Page 1 of 46
European Journal of Applied Sciences – Vol. 12, No. 6
Publication Date: December 25, 2024
DOI:10.14738/aivp.126.18064.
Bender, D. (2024). Classical Quantum Hidden Variable Gravitation. European Journal of Applied Sciences, Vol - 12(6). 635-679.
Services for Science and Education – United Kingdom
Classical Quantum Hidden Variable Gravitation
Darrell Bender
New Mexico Institute of Mining and Technology
903 Sean St, Socorro, New Mexico
ABSTRACT
Years ago, just by thinking about it, we discovered the rate of a unit clock in a
gravitational field (1 −
α
r
)
1
2
, and after that, that the energy that an energy wave
sphere loses, because of the slower clock rate in a gravitational field, becomes its
energy of motion. We calculated the radius of a proton energy wave sphere
contained in an electron energy wave sphere as rp = 4
ħ
mpc
. We extended the
concept of energy wave spheres to imaginary energy wave spheres to explain the
change in frequency of an energy wave sphere. The single expression, (1 −
α
r
)
1
2
, of
which we remarked this is what has to be this way, gives us the relationship
between the wave velocities of the two types of energy wave spheres. The 1 in this
expression for the rate of a unit energy wave sphere clock gives the clock rate
squared of a unit energy wave sphere in the local system and −
α
r
gives the
imaginary energy wave velocity function, dX4, squared of the particle with mass M
at radius r, dX4 being the imaginary energy wave velocity function of the wave in
an imaginary energy wave sphere with radius r and wave velocity ic (
α
r
)
1
2
. For an
energy wave sphere with mass m and radius rm =
ħ
mc
and imaginary energy wave
sphere with mass m with radius αm =
κm
4π
inside that, the velocity, which is i times
the absolute velocity of the energy wave sphere, of the wave in the imaginary energy
wave sphere starts at 0 for infinite r, and for any r > αM, is equal to cdX4(r) =
ic (
αM
r
)
1
2
, and for r = αM, equals dX4
(αM) = i, the imaginary energy wave velocity
function dX4(α) of the wave of the imaginary energy wave sphere, at radius α of a
mass M. It has the same velocity as the wave of the imaginary energy wave sphere,
the kernel, at radius α of a mass M. The energy wave sphere is transformed into the
imaginary energy wave sphere. For r > αM, we have 0 + dX4
2
(r) = 0
2 −
αM
r
= −
αM
r
,
which shows the effect of the imaginary energy wave velocity function of a mass M
on an imaginary energy wave sphere with mass m. The 0 on the left side of the last
equation is clock rate of the imaginary energy wave sphere with mass m for infinite
r and the square of this clock rate since its value is 0. The fraction αM
r
of the energy
that the imaginary energy wave sphere with mass m loses in the sense that dX4
2
(r)
is negative with increasing absolute value is the square (
v
c
)
2
of the ratio of its
velocity −v to c. For r > αM, the absolute velocity, its absolute value, of the energy
wave sphere with mass m increases to the velocity c (
α
r
)
1
2 = v as r ↘ α; at r = α, the
Page 2 of 46
635
Bender, D. (2024). Classical Quantum Hidden Variable Gravitation. European Journal of Applied Sciences, Vol - 12(6). 635-679.
URL: http://dx.doi.org/10.14738/aivp.126.18064
velocity of the wave in the energy wave sphere is 0 with no positive velocity to lose,
thus making it the 0 energy wave sphere in that sense. This occurs just as, or nearly
so, the velocity of the imaginary energy wave sphere, with wave velocity i and
radius αm =
κm
4π
, its wave velocity matching that of the imaginary energy wave
sphere, the kernel, at radius α of a mass M, equals −1 = −c. The notion, as opposed
to the idea of a black hole, of an imaginary energy wave sphere with radius αM =
κM
4π
and wave velocity cdX4(r) = ic (
αM
r
)
1
2
, which for r = αM, equals dX4
(αM) = i, gives
the nature of the kernel with radius α of a mass M.
