EJAS-16240 Camera Ready.pdf

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European Journal of Applied Sciences – Vol. 12, No. 1

Publication Date: February 25, 2024

DOI:10.14738/aivp.121.16240

Bender, D. (2024). Quantum Gravity, Energy Wave Spheres, and the Proton Radius. European Journal of Applied Sciences, Vol -

12(1). 99-165.

Services for Science and Education – United Kingdom

Quantum Gravity, Energy Wave Spheres, and the Proton Radius

Darrell Bender

New Mexico Institute of Mining and Technology

ABSTRACT

We argue, from present considerations and a previous analysis of the

hydrogen atom as a miniature Michelson-Morley experiment in the

Material Point Universe Revisited, that the electron wave velocity in the

hydrogen atom is � and that the fine-structure constant � is the ratio of the

remaining mass of the electron to the initial electron mass, not the ratio of

velocities, as Sommerfield had it. We consider the proton energy wave

sphere, with mass �� and velocity �, so that the radius of the proton energy

wave sphere contained in an electron energy wave sphere is

�� = ħ

��

with �� the remaining mass of the proton equal to one fourth of the initial

proton mass so that

�� = �. �� × ��"�� .

We argue that the hydrogen atom consists of nested energy wave spheres, including those for

the energies lost, so that energy is conserved. With a unit clock rate in the gravitational field

equal to

61 − �

�

;

not its inverse as Einstein had it, we show that energy wave sphere clocks have energy

=61 − �

�

;

h�(

(

so that the energy lost is

@1 − 61 − �

�

;

A=h�( = h� B

�

8 6

�

;

16 6

�

;

)

128 6

�

;

+ ⋯ I

(

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Bender, D. (2024). Quantum Gravity, Energy Wave Spheres, and the Proton Radius. European Journal of Applied Sciences, Vol - 12(1). 99-165.

URL: http://dx.doi.org/10.14738/aivp.121.16240

with the first term, after multiplication by h�, on the right being the Newtonion gravitational

energy of motion of the hydrogen atom. For a light ray moving perpendicular to a radius in a

gravitational field, we obtain

� = ��'

��*

M1 + tan' � = ��'

��*

sec �

With

��'

��*

= U1 − 1

�'

�

and

tan' � =

1 − 61 − 1

�'

�

�;

61 − 1

�'

�

�;

Abstract 1

The fine-structure constant �, also known as the Sommerfield constant, was introduced by

Sommerfield in 1916. In Sommerfeld, A. (1921). Atombau und Spektrallinien (in German)

pp. 241–242, Equation 8, Sommerfield considers � to be

� ,

where �&is velocity of the electron in the first circular orbit of the Bohr model of the hydrogen

atom.

If we did not prove that the mass �+, not �, decreases by the factor � in this case, it is a

reflection that we did not need to prove it since we had already argued the result in the

consideration of the Lorentz contraction of the hydrogen atom and since we are prepared to

argue for it again via the principle that physical reality is comprehensible. Abandoning the

decreasing mass �+ requires abandoning the only rational theory of gravitation that we have.

Somewhere we have placed the rule for not abandoning the correct theory.

For the proton energy wave sphere, we calculated its radius as follows:

“What we see is a matter/energy wave in the form of a sphere, the wave having velocity � with

space-time coordinates along a radius in a quasi-static, spherically symmetric, gravitational

field about a point mass such that the measured velocity of the wave in any direction is always

�. For the proton energy wave, with the mass of the proton �, given by

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European Journal of Applied Sciences (EJAS) Vol. 12, Issue 1, February-2024

�, = 1.67272 × 10"'-��,

we have

�,�' = h�,

so that

�,� = h

�,

�, = h

�,�

where �, is the circumference of the proton energy wave sphere. We calculate the value of

this circumference as

�, = 6.62607015

(1.67272)(2.99792458)

10"&. ��

= 1.321332377387867 × 10"&. ��.

Dividing this last result by 2�, we obtain, for the proton energy wave sphere radius �,,

�, = 0.2102965793286309 × 10"&. ��.

This last value, which is the proton energy wave sphere radius without being inside of the

electron energy wave sphere, is one fourth the measured value of the proton radius with the

proton inside of the electron so that the proton inside of the electron has for energy

h�, = 1

4 h�,/

= 1

2' h�,/

and radius �,! so that

�,! = 4�, = 4

�,� = 0.8411863173145236 × 10"&. ��."

“When we are considering what we see,” which is the proton energy wave sphere or proton,

we know its mass-energy and we know that the velocity of the wave is � so that we know its

momentum, the denominator of the last expression for �,!. Thus, we have the wavelength and

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Bender, D. (2024). Quantum Gravity, Energy Wave Spheres, and the Proton Radius. European Journal of Applied Sciences, Vol - 12(1). 99-165.

URL: http://dx.doi.org/10.14738/aivp.121.16240

radius of the proton energy wave sphere. If what we see is not the correct view, or theory,

then we need to calculate the same radius and obtain the same value for the radius in some

other way, when what we see is already every applicable thing that we know theoretically.

This is not just a coincidental oddity. Similarly, if the measured velocity of the proton energy

wave is not �, then we need to abandon the quantum theory of gravitation that we have given

and replace it when we already know that clocks in a gravitational field slow down by a factor

61 − �

�

;

David Arterburn’s ‘that’s what you need to prove” became “what one must prove,” which in

turn became what must be proved. Theories, e.g., Einstein’s relativity theories, contrived to

imply given results, of physics are commonly taken to be true via the implication. We examine

herein, at the very least, various quantum theories with an eye for missteps, perhaps along

these lines, in logic.

Einstein’s modus operandi included considering an equivalent of what he was proving to be a

postulate, e.g., assuming the constancy of the velocity of light, which became the constancy of

the measured velocity of light in a system with constant velocity � via the first postulate, as a

postulate in order to prove the equations of the Lorentz transformation. As we showed in

“Fast Clocks in the Moving System,” neither result is true.

In Quantum Mechanics (pages 58-59), Albert Messiah, arguing, by “careful analysis” of a

diffraction experiment, for the corpuscular nature of matter as part of the wave-corpuscle

duality, uses the “simplest explanation” claim to reach his conclusion without any further

scrutiny:

“However, the inadequacy of such a pure wave theory is clearly exhibited by a careful analysis

of any diffraction experiment with matter waves. Consider, for instance, a beam of monoërgic

electrons traversing a polycrystalline foil; on a screen suitably placed on the other side of the

foil, one observes a central spot due to the transmitted wave, surrounded by concentric rings

due to the diffracted wave. Suppose, to be specific, that the incident wave is a wave packet

�(�,�) restricted in space; it is obtained by placing (Fig. II.1) an intense source S of monoërgic

cathode rays behind a diaphragm D equipped with a shutter whose opening is fixed once and

for all. When this wave propagates through the foil C, it splits into a transmitted and a

diffracted wave and forms the interference pattern described above, on the screen. Since it is

assumed to be a continuous, classical wave, the observed interference pattern must be

continuous. If one diminishes the intensity of the incident wave, everything else being equal

(for instance, by increasing the distance between the source S and the diaphragm D), the

intensity of the interference spots decreases accordingly, but the interference must remain

continuous. Experiment invalidates these predictions. The observed pattern is actually made

up of a succession of well-localized impacts. If one decreases the intensity of the wave, the

number of the impacts decreases proportionately. In the limit of very low intensity, one

eventually observes just a single impact, either on the central spot, or on one of the diffraction

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rings. The simplest explanation is to attribute each impact to the passage of a corpuscle of

matter: an electron.”

Now, whatever reasoning Messiah might have to argue this claim of “simplest explanation,”

proof does not lie in giving simplest explanations, so this discussion serves nothing more than

to convince the reader of the existence of corpuscles of matter without actually proving it,

upon which, we should insist.

As we shall see, the vast accumulation of physical theory can be discarded if we argue not

from it, but without any knowledge of it. If this be too harsh, then we excuse our ignorance,

which does not put us at any disadvantage.

