**essay: A SEMI-CLASSICAL CALCULATION RE PROTON-RADIUS**

by Mark Creek-water Dorazio, amateur physics enthusiast; MARK.CREEKWATER@gmail.com

**SUMMARY [i.e., “ABSTRACT”]**

Following Dr. Ernest Sternglass **[Refs. 1 + 2]**, one can visualize the **proton** as composed of four **electron-positron pairs** [ep-pairs] and an **unpaired positron at the center.** Using this model, and Sternglass’s “semi-classical” math approach, one can calculate a **theoretical** numeric value which agrees closely with the results of experiments which have been done to determine the proton’s radius. {Most of these have given a numeric value of approx. 8.7 x 10^(-14) cm for proton-radius **[e.g., Ref. 3].**}

Plus, one can propose that this numeric value is **NOT** a measure of the entire proton’s radius, but of the radius of one of the four [4] ep-pairs in Sternglass’s proton-model.

**Key words: Lemaitre, neutron, primeval atom, proton, proton radius, Sternglass;**

**TEXT**

Sternglass **[Ref. 1]** has developed a model which accounts for the origin of protons and neutrons in our universe, and describes their structure.

Using the **“primeval atom”** hypothesis of Georges Lemaitre **[Refs. 4 + 5]** he describes a scenario in which the rotating electromagnetic field of this hypothetical “primeval atom” —**(whose electromagnetic field initially contained all the mass and energy in our universe)**— divided in half, and each of the pieces divided in half, and so on and so forth.

After only 270 generations of such a divide-in-half process, there would be 2^270 tiny pieces, each with the mass of approximately 5 neutrons. At this point, or soon after, there was a “phase transition” [Sternglass’s words, P.11, **Ref. 1**] in which many zillions of the tiny pieces re-configured, in a way which led to the production of zillions of neutrons, some of which quickly “decayed” —producing protons. Many zillions of protons + neutrons emerged from this process, during the last “stage” [Sternglass’s word] of the long divide-in-half scenario, with the release of very large amounts of energy, in the form of high-energy photons [“gamma rays”]; enough energy to power a “big bang”.

Thus Sternglass’s model explains both the Big Bang and the formation of all the protons + neutrons which now exist.

**STERNGLASS’s “TABLE 1”: **Using data from Sternglass’s **“Table 1” [p.234, Ref. 1],** one can derive a **math formula **(below) for the radius of each of the many differently-sized **“cosmological systems”** which participate in the divide-in-half scenario: {Sternglass calls these objects “cosmological systems”, regardless of their size **[p.234, Ref. 1], **and says that “For every system, the mass is proportional to the square of the radius” [p.225, **Ref. 1**]}.

**Here is the math formula: ***(Rs) = ***[*** 2G / c^2 ***]** **[** **[***(Mu)(Ms)***]***^***(1/2)** **] [ 1 / 137.036 ] *** ***(Eqn. 1),**

where ** “Rs”** is the radius of the system’s torus-[donut]-shaped, rotating, electromagnetic field, which I will call the “SYSTEM-RADIUS”;

Please note that this is a modified “Schwarzschild formula”, in which the “local gravity” is greatly increased, according to Sternglass’s theory, which he details in his book **[Ref.#1].**

{ Sternglass says that the tiny systems near the last stage of the long divide-in-half scenario experience a “relativistic shrinkage” by a factor of approx. (137.036), which explains the presence of that number in the formula }

Inspired by Paul Dirac’s so called “large numbers hypothesis” [p.224, **Ref. 4**; pp.73-76, **Ref. 5**], Sternglass derives an ingenious and elegant way to calculate the mass of our universe, theoretically. More regarding this, below.

**COMPTON WAVE-LENGTH AND COMPTON RADIUS: **The physicist Arthur Compton, during the first half of the 20th century, popularized the idea that there is a so called “Compton wave-length” associated with every object which a physicist might want to study; defined as the wave-length of a **photon** whose **energy content** is equivalent to that of the object, and given by a simple math formula:

**(***WL-compton)*** = (***h***) / [(***c***) (***M***)]**

where **“***h***”** is Planck’s constant, **“***c***”** is the speed-of-light, and **“***M***”** is the object’s mass.