INTRODUCTION
If there is any threat to science, the danger of a false theory becoming true in the observances
of the pseudoscientists, the only ones left, first comes to mind here. The local scientists and the
ones far away, the great minds and the lesser minds, somehow turn logic around and twist it
upside down until ridiculous conclusions get accepted with no real scrutiny. Giving analysis, as
the only consciousness left, requires thought; still, with hardly any chance since the only
theories around rely on deceptive trick arguments that get exposed if anyone looks at them, the
first rule of thumb, the pillar of consciousness, is to never consider an argument, but to applaud
it based on how absurdly it reaches, if that be possible. The bog is that bad. The silliness of
discussing this with the only ones left, the pseudoscientists, should be enough to provoke
laughter. No one else is capable of considering it. Of the only ones left, every conscious being
has a field that is not the one under consideration. The existence of the Doppler effect of light
waves, somehow an allowable result, and the concept that velocity is wavelength times
frequency should be enough to establish that the measured velocity of light is not the same in
systems with uniform velocity moving toward and away from the light contradicting Einstein’s
second principle.
In Einstein’s The Foundation of the General Theory of Relativity, the rate of a unit clock in a quasi- static, spherically symmetric gravitational field at radius r is given by
1
g44
1
2
=
1
(1 −
α
r
)
1
2
which is greater than 1, the clock rate in the local system, and does not satisfy the equation, for
the clock rate f,
δf
f0
=
g∆h
c
2
, [i]
that is experimentally verified in “Optical Clocks and Relativity,” Chou, C. W., D. B. Hume, T.
Rosenband, D. J. Wineland, Science Vol 329 24 September 2010: 1630-1633.
Page 3 of 46
Services for Science and Education – United Kingdom 636
European Journal of Applied Sciences (EJAS) Vol. 12, Issue 6, December-2024
The fine-structure constant α, 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 α to be
v1
c
,
where v1is velocity of the electron in the first circular orbit of the Bohr model of the hydrogen
atom. On the other hand, in order to give a description of physical reality, we need α to be
me(a0)
me
,
with me(a0) the mass, corresponding to the Bohr radius orbit, with radius
a0 =
ħ
αmec
,
of the hydrogen atom, that the electron energy wave sphere sheds and to show that this is true.
Similarly, the proton energy wave sphere in the hydrogen atom sheds the mass
1
4
mp,
corresponding to the outer proton radius, the value of which is
rp = 4
ħ
mpc
= 0.8411863173145236 × 10−15 meters.
In “On the Einstein Podolsky Rosen Paradox,” John Stewart Bell sets up and proceeds with the
proof of Bell’s Theorem as follows:
Let this more complete specification be affected by means of parameters λ. It is a matter of
indifference in the following whether λ denotes a single variable or a set, or even a set of functions,
and whether the variables are discrete or continuous. However, we write as if λ were a single
continuous parameter. The result A of measuring a σ⃗⃗⃗1 ∙ a is then determined by a and λ, and the
result B of measuring σ⃗ 2 ∙ b⃗ in the same instance is determined by b⃗ and λ, and
A(a , λ) = ± 1, B(b⃗ , λ) = ± 1. (1)
The vital assumption [2] is that the result B for particle 2 does not depend on the setting a of the
magnet for particle 1, nor A on b⃗ .
If ρ(λ) is the probability distribution of λ then the expectation value of the product of the two
components σ⃗⃗⃗1 ∙ a and σ⃗ 2 ∙ b⃗ is
Page 4 of 46
637
Bender, D. (2024). Classical Quantum Hidden Variable Gravitation. European Journal of Applied Sciences, Vol - 12(6). 635-679.
URL: http://dx.doi.org/10.14738/aivp.126.18064
Ρ(a , b⃗ ) = ∫ dλ ρ(λ) A(a , λ)B(b⃗ , λ) (2)
This should equal the quantum mechanical expectation value, which for the singlet state is
〈σ⃗⃗⃗1 ∙ a σ⃗ 2 ∙ b⃗ 〉 = −a ∙ b⃗ . (3)
But it will be shown that this is not possible. Some might prefer a formulation in which the hidden
variables fall into two sets, with A dependent on one and B on the other; this possibility is contained
in the above, since λ stands for any number of variables and the dependences thereon of A and B
are unrestricted. In a complete physical theory of the type envisaged by Einstein, the hidden
variables would have dynamical significance and laws of motion; our λ can then be thought of as
initial values of these variables at some suitable instant.