On the other hand, we need existing theory, the weaknesses of which are well, if not

commonly, known in order to insist that it is false. Before this work, faults of the theory were

taken with an oh well attitude, making the theory non-falsifiable.

We could literally start anywhere in the wilderness, but we prefer to consider, as we have

already done with Einstein’s relativity theories, the emblems of physical theory. No theory is

too big to be abandoned if there is evidence that contradicts it.

In Quantum Mechanics, page 151, Messiah states,

“The absence of a concrete and coherent representation of the phenomena in Quantum

Theory may therefore in no way be considered as a shortcoming of the theory. Nevertheless, it

is open to criticism from another point of view.

The question arises whether the description of phenomena in Quantum Theory actually

fulfills all the requirements one should expect from a completely satisfactory theory. The first

thing one requires of a theory is, of course, that its predictions be in accord with experimental

observation. It is quite certain that Quantum Theory meets this condition, at least in the

domain of atomic and molecular physics. But a physical theory cannot claim to be complete if

it does not go further than to state what one observes when one does a given experiment. At

the outset of any scientific endeavor, one establishes as a fundamental postulate that nature

possesses an objective reality independent of our sensory perceptions or of our means of

investigation; the object of physical theory is to give an intelligible account of this objective

reality.

Now, all the conclusions of the Quantum Theory can always be put in the following form: ‘One

obtains this or that result if one makes this or that observation’. One may therefore question

whether Quantum Theory actually furnishes a complete description of objective reality.”

“The question is even more legitimate since the predictions of the theory are of a statistical

nature. In Classical Theory, one has recourse to the language and methods of statistics when

the information on the physical systems under study is incomplete. The concepts introduced

in Classical Statistics cannot lead to a complete description of objective reality; they only

allow us to obtain certain average properties and some results concerning the physical

systems under study, despite incomplete information. These results do not apply to one

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Bender, D. (2024). Quantum Gravity, Energy Wave Spheres, and the Proton Radius. European Journal of Applied Sciences, Vol - 12(1). 99-165.

URL: http://dx.doi.org/10.14738/aivp.121.16240

particular system, but to a very large number N of identical and independent systems. In the

same way, Quantum Theory does not generally yield with certainty the result of a given

measurement performed on an individually selected system, but the statistical distribution of

the results obtained when one repeats the same measurement on a very large number N of

independent systems represented by the same wave function.

One would be tempted to conclude that Quantum Theory furnishes a correct description of

statistical distributions of systems of microscopic objects, but that it cannot claim to describe

completely each system when taken individually. According to this view, the knowledge of the

wave function would not suffice to define completely the dynamical state of an individual

system. In order to do so, one should have a certain number of individual data which is

impossible to obtain because of the insufficiency of our means of observation. In other words,

the dynamical state of the physical system should be defined at each instant by a certain

number of hidden variables whose evolution would be governed by some specific laws. The

impossibility to predict with certainty the results of a given measurement would simply come

from our inability to know the precise value of these hidden parameters. The wave function

would not represent the objective state of the system under study; rather it would be a

mathematical object containing the totality of information which one possesses on an

incompletely known system.

Although this opinion is perfectly tenable, the current view holds 1) that Quantum Theory

furnishes a complete description of natural phenomena. This is based on the analysis, due to

Bohr (1927), of the very special conditions of observation on a microscopic scale, and on a

general principle which evolves from Bohr’s analysis — the complementarity principle.”

In Section 16 Description of Microscopic Phenomena and Complementarity, pages 152-

153 of Quantum Mechanics, Messiah continues:

“Any description of natural phenomena — whose objective reality is by no means questioned

here — must inevitably involve at some stage the results of our observations bearing on these

phenomena. Now — this is the first point in Bohr’s analysis — no matter how far the

phenomena transcend the scope of Classical Physics, their account must be expressed in classical

terms. Indeed, to account for an experiment means to give an unambiguous description of the

circumstances of the experiment and of the observed results; it means to state, for instance,

that “this pointer has stopped on the dial at that point and that moment”. The point we wish to

emphasize here is the necessity of unambiguous language, in which no element of uncertainty

on the part of the observer may enter. This is absolutely indispensible, since the experiment

must be reproducible, and its progress must remain completely independent of the observer

who performs it.

However, on the microscopic level, one cannot make the sharp separation required by the

ordinary concept of observation, between the natural phenomenon and the instrument with

which it is observed. To describe the object and the observing instrument as separate entities

is justified only to the extent where the quantum ħ may be considered negligible. This sets a

limit to the analysis of phenomena, when carried out in classical language; any attempt to

push the analysis beyond this limit requires a modification of the experimental arrangement

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which introduces new possibilities of interaction between the object and the measuring

instruments.

As a consequence, evidences obtained under different physical conditions cannot be

comprehended within a single picture. However, they must be regarded as complementary in

the sense that only the totality of the observational results exhausts the possible types of

information about the objects of microscopic physics. Such is the content of the

complementarity principle.”

Bohr’s analysis, or detailed examination, in arriving at the claim that Quantum Theory gives a

complete description of natural phenomena, where we have replaced furnishes with gives, does

not constitute proof of the claim or any point, including the complementarity principle,

obtained along the way since none of the claims can be proven. In A Field Guide to Critical

Thinking, Skeptical Inquirer, Volume 14.4, Fall 1990, James Lett states,

“It may sound paradoxical, but in order for any claim to be true, it must be falsifiable. The rule

of falsifiability is a guarantee that if the claim is false, the evidence will prove it false; and if the

claim is true, the evidence will not disprove it (in which case the claim can be tentatively

accepted as true until such time as evidence is brought forth that does disprove it). The rule of

falsifiability, in short, says that the evidence must matter, and as such it is the first and most

important and most fundamental rule of evidential reasoning.”

Lett does not state here that a claim cannot be proven, only that the evidence will not disprove

it if the claim is true. Similarly, in What is a scientific theory? in Live Science, by Alina

Bradford, Ashley Hamer last updated January 31, 2022, “A scientific theory is based on careful

examination of facts.” An example of a careful examination of facts is Bohr’s analysis above. In

Scientific Proof Is a Myth, Forbes, Nov 22, 2017, after stating, “You've heard of our greatest

scientific theories: the theory of evolution, the Big Bang theory, the theory of gravity. You've

also heard of the concept of a proof, and the claims that certain pieces of evidence prove the

validities of these theories,” Ethan Siegal follows that with “Except that's a complete lie. While

they provide very strong evidence for those theories, they aren't proof. In fact, when it comes

to science, proving anything is an impossibility.”

With parts of the discussion omitted here, Siegal continues as follows:

“Our best theories, like the aforementioned theory of evolution, the Big Bang theory, and

Einstein's General Relativity, cover all of these bases. They have an underlying quantitative

framework, enabling us to predict what will happen under a variety of situations, and to then

go out and test those predictions empirically. So far, these theories have demonstrated

themselves to be eminently valid. Where their predictions can be described by mathematical

expressions, we can tell not only what should happen, but by how much. For these theories in

particular, among many others, measurements and observations that have been performed to

test these theories have been supremely successful.

But as validating as that is — and as powerful as it is to falsify alternatives — it's completely

impossible to prove anything in science.”

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Bender, D. (2024). Quantum Gravity, Energy Wave Spheres, and the Proton Radius. European Journal of Applied Sciences, Vol - 12(1). 99-165.

URL: http://dx.doi.org/10.14738/aivp.121.16240

“This doesn't mean it's impossible to know anything at all. To the contrary, in many ways,

scientific knowledge is the most "real" knowledge that we can possibly gain about the world.

But in science, nothing is ever proven beyond a shadow of a doubt. As Einstein himself once

said:

‘The scientific theorist is not to be envied. For Nature, or more precisely experiment, is an

inexorable and not very friendly judge of his work. It never says "Yes" to a theory. In the most

favorable cases it says "Maybe," and in the great majority of cases simply "No." If an

experiment agrees with a theory it means for the latter "Maybe," and if it does not agree it

means "No." Probably every theory will someday experience its "No"—most theories, soon

after conception.’”