Likewise, the Compton radius is just simply the Compton wave-length divided by [2(*pi*)]:

**(***Rc***) = [***h-BAR***] / [(***c***) (***M***)], (Eqn. 2),**

where **“***Rc***”** is Compton-radius and **“***h-BAR***”** is (Planck’s constant) / (2(*pi*)).

In Sternglass’s model there is initially only ONE “cosmological system” — the “PRIMEVAL ATOM” … Because its mass/energy content so large —(being that of our entire universe !!)— its **Rc** is ridiculously small: because smaller photons contain more energy. As the divide-in-half scenario proceeds, {which I call “the count-down to the Big Bang”}, the masses of the systems, and the sizes of their em-fields, **decrease,** while their compton-radii **increase:** after 270 divide-in-half generations, there are many trillions of tiny systems, and the length of the SYSTEM-radius **[***Rs***]** of each tiny system is almost equal to that of its COMPTON-radius **[***Rc***].**

At some point in this “count-down”, the size of the Compton-radius must equal that of the system-radius — the **“***Rs***”** in **Eqn. 1. **One can use easy maths to calculate both the mass **(***Ms***) **and radius **(***Rs***)** of the system whose system-radius **(***Rs***) **equals its compton-radius **(***Rc***).** By definition, one has:

* Rc*** = [***h-BAR***] / [(***c***) (***Ms***)],** **(Eqn. 2),**

where **“***Rc***”** is compton-radius and **“***Ms***”** is mass-of-system.

Equating this **(***Rc***)** to the **(***Rs***)** in **Eqn. 1,** one has:

**[***h-BAR***] / [(***c***) (***Ms***)] = [ 2***G ***/ ***c***^2 ] [ (***Mu***) (***Ms***)** **]^(1/2) ] [1/137.036];**

Solving this equation for (*Ms*):

**(***Ms***) = { [ (***c***) (***h-BAR***) / 2***G ***] [ (137.036) / (***Mu***)^(1/2) ] }^(2/3) (Eqn. 3);**

**MASS OF OUR UNIVERSE ??**

Sternglass’s formula for the mass of our universe, (using the identity **(***e***^2) = (***K***) (***Qe***) (***Qe***), **and neglecting a **typo **in the book), appears on p.265, **Ref. 1,** as:

**(***Mu***) = [ [ (***K***) (***Qe***) (***Qe***) ]^2 ] / [ (***G***)***^2*** (***Me***)^3 ] (Eqn. 4),**

where ** “Me”** is the mass of an electron;

Re-arrangeing this, to make it more beautiful + “elegant”:

**(***Mu***) = [ { ( ***K**Qe Qe*** ) / ( ***G**Me Me*** ) }^2 ] [***Me***] (Eqn. 4a);**

{NOTE: this gives a numeric value of *Mu*** = approx. 1.581 x 10^58 grams,** which is approx. 100x greater than the mass of our universe which one usually sees in books, papers, and essays which address this issue. As Sternglass says, this is **“consistent with the evidence that only about one percent of the mass of the universe is in visible form” [p.210, Ref.1]**}

Using this expression for **(***Mu***)** In **Eqn. 3,** one has:

** (***Ms***) = { [ (***c***) (***h-BAR)*** / (2***G***) ] [ (137.036) / (***Me***)^(1/2) ] [ ( ***G**Me Me*** ) / ( ***K**Qe Qe*** ) ] }^(2/3)**;

**NOTE: **the “**^(2/3)**” at the end means that one **squares **the entire expression and then calculates the** cube-root **of the result.

Using numeric values, *c *= 2.9979 x 10^(10) cm/sec, *h-BAR* = 1.0546 x 10^(-27) gram.cm.cm/sec, *G* = 6.673 x 10^(-8) cm^3 / gram.sec^2, (*Me*) = 9.1094 x 10^(-28) gram, and [*K* *Qe Qe*] = 2.3071 x 10^(-19) (gram.cm^3)/sec^2 ;

one calculates that **(***Ms***) = 4.0543 x 10^(-25) gram.**

**NOTE:** this mass is somewhere between that of two pi-mesons and one pi-meson: it’s the theoretical mass of the “cosmological system” in Sternglass’s model whose system-radius [*Rs*] is equal to the radius of a photon which contains the same amount of energy; i.e., is equal to the system’s so-called “compton-radius” [*Rc*].

