, so trivial in fact that it is difficult to believe it at the first glance. Also since the calculation that is the algebra, trigonometry, geometry and calculus is lengthy even though elementary and therefore time consuming to type every thing in all its gory details therefore I have left it as an exercise for the readers. Finally I have not uttered the final word about the electron, in fact no body can do that, yet I am ready to answer the queries from the readers if any to the best of my capacity.

The symbol-meanings in this paper.

U, V= any speed less than that of light. C=speed of light in vacuum. *= a mathematical operation meaning generalized addition and initially modeling infinity and later zero by a finite number. r1, r2, r0 = arithmetic numbers having various geometrical interpretations depending on the context. `=sign of single differentiation, `` =sign of double time differentiation with respect to time etc. a’= prime on a. α,β,γ = constants. β= also an angle in a later context down the paper. ~ =approximately equal to. ά=infinity = ∞ ө=angle. €0='epsilon not', the electric permittivity of free space that is vacuum. r0 = initially stands for analogue of what we call the radius of observable universe and later on for what we call the classical electron radius. c= initially a constant and later also the speed of light in vacuum. φ= a constant. П= pie=22/7=3.141592654……… e = exponential =2.718281828……. in the classical equation of motion of the electron. e= also equal to q^2/4П€0 else where, where q = charge on the model electron. ‘ma’= Newtonian product of (mass) and (acceleration) that is Force. z, z’=complex numbers and in general having both an real and imaginary part. i= an pure imaginary number equal to square root of ‘-1’ ρ=electric charge density that is charge per unit volume. E= electric field intensity. dF’=differential force. t0= time constant corresponding to the space constant r0 related to the model electron. ^ = raised to the power. z0’= a point in the complex z’ plane. The remaining symbols have their meaning specified explicitly or already apparent as and when it occurs in the paper u,v =speeds less than that of light in vacuum. x = sign of multiplication. Q.E.D= QUANTUM ELECTRODYNAMICS.
 * 1) =a generalized addition modeling the number zero by a finite number.
 * 1) = a generalized difference also modeling infinity by a finite number and later modeling zero by a finite number.