Siegal’s analysis, which comprises much of what was not included previously, is as follows:

“In science, at its best, the process is very similar, but with a caveat: you never know when

your postulates, rules, or logical steps will suddenly cease to describe the Universe. You never

know when your assumptions will suddenly become invalid. And you never know whether

the rules you successfully applied for situations A, B, and C will successfully apply for situation

It's a leap of faith to assume that it will, and while these are often good leaps of faith, you

cannot prove that these leaps are always valid. If the laws of nature change over time, or

behave differently under different conditions, or in different directions or locations, or aren't

applicable to the system you're dealing with, your predictions will be wrong. And that's why

everything we do in science, no matter how well it gets tested, is always preliminary.

Even in theoretical physics, the most mathematical of all the sciences, our "proofs" aren't on

entirely solid ground. If the assumptions we make about the underlying physical theory (or its

mathematical structure) no longer apply — if we step outside the theory's range of validity —

we'll "prove" something that turns out not to be true. If someone tells you a scientific theory

has been proven, you should ask what they mean by that. Normally, they mean "they've

convinced themselves that this thing is true," or they have overwhelming evidence that a

specific idea is valid over a specific range. But nothing in science can ever truly be proven. It's

always subject to revision.”

If there is a kernel of proof in Siegal’s argument, it fails to prove that nothing in science can be

proven if for no other reason than nothing is set up rigorously. We could seek a source that

proves this with no guarantee that this is possible. On the other hand, it is possible to explain

here, if we are up to it, why this is the case.

By the expression if p, then q, one means that if p is true, then q is true. Suppose q is true. Even

if p is false, if p were true, then q is true so that if p, then q is true. Thus, p implies q being true

does not imply that p is true; yet, incredibly, Siegal claims that “measurements and

observations that have been performed to test these theories have been supremely

successful.” When he states, “But as validating as that is . . .,” he should state but as validating

as that is not.

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Einstein’s relativity theories are contrived to imply the evidence; in the case of General

Relativity, Einstein asserts that the field equations in the absence of matter in combination

with the equations of the geodetic line give to a first approximation Newton’s law of attraction

and to a second approximation the explanation of the motion of the perihelion of the planet

Mercury. In Einstein’s opinion, “these facts must be taken as convincing proof of the

correctness of the theory.” In the case of special relativity, Einstein postulates the constancy of

the velocity of light in deriving the equations, which cannot possibly be true, of the Lorentz

Transformation. In neither case, as we have shown extensively in On the Nature of Being:

Gravitation and Fast Clocks in the Moving System: An Equivalent to the Lorentz Transformation

Yields a Counterexample, is the theory, the p part in p implies q, true. In the latter case, neither

result is true.

We stated in the abstract to On the Nature of Being: Gravitation,

“The experimental result for the rate of a clock in a gravitational field is given in the paper,

“Optical Clocks and Relativity” by C. W. Chou et al., Science 329, 1630 (2010). The clock rate �

satisfies

��

�/

= �∆h

�' . [�]

By elementary functional analysis, Einstein’s clock rate,

� = (�**)

' = 61 − �

�

;

, [��]

from The Foundation of the General Theory of Relativity is greater than 1, the clock rate in flat

space-time, and does not satisfy, as we show in Part 2, the equation that is experimentally

verified. On the other hand, the multiplicative inverse of Einstein’s clock rate,

� = 61 − �

�

;

, [��]

gives a clock rate that is less than 1, thus a slower rate than that in flat space-time, and

satisfies the equation that is experimentally verified.”

The clock rate in the Chou paper is not the clock rate from The Foundation of the General

Theory of Relativity, but rather, the clock rate from Einstein’s 1911 paper, On the Influence of

Gravitation on the Propagation of Light. The true clock rate is not the one implied by the

General Theory of Relativity; but, if General Relativity is true, then the clock rate arrived at by

assuming General Relativity to be true must be true for General Relativity to imply it. More

generally, no true statement implies one that is false. On the other hand, a false statement

implies one that is true; moreover, assuming statements to be true, for example, the statement

that the clock rate in a gravitational field is Einstein’s clock rate, does not prove the premises

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Bender, D. (2024). Quantum Gravity, Energy Wave Spheres, and the Proton Radius. European Journal of Applied Sciences, Vol - 12(1). 99-165.

URL: http://dx.doi.org/10.14738/aivp.121.16240

or theory from which the statements follow; yet Einstein, referring to the facts given above,

claims proof: “these facts must be taken as convincing proof of the correctness of the theory.”

In the situation in which no theory of science can be proven, Einstein claims proof based on

his fallacious logic. When no proof can be given, Einstein steps forward with a proof of his

theory. No one is surprised and no one cares enough to challenge it; certainly, no one who is

convinced is going to consider obvious evidence that contradicts the theory. This is a logical

crime that cries out hysterically into the darkness of thought. No one, laziness and ignorance

come to mind as applicable traits, knows the theory and no one cares, just that the theory is

proven. The obvious contradictory evidence must be ignored. The experiments that find that q

is true are repeatedly done, with increasing precision perhaps, as if they constitute proof of

the theory.

The matter of actually proving a scientific theory, as opposed to these theories, which fail to

be scientific, lies in the observation that the theory and the theoretical world that the theory

describes must be logically equivalent. The only grasping of that world is via experiment,

which serves to observe it. The world must imply the theory, that is, q must imply p, as well as

the other way around. The inexhaustible totality of experiments must imply the theory, and

this serves to explain why theories cannot be proven. This is not intended to be a proof and

cannot as a proof be a part of a theory; otherwise, a contradiction would result.

In mathematics, statements are taken to be true via introduction as axioms or postulates and

proofs are made through logical arguments on defined terms and what was assumed to be

true. In science, whatever was assumed to be true must be proven although, for example,

Einstein’s second postulate, “that light is always propagated in empty space with a definite

velocity c which is independent of the state of motion of the emitting body,” as we show in

Fast Clocks in the Moving System, is false if taken to mean, via the first postulate, that the

measured velocity is independent of the state of motion of the system according to which it is

measured. For the sake of considering Einstein’s theories, Einstein makes for a good quote; on

the other hand, Einstein is not a reliable witness or source of logic.

Even if the current view, based on Bohr’s analysis as described above, that Quantum Theory

furnishes a complete description of natural phenomena is true, this does not make Quantum

Theory true since the elucidation of furnishes is gives or implies. One might guess that there is

no proof of the current view although it might serve to convince anyone, who does not realize

that it does not make Quantum Theory true, that Quantum Theory is true.

In Messiah’s discussion of whether the description of phenomena in Quantum Theory actually

fulfills all the requirements one should expect from a completely satisfactory theory, he states,

“The first thing one requires of a theory is, of course, that its predictions be in accord with

experimental observation.” By predictions, he means that which the theory predicts, which

means foretells, or, here, since any foretelling is arguable, that which the theory implies. In an

effort to provide a way to go further “than to state what one observes when one does a given

experiment,” otherwise, the “theory cannot claim to be complete,” he remarks, “At the outset

of any scientific endeavor one establishes as a fundamental postulate that nature possesses an

objective reality independent of our sensory perceptions or of our means of investigation; the

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object of physical theory is to give an intelligible account of this objective reality.” He thus

introduces a fundamental postulate, just like that; moreover, Messiah uses, once again, the

word gives, which here means imply. A theory giving or implying that which is true does not

imply that the theory is true. He fails to require that the theory be necessary for, that is, be

implied by, objective reality, which corresponds to our theoretical world.

We know, when it comes to any claim that a scientific theory has been proven, that no such

claim is possible and that the modus operandi is to assert that the theory furnishes, gives, or

implies the evidence that has supposedly been verified. We should know enough to question

and be on alert for any theory that claims to furnish a complete description of natural

phenomena.