CONTRADICTION
The main result will now be proved. Because ρ is a normalized probability distribution,
∫ dλρ(λ) = 1 (12)
and because of the properties (1), Ρ in (2) cannot be less than - 1. It can reach - 1 at a = b⃗ only if
A(a , λ) = −B(a , λ) (13)
except at a set of points A of zero probability. Assuming this, (2) can be rewritten
Ρ(a , b⃗ ) = − ∫ dλ ρ(λ) A(a , λ)A(b⃗ , λ). (14)
It follows that c is another unit vector
Ρ(a , b⃗ ) − Ρ(a , c ) = − ∫ dλρ(λ) [A(a , λ)A(b⃗ , λ) − A(a , λ)A(c , λ)]
= ∫ dλ ρ(λ) A(a , λ) A(b⃗ , λ)[A(b⃗ , λ)A(c , λ) − 1]
using (1), whence
|Ρ(a , b⃗ ) − Ρ(a , c )| ≤ ∫ dλ ρ(λ)[1 − A(b⃗ , λ)A(c , λ)]
The second term on the right is Ρ(b⃗ , c ), whence
1 + Ρ(b⃗ , c ) ≥ |Ρ(a , b⃗ ) − Ρ(a , c )| (15)
Page 5 of 46
Services for Science and Education – United Kingdom 638
European Journal of Applied Sciences (EJAS) Vol. 12, Issue 6, December-2024
For both equations (13) and (14), the assumption is made that a = b⃗ . The equality
Ρ(a , b⃗ ) − Ρ(a , c ) = − ∫ dλρ(λ) [A(a , λ)A(b⃗ , λ) − A(a , λ)A(c , λ)]
uses equation (14) twice so that the assumption is made that
a = b⃗ = c .
Bell does not explain how he obtains the inequality
|Ρ(a , b⃗ ) − Ρ(a , c )| ≤ ∫ dλ ρ(λ)[1 − A(b⃗ , λ)A(c , λ)]
from
Ρ(a , b⃗ ) − Ρ(a , c ) = ∫ dλ ρ(λ) A(a , λ) A(b⃗ , λ)[A(b⃗ , λ)A(c , λ) − 1].
Since
Ρ(a , b⃗ ) = Ρ(a , c ) = Ρ(b⃗ , c ) = Ρ(a , a ),
both sides of this last equation are just 0, so that we have
0 = 0.
While this does imply that
0 ≤ 0,
this last equation does not imply
0 < 0.
Inequality (15), since greater than never applies, should be equality (15):
1 + Ρ(b⃗ , c ) = Ρ(a , b⃗ ) − Ρ(a , c ) (15)
or
Ρ(a , b⃗ ) = Ρ(a , c ) = Ρ(b⃗ , c ) = Ρ(a , a ) = −1,
which we knew all along.
Page 6 of 46
639
Bender, D. (2024). Classical Quantum Hidden Variable Gravitation. European Journal of Applied Sciences, Vol - 12(6). 635-679.
URL: http://dx.doi.org/10.14738/aivp.126.18064
For the purpose of obtaining a contradiction, from equation (3) Bell obtains
|a ∙ c − a ∙ b⃗ | ≤ 1 − b⃗ ∙ c .
For a ∙ c = 0, a ∙ b⃗ = b⃗ ∙ c = 1⁄√2 we have, or Bell obtains,
1
√2
≤ 1 −
1
√2
,
or
√2 ≤ 1,
supposedly violating Bell’s inequality. Of course, since the conditions for which equation (15)
holds are not met, there is no contradiction to equation (15). Any proof of equation (15) implies
equation (15) or whatever inequality one picks as long as the assumptions, including
a = b⃗ = c ,
hold. No strict inequality, without the equality attached, holds. Thus, Bell’s Theorem serves as
another example of a deceptive trick argument that gets exposed if anyone looks at it.
Yet, in “Bells Theorem,” February 1999, by David M. Harrison, Department of Physics,
University of Toronto, Harrison, at the beginning of the introduction, states,
“In 1975 Stapp called Bell's Theorem "the most profound discovery of science." Note
that he says science, not physics. I agree with him.”
Bell’s error in logic, as an attempt to deceive or not, lies in assuming that his inequality, which
was never an inequality in the first place, still holds if
a = b⃗ = c
is false. Particularly since Bell is held with such high esteem for this deceptive trick argument
that gets exposed if anyone looks at it, we add Bell’s Theorem to the list of theories, Einstein’s
relativity theories and theories of the nature of mass in particular, that we have taken apart via
logic.