One can now use **Eqn. 2** to calculate this radius:

**(***Rc***) = (***Rs***) = [ ***h-BAR ***] / [ (***c***) (***Ms***) ] =**

**[1.0546×10^(-27) gram.cm.cm/sec] / [ (2.9979×10^(10) cm/sec) x (4.0543 x 10^(-25) gram) ] = 8.677 x 10^(-14) cm.**

**Note #1: ** this is very close to the MEASURED “radius of the proton”, which experiments have determined, by a variety of methods, to be somewhere in the neighborhood of between approx. 8.42 x 10^(-14) cm AND 8.97 x 10^(-14) cm **[Ref. 3].**

**Note #2: ** by this method, one calculates the numeric value [8.677 x 10^(-14) cm] in a “semi-classical” way, from Sternglass’s model, using none of the fiendishly difficult maths for which quantum mechanics is famous.

**~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~**

As for the calculated mass, (*Ms*) = 4.0543 x 10^(-25) gram: WHAT MIGHT THIS BE ??

Well, in Sternglass’s model, the proton consists of four [4] electron-positron pairs, and an unpaired positron at the center [p.250, **Ref. 1**].

Perhaps each of these 4 [four] pairs has a mass of approx. 4.0543 x 10^(-25) gram, and a radius of approx. 8.677 x 10^(-14) cm ??

Consider: 4 x [4.0543 x 10^(-25) gram] = 16.217 x 10^(-25) gram = 1.6217 x 10^(-24) gram, which is almost the known mass of the proton !! Perhaps the positron at the center provides the remaining mass ??

Perhaps the “proton-radius” which experiments determine to be approx. 8.7 x 10^(-14) cm { CODATA value is given as approx. 8.768 x 10^(-14) cm **[Ref. 3]** } might actually be measurements of the radius of this particular {[pun intended]} Sternglass cosmological system, whose COMPTON-radius ** [Rc]** just happens to be equal to its SYSTEM-radius

**PERHAPs THIS IS WHY PROTONS ARE SO STABLE ???**

**CONCLUSION**

Using only Sternglass’s model, and some of his numbers, one can solve two easy algebraic equations [Eqns. # 1 and #2, above] simultaneously, to calculate theoretical values for the mass and the radius of one of the four [4] electron-positron pairs which constitute most of the proton’s mass in Sternglass’s proton model. **The calculated numeric value for radius obtained by this method is so near to that of the “proton-radius” which has been measured by many teams of experimental physicists that one suspects that the experimenters have been measuring the radius of one of the four ep-pairs, rather than that of the entire proton.**

Note that this numeric value [8.677 x 10^(-14)] is calculated with no reference to any of the fiendishly difficult mathematics for which quantum field theory is famous.

Plus: the calculated numeric value for mass obtained by this method is slightly less than one fourth [¼] the mass of a proton, which supports Sternglass’s idea that there are four [4] electron-positron pairs in a proton (or neutron), and that these four ep-pairs constitute most of the mass of the proton or neutron.

**REFERENCEs**

1) Sternglass, Ernest; **Before the Big Bang **(1997, 2001), New York, Four Walls Eight Windows;

2) Sternglass, Ernest, “Relativistic Electron-pair Systems and the Structure of Neutral Mesons”, **Phys Rev** (v.123) pp. 391-398, (July 1, 1961);

3) Antognini, A., + others, “The Proton Radius Puzzle”, **Journal of Physics: Conference Series,** v.312, n.3 (2011);

4) Kragh, Helge, **Dirac: a Scientific Biography** (1990), Cambridge, Cambridge University Press;

5) Dirac, Paul; **Directions in Physics **(1978), New York, John Wiley & Sons;

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