In The Idea That a Scientific Theory Can Be ‘Falsified’ Is a Myth, Scientific American, by Mano

Singham on September 7, 2020, Singham writes:

“Falsification is appealing because it tells a simple and optimistic story of scientific progress,

that by steadily eliminating false theories we can eventually arrive at true ones. As Sherlock

Holmes put it, “When you have eliminated the impossible, whatever remains, however

improbable, must be the truth.” Such simple but incorrect narratives abound in science

folklore and textbooks. Richard Feynman in his book QED, right after “explaining” how the

theory of quantum electrodynamics came about, said, "What I have just outlined is what I call

a “physicist’s history of physics,” which is never correct. What I am telling you is a sort of

conventionalized myth-story that the physicists tell to their students, and those students tell

to their students, and is not necessarily related to the actual historical development which I

do not really know!"”

Together with whatever arguments Singham gives previously, including “J.B.S. Haldane, one of

the founders of modern evolutionary biology theory, was reportedly asked what it would take

for him to lose faith in the theory of evolution and is said to have replied, “Fossil rabbits in the

Precambrian.” Since the so-called “Cambrian explosion” of 500 million years ago marks the

earliest appearance in the fossil record of complex animals, finding mammal fossils that

predate them would falsify the theory.

But would it really?

The Haldane story, though apocryphal, is one of many in the scientific folklore that suggest

that falsification is the defining characteristic of science. As expressed by astrophysicist Mario

Livio in his book Brilliant Blunders: "[E]ver since the seminal work of philosopher of science

Karl Popper, for a scientific theory to be worthy of its name, it has to be falsifiable by

experiments or observations. This requirement has become the foundation of the ‘scientific

method.’”

But the field known as science studies (comprising the history, philosophy and sociology of

science) has shown that falsification cannot work even in principle. This is because an

experimental result is not a simple fact obtained directly from nature. Identifying and dating

Haldane's bone involves using many other theories from diverse fields, including physics,

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URL: http://dx.doi.org/10.14738/aivp.121.16240

chemistry and geology. Similarly, a theoretical prediction is never the product of a single

theory but also requires using many other theories. When a “theoretical” prediction disagrees

with “experimental” data, what this tells us is that that there is a disagreement between two

sets of theories, so we cannot say that any particular theory is falsified.

Fortunately, falsification—or any other philosophy of science—is not necessary for the actual

practice of science. The physicist Paul Dirac was right when he said, "Philosophy will never

lead to important discoveries. It is just a way of talking about discoveries which have already

been made.” Actual scientific history reveals that scientists break all the rules all the time,

including falsification. As philosopher of science Thomas Kuhn noted, Newton's laws were

retained despite the fact that they were contradicted for decades by the motions of the

perihelion of Mercury and the perigee of the moon. It is the single-minded focus on finding

what works that gives science its strength, not any philosophy. Albert Einstein said that

scientists are not, and should not be, driven by any single perspective but should be willing to

go wherever experiment dictates and adopt whatever works.”

Singham concludes, “But if you propagate a “myth-story” enough times and it gets passed on

from generation to generation, it can congeal into a fact, and falsification is one such myth- story.” This comes from a man who wrote, “Actual scientific history reveals that scientists

break all the rules all the time, including falsification.”

Refuting or falsifying a theory is as simple as noticing evidence that contradicts the theory,

and we have done that repeatedly in the case of Einstein’s relativity theories. Singham’s claim

of falsification being a myth is simply nonsense. Singham, conspicuously ignorant of all facts

and reason, brings up the notion of consensus of experts as follows:

“Science studies provide supporters of science with better arguments to combat these critics,

by showing that the strength of scientific conclusions arises because credible experts use

comprehensive bodies of evidence to arrive at consensus judgments about whether a theory

should be retained or rejected in favor of a new one. These consensus judgments are what

have enabled the astounding levels of success that have revolutionized our lives for the better.

It is the preponderance of evidence that is relevant in making such judgments, not one or even

a few results.”

This is similar to the view that all of the great minds have looked at a theory and have found it

to be true, so, therefore, the theory is true or, here, that the consensus of experts judge that the

theory be retained while experiments are done that supposedly validate the theory, creating

the impression that the theory is proven.

Singham, writing about his book, The Great Paradox of Science: Why Its Conclusions Can Be

Relied Upon Even Though They Cannot Be Proven, with a consensus five-star rating on Amazon,

states:

“The Great Paradox of Science argues that to better counter such anti-science efforts require

us to understand the nature of scientific knowledge at a much deeper level and dispel many

myths and misconceptions. It is the use of scientific logic, the characteristics of which are

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elaborated on in the book, that enables the scientific community to arrive at reliable

consensus judgments in which the public can retain a high degree of confidence. This

scientific logic is applicable not just in science but can be used in all areas of life. Scientists,

policymakers, and members of the general public will not only better understand why science

works: They will also acquire the tools they need to make sound, rational decisions in all areas

of their lives.”

Evidently, Singham has the ability to twist, as in misrepresent, things better than most

politicians.

In Conceptual Problems in Quantum Gravity and Quantum Cosmology, International Scholarly

Research Notices, Claus Kiefer, 2013, Kiefer, in the Abstract, writes,

“The search for a consistent and empirically established quantum theory of gravity is among

the biggest open problems of fundamental physics. The obstacles are of formal and of

conceptual nature. Here, I address the main conceptual problems, discuss their present status,

and outline further directions of research. For this purpose, the main current approaches to

quantum gravity are briefly reviewed and compared.”

Conceptual Problems, in this case, are, “search(es) for a consistent and empirically established

quantum theory of gravity” or, evidently, searches for theories, none of which are provable if

found. Empirically established means that the theory searched for implies, once again, the

result of experiments. Evidently, empirically is Kiefer’s word of choice among possible

synonyms since he uses the adjectival form empirical three times in the first paragraph, of 1.

Quantum Theory and Gravity-What Is the Connection? that we give here: “According to our

current knowledge, the fundamental interactions of nature are the strong, the

electromagnetic, the weak, and the gravitational interactions. The first three are successfully

described by the Standard Model of particle physics, in which a partial unification of the

electromagnetic and the weak interactions has been achieved. Except for the nonvanishing

neutrino masses, there exists at present no empirical fact that is clearly at variance with the

Standard Model. Gravity is described by Einstein’s theory of general relativity (GR), and no

empirical fact is known that is in clear contradiction to GR. From a pure empirical point of

view, we thus have no reason to search for new physical laws. From a theoretical

(mathematical and conceptual) point of view, however, the situation is not satisfactory.

Whereas the Standard Model is a quantum field theory describing an incomplete unification of

interactions, GR is a classical theory.” Now the idea that “no empirical fact is known that is in

clear contradiction to GR” is simply not true since we have already shown here and in On the

Nature of Being: Gravitation that Einstein’s clock rate in a gravitational field is not the one that

is experimentally, or empirically, verified. Einstein’s explanation of the motion of a material

body in a gravitational field is that it moves in “a four-dimensional straight line, i.e., a geodetic

line.”

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Scanned image from The Foundation of the General Theory of Relativity from The Principle of

Relativity, Dover Publications, Inc., 1952, pages 142, 143.

In § 9. The Equation of the Geodetic Line. The Motion of a Particle of The Foundation of

the General Theory of Relativity, Einstein defines a geodetic line as follows:

“As the linear element ds is defined independently of the system of co-ordinates, the line

drawn between two points P and P' of the four-dimensional continuum in such a way that ∫ds

is stationary a geodetic line has a meaning which also is independent of the system of co- ordinates.”

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Einstein solves the equation of the geodetic line for approximate space-time coordinates in

the case of a quasi-static, spherically symmetric gravitational field for a field-producing point

mass at the origin of the coordinates.

In order that no one has to look it up and we can get on with showing, once and for all,

without spending any further time on development, that General Relativity cannot possibly be

true, in On the Nature of Being: Gravitation, Part 6, we wrote:

“In Part 5 we showed that by assuming that a material point moves in a geodetic line and that

the coordinate acceleration 0"1#

02" along a radial path in space is given by

�'�&

��' = − �

�' , [6.1]

instead of assuming that

�'�&

��*

' = − �

�' , [6.2]

“Einstein guarantees that his space-time coordinates correspond to a faster clock and a larger

unit of distance and that the measured velocity and acceleration of a material point in a

gravitational field are what we just gave.”