We choose not to repeat the entirety of our work here at this time for the purpose of presenting
the great wealth of examples of similarly ridiculous results that cannot last. The time is right, if
not ripe, for a discussion of a sort of classical quantum gravitation. In order to set this up, we
quote in italics from Mystery’s End: Analysis of Bell’s Theorem as follows:
In Quantum Gravity, Energy Wave Spheres, and the Proton Radius, we considered electrons and
protons in the hydrogen atom as energy wave spheres that shed mass corresponding to, in the
Page 8 of 46
641
Bender, D. (2024). Classical Quantum Hidden Variable Gravitation. European Journal of Applied Sciences, Vol - 12(6). 635-679.
URL: http://dx.doi.org/10.14738/aivp.126.18064
we have for the radius of the shed proton energy wave sphere inside of an electron energy wave
sphere
rpeshed
=
ħ
mpeshed
c
= 4
ħ
mpc
= 4
1
2π
6. 62607015
(1. 67272)(2. 99792458)
10−15 meters
= 0. 8411863173145236 × 10−15 meters.
The concept of energy remaining arose from the consideration of clock rates in a gravitational
field and the realization that energy wave spheres are clocks such that the frequency of the
energy wave is the clock rate if the wavelength is the circumference of the energy wave sphere.
In Quantum Gravity, Energy Wave Spheres, and the Proton Radius, pages 9-10, our copy, we
wrote,
“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 f
satisfies
δf
f0
=
g∆h
c
2
. [i]
By elementary functional analysis, Einstein’s clock rate,
f = (g44)
−
1
2 = (1 −
α
r
)
−
1
2
, [ii]
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,
f = (1 −
α
r
)
1
2
, [iii]
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
Page 12 of 46
645
Bender, D. (2024). Classical Quantum Hidden Variable Gravitation. European Journal of Applied Sciences, Vol - 12(6). 635-679.
URL: http://dx.doi.org/10.14738/aivp.126.18064
(1 −
α
r
)
1
2
ν
h
c
= (1 −
α
r
)
1
2
cm.
Thus
1
λ
h
c
= m,
which is the same value for m that it had in the absence of gravitation. Any change in m by the
multiplicative factor k occurs if and only if the circumference λ changes by the multiplicative
factor 1
k
. By the law of conservation of mass, as well as Newton’s law of gravitation, neither
changes. Moreover, we have
(1 −
α
r
)
1
2
ν
(1 −
α
r
)
1
2
c
=
1
λ
,
so that the velocity of the energy wave in the energy wave sphere at the distance r is
(1 −
α
r
)
1
2
c.
If we consider the velocity of the energy wave sphere along a radius to be
dx1
dx4
=
g44
1
2dX1
dX4
= − (1 −
α
r
)
−
1
2
c (1),
then the absolute value of this velocity is the multiplicative inverse of the velocity of the energy
wave in the energy wave sphere since c = 1. Since the absolute value of this velocity is greater
than c and increasing, equation (1) does not give the velocity of the energy wave sphere along
a radius.
In On the Nature of Being: Gravitation, pages 69-70, our copy, we wrote,
“If the distance coordinate x1, aligned along a radius in the gravitational field, does not vary so
that g11 = 1 and the clock rate is that above for
g44 = (1 −
α
r
)
−1
, [6. 13]
then, if a material point moves in a geodetic line with these coordinates, we have, by the chain
rule,
Page 13 of 46
Services for Science and Education – United Kingdom 646
European Journal of Applied Sciences (EJAS) Vol. 12, Issue 6, December-2024
d
2x1
dx4
2 = −
1
2
α
r
2
(1 −
α
r
)
−
3
2 dX1
dX4
dx1
dx4
= −
1
2
α
r
2
(1 −
α
r
)
−2
(
dX1
dX4
)
2
. [6. 14]
Thus, if a material point moves in a geodetic line with these coordinates, in order for the first
order approximation to be −
α
2r
2
, which is the value of the measured acceleration due to gravity,
we must have
(
dX1
dX4
)
2
= 1, [6. 15]
or
dX1
dX4
= −c. [6. 16]"
If we consider the energy wave sphere to have energy
(1 −
α
r
)
−
1
2
hν,
which is the energy that it should have with Einstein’s clock rate,
(1 −
α
r
)
−
1
2
,
then the velocity of the energy wave is greater than c and increasing as r decreases, thus
placing Einstein’s clock rate on the scrap pile of potential clock rates.