Einstein’s assumption, which readily suggested itself, that a material point moves in a

geodetic line is the basis of relativity theory and is one of the assumptions that gets “verified”

as a result of the “truth” of the implications of the theory.

In Influence, Einstein argues, via the Doppler principle, that in a uniformly accelerated system

�′, light emitted at �' with frequency �' has frequency �& with respect to an identical clock at

�& upon its arrival at �&, the frequency given to a first approximation by

�& = �' U1 + �

�'W. [6.3]

By the principle of equivalence, this same equation holds to a first approximation in a system

of coordinates � at rest in a homogeneous gravitational field. For �h, Einstein substitutes the

gravitational potential Φ of �', with the gravitational potential of �& taken as zero, obtaining

�& = �' U1 +

�'W. [6.4]

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Image 18: Scanned image from On the Influence of Gravitation on the Propagation of Light from

The Principle of Relativity, Dover Publications, Inc., 1952, page 104.

In Part 2, we showed the equivalence of the equation that Chou, Wineland verify with

Einstein’s equation (2),

�& = �' U1 + �

�'W, [6.5]

from Influence so that

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�

��

�� = − 1

�

��

�� = �

�' . [6.6]

For �, � in these last two equations

� = � > 0. [6.7]

In Part 2 we considered the clock rate �′ if �** is the multiplicative inverse, i.e.,

�** = 61 − �

�

;

, [6.8]

of Einstein’s value. By our previous result in Part 2, we have

�′

��′

�� = − 1

�

��

�� = �

�'

61 − �

�;

, [6.9]

where � is Einstein’s clock rate in Foundation. Thus, the clock rate �′ is one first-order

approximate solution of

�′

��′

�� = − 1

�

��

�� = �

�' . [6.10]

Another such approximate solution is given by, for example,

�3 = 61 − �

2�

; ; [6.11]

we have

�′

��′

�� = �

�'

61 − �

2�;

. [6.12]

If the distance coordinate �&, aligned along a radius in the gravitational field, does not vary so

that �&& = 1 and the clock rate is that above for

�** = 61 − �

�

;

; [6.13]

then, if a material point moves in a geodetic line with these coordinates, we have, by the chain

rule,

�'�&

��*

' = − 1

�

�' 61 − �

�

;

' ��&

��*

��&

��*

= − 1

�

�' 61 − �

�

;

��&

��*

. [6.14]

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Thus, if a material point moves in a geodetic line with these coordinates, in order for the first

order approximation to be − 4

'5", which is the value of the measured acceleration due to

gravity, we must have

��&

��*

= 1, [6.15]

��&

��*

= −�. [6.16]

A material point, with a path in space aligned along a radius, does not move in a geodetic line

just because Einstein assumed that it moves in a geodetic line and Einstein claimed that the

theory implies Newton’s law of gravitation and the precession of the perihelion of the orbit of

the planet Mercury. We have shown repeatedly that Einstein’s theory does not imply

Newton’s law of gravitation; even if Einstein’s theory implies every result that Einstein states

that it implies, the validity of Einstein’s assumptions is not established by the implications of

these assumptions being true.

Such is the nature of relativity, developed, as Einstein’s “main object,” “in such a way that the

reader will feel that the path we have entered upon is psychologically the natural one, and

that the underlying assumptions will seem to have the highest possible degree of security,”

that we somehow consider, in the face of proof that, with Einstein’s space-time coordinates, the

inverted Schwarzschild coordinates, or with the coordinates just given in which only the time

coordinate varies, a material point cannot possibly move in a geodetic line, that a material

point moves in a geodetic line in the space-time coordinates of a gravitational field.

The experimentally verified, by Chou, Wineland, clock rate, which is the same thing as clock

frequency, is any clock rate that satisfies, to a first approximation,

�′

��′

�� = − 1

�

��

�� = �

�' . [6.17]

Thus, we have given above two such approximate solutions. The experimental result is that

the clock frequency in a gravitational field is given by something like, or approximately,

�3 = 61 − �

2�

;. [6.18]

We have yet to reflect on, other than in the consideration of the Schwarzschild coordinates

and the inverted Schwarzschild coordinates as solutions of the equation of the geodetic line,

the change in the distance coordinate, aligned along a radius, in a gravitational field. If there

exists an experimentally verified result, we have yet to consider that either.

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Rather than considering Einstein’s premises or results in order to determine the change, if

any, in the distance coordinate along a radius, we can argue logically on this matter by

considering a simple clock, the hydrogen atom, in the gravitational field. For the lowest

energy level, � = 1, the circumference of the atom is the wavelength �6 of the de Broglie wave

associated with the electron. The frequency �6 of this de Broglie wave is the clock frequency.

Using the experimental result that, approximately,

�3 = 61 − �

2�

;, [6.19]

a decrease, corresponding to the unit of distance getting smaller, in the Bohr radius of the

atom by some factor �, forces the de Broglie wave velocity �6 to decrease by both � and

61 − 4

; since the wave velocity is given by

�6 = �6�6. [6.20]

Although there is no known condition, other than by premise, that implies that this decrease

in the Bohr radius, and corresponding change in the de Broglie wave velocity by both � and

61 − 4

;, does not occur, we assume for the purpose of argument that there is no change in the

unit of distance along a radius in a gravitational field.

In making this last assumption, we note that we seek the nature of reality for the gravitational

force; we adhere to the concept that a rational explanation for the gravitational force, and

reality in general, exists. Otherwise, the endeavor that we call physics is a waste of time. In a

sense, Einstein wasted our time for a hundred years. As far as replacing Einstein’s theory by a

better one, we note that we have proved that every result, obtained by assumption or

otherwise, of the theory to be false.”

In Part 5 of On the Nature of Being: Gravitation, we considered a material point moving in a

geodetic line with Einstein’s space-time coordinates along a radius in the direction of a point

mass as follows:

“Einstein, of course, assumes that material points have geodetic lines as paths since that

behavior of their motion “readily suggests itself.” Einstein, claiming proof on the basis of

verification of certain implications of the theory, considers not a single point of what we have

written in these first fifty pages. Metaphorically, the fat lady is ready to sing and the removal

of the furniture is at hand. If the positive �& axis is aligned along a radius, �& increasing with

increasing � , and the �& and �* axes in the gravitational field are aligned with the

corresponding axes in flat space-time, we have, for the velocity 01#

01$

of the material point in the

gravitational field,

��&

��*

= { �**

−�&&

��&

��*

= 61 − �

�

;

��&

��*

. [5.1]

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We note that there is no square-root in the expression on the right. By elementary functional

analysis, the expression 61 − 4

;, which is the ratio of the velocity of the material point in the

gravitational field to the velocity of the material point in flat space-time, decreases as �

decreases. This contradicts the acceleration, an increase in the absolute value of the velocity,

of the material point in the gravitational field as � decreases.

Thus, we have proved that if a material point moves in a geodetic line, it does not do it with

Einstein’s space-time coordinates, which we have previously shown to be incorrect. According

to Einstein’s theory, the expression 01#

01$

must correspond to the measured velocity of the

material point in the gravitational field.

If we consider the acceleration, 0"1#

01$

", which must equal the measured acceleration in the

gravitational field, we have, once again, by the chain rule,

�'�&

��*

' = �

�'

��&

��*

��&

��*

= �

�' 61 − �

�

; U

��&

��*

. [5.2]

This last expression is positive, which is the wrong sign, and in order for the first order

approximation to be 4

'5", which is the absolute value of the measured acceleration due to

gravity, we must have

��&

��*

= 1

, [5.3]

��&

��*

= − �

√2 [5.4]

since the unit of time is chosen so that the velocity of light is equal to one.”