The notion that the energy wave spheres lose the energy that they no longer have, the energy
that the remaining energy lost, is an essential feature of gravitational theory. We arrived at it
by searching for the rule that explains and is explained by the motion of an energy wave sphere
in a gravitational field. We found the rule by thinking about it. We want and encourage the
possibility that some other explanation exists when the Newtonian energy of motion of an
energy wave sphere is the first order term of the energy lost. Specifically, the energy lost,
hν − (1 −
α
r
)
1
2
hν = (1 − (1 −
α
r
)
1
2
) hν = hν (
1
2
α
r
+
1
8
(
α
r
)
2
+
1
16 (
α
r
)
3
+
5
128 (
α
r
)
4
+ ⋯ ),
is such that that, after multiplication by hν, the first term of the last expression is the Newtonian
value for the gravitational energy of motion of the energy wave sphere. The first term of this
last series,
Page 14 of 46
647
Bender, D. (2024). Classical Quantum Hidden Variable Gravitation. European Journal of Applied Sciences, Vol - 12(6). 635-679.
URL: http://dx.doi.org/10.14738/aivp.126.18064
1
2
α
r
hν,
is just the kinetic energy,
1
2
mv
2
,
that an energy wave sphere gains in moving from r = ∞ to some finite r. Since v is not
constant, the energy wave sphere does not move in a geodetic line as Einstein would have it.
Somehow, the collection of falsehoods that comprise Einstein’s relativity theories is an
ignorable fault in the game of science.
Somehow, where we ended this lengthy quotation, which set things up, was where we wanted
to be in that sense that what we wanted to discuss was where we left off. The consideration
here is just how a material point moves in a gravitational field with changing velocity as
measured with the space-time coordinates there, apparently accelerating, blatantly perhaps so
that there is no question that it is not moving in a geodetic line. The unit clock rate in the
gravitational field at the radius r is simply the rate of the unit energy wave spheres,
(1 −
α
r
)
1
2
,
at radius r. For the geodetic line in the local system with dX1 aligned along a radius and dX4
such that
dX1
dX4
= −1,
the velocity of the material point in the gravitational field, according to Einstein’s The
Foundation of the General Theory of Relativity, with the xi axis aligned with the Xi axis,
dx1
dx4
= −
g44
1
2
g11
1
2
= −g44
1
2 = − (1 −
α
r
)
−
1
2
, (1)
where we have used the actual values for the gii. The velocity
dx1
dx4
= − (
α
r
hν
m
)
1
2
= − (
α
r
c
2
)
1
2
is not the velocity (1) just given so that the material point does not move in the geodetic line
with the property that
Page 15 of 46
Services for Science and Education – United Kingdom 648
European Journal of Applied Sciences (EJAS) Vol. 12, Issue 6, December-2024
dX1
dX4
= −1
since the velocity (1) is not the velocity of the material point.
At the beginning of the lengthy quote, which ended on the previous page, from Mystery’s End:
Analysis of Bell’s Theorem, we considered electrons and protons, in the hydrogen atom here,
to be energy wave spheres:
In Quantum Gravity, Energy Wave Spheres, and the Proton Radius, we considered electrons and
protons in the hydrogen atom as energy wave spheres that shed mass corresponding to, in the
case of the electron wave sphere, the Bohr radius electron orbit and, in the case of the proton
wave sphere, the outer proton radius so that the corresponding radius of the shed mass is given
by
r =
ħ
mc
.
The energy wave sphere, which exists, along with any other attributes the particle might have,
is the particle corresponding to the energy wave sphere. The clock rate, or frequency, of the
energy wave in the energy wave sphere multiplied by Plank’s constant h is its energy hν.
The next step, following the concept that the energy wave sphere is the particle, is to consider
the clock rate function everywhere to be the particle. Since this frequency determines the
gravitational motion of a material point, the matter of how the particle, which is now a
frequency function, achieves this is explainable by considering the particle to be the frequency
function.
In On the Nature of Being: Gravitation, page 84, our copy, we wrote:
The mechanism for slowing these de Broglie wave clocks is not known, redundantly, the
manner by which the de Broglie waves are slowed constituting another open question. This
change in clock frequency is driven by the difference in clock rates along a radius passing
through the atom; the outer clock rate slows, by decreasing the radius, in an endless attempt to
equalize the wave frequency.
Similarly, in Quantum Gravity, Energy Wave Spheres, and the Proton Radius, page 54, our copy,
we wrote:
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 this situation, with
Page 22 of 46
655
Bender, D. (2024). Classical Quantum Hidden Variable Gravitation. European Journal of Applied Sciences, Vol - 12(6). 635-679.
URL: http://dx.doi.org/10.14738/aivp.126.18064
then the series (4) converges to 0 if and only if
1 = (
1
2
+
1
8
+
1
16 +
5
128 + ⋯ ).
The sequence
{
1
2
,
1
8
,
1
16 ,
5
128 , ⋯ }
is decreasing and converges to 0 with nth term
an =
(2n − 3)(2n − 5)(2n − 7) ... 1 × 1
2
nn!
. (5)
Thus
an+1
an
=
2n + 2 − 3
2(n + 1)
= 1 −
3
2(n + 1)
,
so that
limn→∞
inf |
an+1
an
| ≤ 1 ≤ limn→∞
sup |
an+1
an
|
and the ratio test is inconclusive; however, since
limn→∞
sup √|an
|
n < 1,
the series
∑an
∞
n=1
with an as in equation (5) converges absolutely by the root test.
For r < α, just as in the case for r > α,
dX4 = i (
α
r
)
1
2
= i
v
c
,
so that equation (3) gives, upon multiplication by i, the ratio of the velocity iv of the imaginary
wave velocity function wave or imaginary energy wave sphere wave at radius r to c and, upon
multiplication by −1 since the velocity is negative, the ratio of the velocity v of the energy wave
Page 24 of 46
657
Bender, D. (2024). Classical Quantum Hidden Variable Gravitation. European Journal of Applied Sciences, Vol - 12(6). 635-679.
URL: http://dx.doi.org/10.14738/aivp.126.18064
dx4
2 = 1 + dX4
2
.
The energy wave sphere, the one with radius
r =
ħ
mc
,
which varies inversely with the mass m, differs from the kernel, with radius
α =
κM
4π
,
which varies directly with the mass M. For an energy wave sphere with mass M, the mass of the
kernel, the radius r is less than α if
ħ
Mc <
κM
4π
or
M2 >
4πħ
κc
. (6)
The problem is that if
r =
ħ
Mc
for the energy wave sphere with mass M, then if the velocity of the energy wave in the energy
wave sphere is constant, the frequency dx4 of energy wave increases, if r decreases, by the
factor
1
2πr
,
which is equal to
Mc
h
.
In this case, the clock rate dx4, which we should have set up to be imaginary, of the energy wave
sphere does not increase because the absolute value of the velocity of the energy wave is
increasing, but, rather, because the radius of the energy wave sphere is decreasing with
constant velocity of the energy wave.
Page 32 of 46
665
Bender, D. (2024). Classical Quantum Hidden Variable Gravitation. European Journal of Applied Sciences, Vol - 12(6). 635-679.
URL: http://dx.doi.org/10.14738/aivp.126.18064
of the energy that the imaginary energy wave sphere with mass m loses in the sense that
dX4
2
(r) is negative with increasing absolute value is the square (
v
c
)
2
of the ratio of its velocity
−v to c.
For r > αM, the absolute velocity, its absolute value, of the energy wave sphere with mass m
increases to the velocity c (
α
r
)
1
2 = v as r ↘ α; at r = α, the velocity of the wave in the energy
wave sphere is 0 with no positive velocity to lose, thus making it the 0 energy wave sphere in
that sense. This occurs just as, or nearly so, the velocity of the imaginary energy wave sphere,
with wave velocity i and radius
αm =
κm
4π
,
its wave velocity matching that of the imaginary energy wave sphere, the kernel, at radius α of
a mass M, equals −1 = −c.
The notion, as opposed to the idea of a black hole, of an imaginary energy wave sphere with
radius
αM =
κM
4π
and wave velocity
cdX4(r) = ic (
αM
r
)
1
2
,
which for r = αM, equals
dX4
(αM) = i,
gives the nature of the kernel with radius α of a mass M.
In The Foundation of the General Theory of Relativity, The Principle of Relativity, § 21. Newton’s
Theory as a First Approximation and § 22. Behaviour of Rods and Clocks in the Static
Gravitational Field. Bending of Light-rays. Motion of the Perihelion of a Planetary Orbit, pages
158-160, Einstein obtains, for a field-producing point mass at the origin of co-ordinates, to the
first approximation, the radially symmetrical solution for the gμν as follows:
If in addition we suppose the gravitational field to be a quasi-static field, by confining ourselves
to the case where the motion of matter generating the gravitational field is but slow (in
comparison with the velocity of propagation of light), we may neglect on the right-hand side
differentiations with respect to time in comparison with those with respect to the space co- ordinates, so that we have