Similarly, if the distance coordinate �&, aligned along a radius in the gravitational field, does

not vary so that �&& = 1 and the clock rate is that above for

�** = 61 − �

�

;

; [6.13]

then, if a material point moves in a geodetic line with these coordinates, we have

��&

��*

= { �**

−�&&

��&

��*

= (1 − �/�)

��&

��*

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They may nevertheless be of some value in an approximate way. Independent of the problems

with (4), one can try to test them in a simple setting such as the Schrodinger-Newton

equation; it seems, however, that such a test is not realizable in the foreseeable future [5].

This poses the question of the connection between gravity and quantum theory [6].

Despite its name, quantum theory is not a particular theory for a particular interaction. It is

rather a general framework for physical theories, whose fundamental concepts have so far

exhibited an amazing universality. Despite the ongoing discussion about its interpretational

foundations (which we shall address in the last section), the concepts of states in Hilbert

space, and in particular the superposition principle, have successfully passed thousands of

experimental tests.”

The last claim, that “the concepts of states, and in particular the superposition principle, have

successfully passed thousands of experimental tests,” does not prove that the general

framework of quantum theory holds true since no proof is possible. Not one detail of the

thousands of experimental tests, perhaps of the same experiment, is given. The absolute value

of a wave function is the probability density of the variable for which it is the wave function

according to the framework. Measuring this probability density, if possible, does not imply

equivalence of the absolute value of the wave function to the probability density.

Rather than, following Kiefer or otherwise, considering the main current approaches to

quantum gravity, we give a quantum theory of gravity that begins with the proper clock rate

in a gravitational field and the revelation that the motion of a material point in a gravitational

field is not a geodetic line. The energy of motion of a clock in a gravitational field is not the

energy that the clock has, but rather, the energy that the clock lost so that the total energy is

constant. The source of the gravitational potential energy is the energy of the clock.

In On the Nature of Being: Gravitation, pages 79-80, our copy, we considered, assuming the

hydrogen atom to be a sphere, both a light clock and the electron de Broglie wave clock of the

hydrogen atom, with the length of the light clock aligned along a radius and the Bohr radius of

the hydrogen atom, once again, along a radius, shorter by a factor �, as follows:

“If the length, aligned along a radius in a gravitational field, of the light clock is shorter than

the length it should have in flat space-time by a factor of �, then, in order that the light clock

slow down, with respect to a clock in flat space-time, by a factor of

�3 = 61 − �

�

;

, [6.39]

which is inverted Schwarzschild clock rate, the velocity of the light in flat space-time, that is,

as measured by measuring rods and clocks of flat space-time, must decrease by both

multiplicative factors � and �′.

If we assume that the Bohr radius of the hydrogen atom decreases by a factor �, with

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Similarly, we have

��' = � − ��' or �' = �⁄(� + �).

This gives, for the total time,

�& + �' = 'A@

@"" B" = 'A⁄@

&"B" @⁄ ". (1)

For the time t3 for the light to go from B to C, noting that in the same time the mirror C moves

to the right a distance ut3 to the position C’, the light travels a distance ct3 along the

hypotenuse BC' of a right triangle, Feynman obtains

(��))' = �' + (��))'

�' = (��))' − (��))' = (�' – �')�)

which gives

�) = � √�' − � ⁄ '.

Noting that, for the return trip from C', the distance is the same and, hence, the time is also the

same, Feynman gets

2�) = 'A

√@""B" = 'A⁄@

E&"B" @⁄ ". (2)

Feynman remarks that “Michelson and Morley oriented the apparatus so that the line BE was

nearly parallel to the earth’s motion in its orbit (at certain times of the day and night). This

orbital speed is about 18 miles per second and the “ether drift” should be at least that much at

some time of the day or night and at some time during the year. The apparatus was amply

sensitive to observe such an effect, but no time difference was found — the velocity of the

earth through the ether could not be detected. The result of the experiment was null.”

Feynman notes that it was Lorentz who had “the first fruitful idea for finding a way out of this

impasse,” the idea being that moving material bodies contract in the direction of motion and

only in the direction of motion, and also, that the contracted length, L||, is given by

�|| = �/M1 − �' �⁄ ',

where L0 is the rest length of a body and u the speed parallel to its length. Applying this result

to the Michelson-Morley experiment, the distance from B to C is unchanged, but the distance

from B to E is shortened to �M1 − �' �⁄ '. This gives

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�& + �' = 2�),

so that the Lorentz contraction provides “a way of understanding why the Michelson-Morley

experiment gives no effect at all.” Feynman states, “Although the contraction hypothesis

successfully accounted for the negative result of the experiment, it was open to the objection

that it was invented for the express purpose of explaining away the difficulty, and was too

artificial.””

Now that this is set up, we consider in The Material Point Universe, Part 4, pages 10-11, our

copy, the Lorentz contraction:

“The apparatus in the Michelson-Morley experiment consists of a clock, or really, two clocks

and a way of comparing the periods of those two clocks. At rest each arm has a period of

�/ = 2�

� .

If the apparatus has a velocity v, the parallel arm has a period, in the rest system, if there is no

change in l with respect to the rest system, of

�G,,I5IJJ+J = 2�

�

1 − �'

�'

Similarly, if the apparatus has a velocity v, the perpendicular arm has a period, in the rest

system, if there is no change in l with respect to the rest system, of

�G,,+5,+K0(@BJI5 = 2�

�

è1 − �'

�'

Thus, in the parallel direction, the moving clock unavoidably slows down by a factor of

1 − �'

�' ;

in the perpendicular direction, by a factor of

{1 − �'

�'.

The result of the Michelson-Morley experiment is that these times are equal. So, the only way

to get both clocks to slow down, the experimental result, by a factor of è1 − G"

@" is for the

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= @

�'

�' +

8 B

�'

�'I

16 B

�'

�' I

)

128 B

�'

�'I

+ ⋯ A �OP05QR+K�'

= �OP05QR+K�' − B1 − �'

�'I

�OP05QR+K�'. [6.56]

The expression,

B1 − �'

�'I

�OP05QR+K�', [6.57]

gives the remaining energy of the de Broglie wave clocks of the hydrogen atom at radius �

with

�

� = �'. [6.58]"

Extending the result for the electron to the proton, which is the rest of the atom, is a matter of,

to rephrase David Arterburn, “seeing what is going on” and applying this idea to obtain a

theory of gravitation that does not require some other idea for the proton since it would be

foolish not to consider it given the lack of alternatives. In a sense, never bypass the correct

theory.

What we see is a matter/energy wave in the form of a sphere, the wave having velocity � with

space-time coordinates along a radius in a quasi-static, spherically symmetric, gravitational

field about a point mass such that the measured velocity of the wave in any direction is always

�. For the proton energy wave, with the mass of the proton �, given by

�, = 1.67272 × 10"'-��,

we have

�,�' = h�,

so that

�,� = h

�,

�, = h

�,�

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where �, is the circumference of the proton energy wave sphere. We calculate the value of

this circumference as

�, = 6.62607015

(1.67272)(2.99792458)

10"&. ��

= 1.321332377387867 × 10"&. ��.

Dividing this last result by 2�, we obtain, for the proton energy wave sphere radius �,,

�, = 0.2102965793286309 × 10"&. ��.

This last value, which is the proton energy wave sphere radius without being inside of the

electron energy wave sphere, is four times the measured value of the proton radius with the

proton inside of the electron radius so that the proton inside of the electron has for energy

h�, = 1

4 h�,/

= 1

2' h�,/

and radius �,! so that

�,! = 4�, = 0.8411863173145236 × 10"&. ��.

Similarly, for the electron with mass,

�+ = 9.1093837015 × 10 ")&��,

we have, for the electron not containing the proton,

�+ = 6.62607015

(9.1093837015)(2.99792458)

10"&& ��

= 0.2426310238683092 × 10"&& ��,

so that

�+ = 1

2�

0.2426310238683092 × 10"&& ��

= 0.386159267960865 × 10"&' ��.

For the electron containing the proton, the equation for the Bohr radius of the hydrogen atom,

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�/ = 4��/ħ'

�'�+

= �/h'

��'�+

= ħ

�+��,

where �/ is the permittivity of free space, � is the fine-structure constant, �+ is the electron

mass, and ħ is the reduced Planck constant, gives the value that we seek. The equation,

�+ = ħ

�+�

which we used to calculate the electron radius for the electron not containing the proton,

contains two constants, ħ and �, so that �+ varies if and only if �+ varies, each variable being

inversely proportional to the other. We consider later the case that � varies, so the constancy

of � here is an assumption.

Rewriting �/ as �+/ and equating that with the right-hand most expression for �/ so that we

have

�+/ = ħ

��+�

the idea that ��+ is mass that the electron energy wave sphere has left after the radius

increases from �+ to �+/ with the mass of the electron energy wave sphere losing the mass

(1 − �)�+ is intuitive.

Solving the previous expressions of �/ for �, we obtain

� = �'�+

4��/ħ'

�+�

= �'

4��/ħ�

= �'

2�/h�

Thus

� = �'

4��/ħ�

= �'

4��/

�

2�

h�

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that is, the energy of the electron energy wave sphere containing a proton energy wave

sphere in a hydrogen atom is � times the energy of the electron energy wave sphere not

containing a proton energy wave sphere.

After dividing both sides of equation (3) by �/ in equation (4), dividing � by �/ on the right- hand side of equation (4) gives a second factor of �. As long as we are aware that �/ is the

Bohr radius and multiplication by &

is commutative, avoiding dividing � by �/ in the right- hand side of equation (4), we have

�'

4��/

�/

= �'h�. (5)

From the equation for the Bohr radius of the hydrogen atom

�/ = ħ

�+��,

which is the only expression, of the original three, for �/ that contains �, we have

�+� = ħ

��/

so that, since ��/ is the radius of the electron energy wave sphere, �+ and � represent the

mass and wave velocity, respectively, of the electron energy wave sphere not containing a

proton energy wave sphere. Since

��+� = ħ

�/

if � is constant, then ��+ is the mass and ��+�' is the energy of the electron energy wave

sphere in the case that the electron energy wave sphere contains the proton energy wave

sphere. If the wave velocity � of the electron energy wave sphere is equal to �� and �+ is

constant in the case that the electron energy wave sphere contains the proton energy wave

sphere, then

�+� = �+�� = ħ

�/

In Erick Weisstein’s World of Physics, Electron Radius, Weisstein defines the so-called

"classical" electron radius �/, also called the Compton radius, “by equating the electrostatic

potential energy of a sphere of charge e and radius �/ with the rest energy of the electron” by

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Bender, D. (2024). Quantum Gravity, Energy Wave Spheres, and the Proton Radius. European Journal of Applied Sciences, Vol - 12(1). 99-165.

URL: http://dx.doi.org/10.14738/aivp.121.16240

The only energy wave spheres of which we consider in the case of the hydrogen atom are

those for the proton and the electron, there being two energy wave spheres for the electron,

the free one and bound one. For the free proton wave sphere radius, we have from the

generalization of the relation for the electron

� = ħ�

h�

= ħ

�(�)�

�, = 0.2102965793286309 × 10"&. ��.

The radius of the free electron wave sphere is the ratio of the masses,

�,

�+

= 1836.15267343,

times �,, or

. 3861366263474497 × 10"&'��.

For the bound electron energy wave sphere, the one in the hydrogen atom, the radius of the

electron wave sphere, the Bohr radius, is the ratio of the masses

�+(��/)

�+(�/) = �+(�/)

�+(�/) = 1

�

times �/.

Curiously, experiments that attempt to measure the proton radius do so operating on the

hydrogen atom with a proton and electron that are not free, but bound. Both the proton and

electron energy wave spheres lose mass and have radii that expand as a consequence. In The

Rydberg constant and proton size from atomic hydrogen, Science, 6 October 2017, Vol 358,

Issue 6359, pp. 79-85, Axel Beyer et al, the discussion begins as follows:

“The study of the hydrogen atom (H) has been at the heart of the development of modern

physics. Precision laser spectroscopy of H is used today to determine fundamental physical

constants such as the Rydberg constant R∞ and the proton charge radius rp, defined as the

root mean square (RMS) of its charge distribution. Owing to the simplicity of H, theoretical

calculations can be carried out with astonishing accuracy, reaching precision up to the 12th

decimal place. At the same time, high-resolution laser spectroscopy experiments deliver

measurements with even higher accuracy, reaching up to the 15th decimal place in the case of

the 1S-2S transition (1, 2), the most precisely determined transition frequency in H.

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The energy levels in H can be expressed as

�KJU = �V U− &

K" + �KJU U�,

, ... W + �l/

X+,

K- �Y

'W (1)

where n, l, and j are the principal, orbital, and total angular momentum quantum numbers,

respectively. The first term describes the gross structure of H as a function of n and was first

observed in the visible H spectrum and explained empirically by Rydberg. Later, the Bohr

model, in which the electron is orbiting a pointlike and, in simplest approximation, infinitely

heavy proton, provided a deeper theoretical understanding.

The Rydberg constant R∞ = meα2c/2h links the natural energy scale of atomic systems and the

SI unit system. It connects the mass of the electron me, the fine structure constant α, Planck’s

constant h, and the speed of light in vacuum c. Precision spectroscopy of H has been used to

determine R∞ by means of Eq. 1 with a relative uncertainty of 6 parts in 1012, making it one of

the most precisely determined constants of nature to date and a cornerstone in the global

adjustment of fundamental constants (3).

The second term in Eq. 1, �KJU U�,

, ... W = �'/�' + �)/�) + �)&�) ln(�) + �*/�* + ⋯+...,

accounts for relativistic corrections, contributions coming from the interactions of the bound- state system with the quantum electrodynamics (QED) vacuum fields, and other corrections

calculated in the framework of QED (3). The electron-to-proton mass ratio me/mp enters the

coefficients X20, X30, ... through recoil corrections caused by the finite proton mass.

The last term in Eq. 1 with coefficient CNS is the leading-order correction originating from the

finite charge radius of the proton, rp (3). It only affects atomic S states (with l = 0) for which

the electron’s wave function is nonzero at the origin. Higher-order nuclear charge distribution

contributions are included in �KJU U�,

, ... W.”

Equation (1) shows that the energy levels in the hydrogen atom can be expressed as Bohr

model energy levels plus relativistic and quantum dynamical corrections. The exact

calculations for the proton radius from this expression are not given. The measurement of the

proton radius is not direct just as our measurement of the proton radius comprises evaluating

the mass and using that to calculate the radius via the equation

� = ħ�

h�

= ħ

�,�

Using the mass of the proton obtained from the measurements of complete strangers, we

obtained, giving calculations, the radius of the proton as

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Bender, D. (2024). Quantum Gravity, Energy Wave Spheres, and the Proton Radius. European Journal of Applied Sciences, Vol - 12(1). 99-165.

URL: http://dx.doi.org/10.14738/aivp.121.16240

�, = 0.2102965793286309 × 10"&. ��

for the free proton and

�,! = 2'�, = 0.8411863173145236 × 10"&. ��

for the proton contained in an electron, as in the hydrogen atom.

The Bohr model, as noted, provided a deeper theoretical understanding, but not enough for

the authors to understand that it is the mass, not the velocity, that decreases as the radius

increases. Scientists can be puzzled by this all they want, but physical reality is not in itself

puzzling here. The comment in the article about the astonishing accuracy of calculations and

even higher accuracy of measurements assures us somewhat although the accuracy of the

understanding of physical reality does not.

Both the proton and electron energy wave spheres lose mass and, hence, energy from the

electron swallowing the proton. The energy lost must go somewhere.

In the gravitational case, the energy wave spheres are in an environment where the clock rate

function of a mass is less on the side of the wave sphere that is closer to the mass so that the

far side of the wave has to slow down to match up with the near side of the wave, which is

endlessly pushed closer to the mass. In a sense, the energy wave sphere has to move, giving

up energy that goes into the energy of motion of the wave sphere so that the wave frequency

is the same in any direction.

In the case of the electron and proton energy wave spheres, if the energy lost stays in the

system, we expect energy wave spheres with energy equal to the energy lost. For the proton

energy wave sphere, this is an energy wave sphere with mass )

�, and radius

3 �, = 0.2102965793286309 × 10"&. ��

= 0.2803954391048412 × 10"&. ��.

Yet another radius, the Zemach radius, for the proton energy wave sphere is at

5�, = 1.051482896643155 × 10"&. ��.

For the electron energy wave sphere, this is an energy wave sphere with mass (1 − �)�+ and

radius

1 − �

�+ = 1

0.99270264744 0.386159267960865 × 10"&' ��

= 0.3889979229497369 × 10"&' ��.

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Shockingly, what was taken to be the energy lost turns out to be the free electron energy wave

sphere, with nearly the same radius and energy and with the energy that it loses being the

energy of the electron energy wave sphere at the Bohr radius. A similar statement applies to

the free proton energy wave sphere.

As noted previously in On the Nature of Being: Gravitation, pages 83-86, our copy, we

extended this result to the rest of the hydrogen atom, considering the energy that the de

Broglie wave clocks lose to be negative since that energy is the change in potential energy, as

follows:

“The energy that the de Broglie wave clocks lose becomes the energy of motion of the

hydrogen atom so that energy is conserved. We have

=61 − �

�

;

h�(

(

− =h�(

(

= =@61 − �

�

;

' − 1A h�(

(

= =B− 1

�

� − 1

8 6

�

;

− 1

16 6

�

;

)

− 5

128 6

�

;

− ⋯ I

(

h�( [6.51]

since

(1 − �) − 1 = −�.

Thus, we have shown that for the proton in the hydrogen atom, the proton energy wave clocks

exist and that it is possible to extend this result to the rest of the hydrogen atom, the result

being as follows:

“The energy that the electron de Broglie wave of the hydrogen atom has left, for h� the energy

at infinite �, at radius � in a gravitational field is 61 − 4

;

" h� so that the energy lost is

h� − 61 − �

�

;

h� = h� @1 − 61 − �

�

;

= h� B

�

8 6

�

;

16 6

�

;

)

128 6

�

;

+ ⋯ I.

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ù1 − U1 − 1

�'

�

' ��3

�� ûh�, [6.86]

becomes the component in the radial direction of the energy of the resulting light ray.”

As we note in On the Nature of Being: Gravitation, such a light ray, counter to physical reality

perhaps, which is enough to cancel the deal, is invariably red-shifted:

“More generally, for a light ray moving perpendicularly to a radius, if the frequency of the light

in the perpendicular direction decreases by a factor of

�(�) [6.87]

With

ü1 − �(�)†h�, [6.88]

the energy lost, becoming the component in the radial direction of the energy of the resulting

light ray. While energy is conserved in the sense that

�(�) + ü1 − �(�)† = 1, [6.89]

the energy,

61 − 2�(�)+2ü�(�)†

;

' h�, [6.90]

of the resulting light ray is not.

If �(�) is the clock rate at �, then the frequency of the component of light in the radial

direction, as measured by the clock at �, is given by

U 1

�(�) − 1W �; [6.91]

moreover, the factor of the frequency,

U 1

�(�) − 1W [6.92]

is also, approximately, for sufficiently large � and �(�) well-behaved, e.g.,

�(�) = U1 − 1

�'

�

' ��3

�� , [6.93]

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Before further consideration of whether this account gives the explanation of the bending of

light in a gravitational field, we continue with Einstein’s development as given in On the

Nature of Being: Gravitation:

“Einstein has space-time coordinates in the gravitational field given by a clock with rate

� = U1 − 1

�'

�

' ��3

�� , [6.99]

the multiplicative inverse of the actual clock rate, and a distance coordinate, for the �

coordinate aligned along a radius, given by, for a unit measure of length in flat space –time,

�� = 1 − �

2�

, [6.100]

which gives a greater unit of distance rather than the smaller one claimed by Einstein or the

actual unit of distance that does not change. Thus equipped, Einstein obtains his value, which

we considered in Part 4, for the curvature of a ray of light.

Einstein, without giving an argument, easily recognizes that the course of the light ray must be

bent with regard to the system of coordinates if the �;< are not constant, the curvature, by the

Huyghens principle, of the light ray is given by

− ��⁄��, [6.101]

where

��&

��*

+ U

��'

��*

+ U

��)

��*

= �, [6.102]

and � is a direction perpendicular to the propagation of light.

For “the curvature undergone by a ray of light passing by a mass � at the distance Δ. If we

choose the system of co-ordinates in agreement with the accompanying diagram, the total

bending of the ray (calculated positively if concave towards the origin) is given in sufficient

approximation by

while (73) and (70) give

1 . 2

1 2

44 ÷

æ = - + ÷

æ = r

g r

g a

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Bender, D. (2024). Quantum Gravity, Energy Wave Spheres, and the Proton Radius. European Journal of Applied Sciences, Vol - 12(1). 99-165.

URL: http://dx.doi.org/10.14738/aivp.121.16240

Carrying out the calculation, this gives

Fig: 8

Since, by the chain rule for partial derivatives,

��

��&

= �

2 B 1

�' + 3

�'

�* I ��

��&

, [6.103]

the integral,

��

��&

��', [6.104]

is equal to

�

2 B 1

�' + 3

�'

�* I

�

�&��'. [6.105]

For the integral,

�

�) �&��'

, [6.106]

of the first term, we have, for example,

2 �

(Δ' + �'

)

Δ��'

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= �Δ 1

(Δ' + �'

)

��'

= �Δ lim1"→V

�'

Δ'M�'

' + Δ'

= �

Δ, [6.107]

with units that of

��

�� = (��)'. [6.108]

One might argue that Einstein obtains the correct result, the experimental verification of

which we have not considered, for the bending of light by considering space-time coordinates

with his �;<’s, inverted in the case of �**, by calculating

��

��&

[6.109]

from the wrong direction, that is, in the gravitational field.

We must consider any bending of light around a mass to be observable in flat space-time. On

the other hand, since

��&

��'

= {

�''

�&&

��&

��'

, [6.110]

the tangent of the angle that a light ray, perpendicular to a radius, is bent, this bending

seemingly contradicting the perpendicularity, is not the same, for non-zero

��&

��'

, [6.111]

unless

�&& = �''. [6.112]

If we multiply

��

��&

[6.113]

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� = {U

��&

��*

+ U

��'

��*

+ U

��)

��*

= {

�**

�''

, [6.121]

he considers ��& to be zero, whereas, in the expression for ��* and the expression for ��', �

varies, which makes � a function of �&, which varies, so that ��& cannot be zero, contradicting

the assumption that it vanishes.

With Einstein’s inverted �;;’s, the velocity,

� = ��'

��*

= {

�**

�''

��'

��*

, [6.122]

has the same form as

��'

��*

= {

�**

�''

��'

��*

= {�''

�**

��'

��*

, [6.123]

where the �;;

3 are the actual, non-inverted �;;. Thus, Einstein does not calculate

�

��&

��'

��*

= �

��&

ù{

�**

�''û ��'

��*

�

��&

��'

��*

W {

�**

�''

[6.124]

since, using the chain rule for partial derivatives on the right side, the partial derivative on the

left vanishes. Thus

�

��&

Uè�**

�''W

è�**

�''

= −

�

��&

��'

��*

;

��'

��*

. [6.125]

Even if the �;; are not the multiplicative inverse of the actual �;;

3 ’s everywhere, or, anywhere,

the last equation holds. In fact, for �;;

3 are the actual, non-inverted �;;, we must have

�

��&

ù{�''

�**

3 û

{�''

�**

= −

�

��&

��'

��*

;

��'

��*

; [6.126]