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Journal of Applied Mathematics and Physics, 2020, 8, 1135-1154
https://www.scirp.org/journal/jamp
ISSN Online: 2327-4379
ISSN Print: 2327-4352
DOI:
10.4236/jamp.2020.86086 Jun. 12, 2020 1135
Journal of Applied Mathematics and Physics
Precursive Time, the Hidden Variable
Renè Burri
Free Academy of Modern Physics, Lugano, Switzerland
Abstract
In this paper,
a new complex variable defined as “precursive time” able to
correlate general relativity (GR) and quantum field theory (QFT) in a single
principle was characterized. The thesis was elaborated according to a hypo-
thesis coherent with the “Einstein’s General Theory of Relativity”,
making use
of a new mathematical-topological variety called “time-space”
developed on
the properties of the hypersphere and explained mathematicall
y through the
quaternion of Hurwitz-Lipschitz algebra. In this
publication we pay attention
to the interaction between the weak nuclear force theory (EWT) and the nuc-
lear mass of the Standard Model.
Keywords
Time Curvature, Precursive Time, Hidden Variable, Timespace Manifold,
Chronotope, Quantum Compensation Spacetime
1. Introduction
In the previously published study (HDTSS 2016)1, we have exposed the thesis of
a plausible “time curvature” induced by spacetime modifications. The hypothesis
is coherent with GR which assumes that time curves jointly with space this is by
definition.
The plausibility of the “time curvature” finds its theoretical hypothesis in the
different classical representations of the “curvature” of spacetime, differently in-
terpreted as a dynamic of a “twisting” process of spacetime.
From aconceptual point of view, the twisting dynamic implies a contraction
effect of spacetime; the theoretical difference between “curvature” and “twist
(torsion)” is decisive in the analysis of the topological variety of the system. The
representation of space-time curvature is specifically described through the ten-
1
Burri, R. (2016) Vacuum-Matter Interaction through Hyper-Dimensional Time-Space Shifting.
Journal of High Energy Physics, Gravitation and Cosmology, 2, 432-446.
http://dx.doi.org/10.4236/jhepgc.2016.23037.
How to cite this paper:
Burri, R. (2020
)
Precursive Time, the Hidden Variable
.
Jou
r-
nal of Applied Mathematics and Phy
sics
,
8,
1135
-1154.
https://doi.org/10.4236/jamp.2020.86086
Received:
March 16, 2020
Accepted:
June 19, 2020
Published:
June 22, 2020
Copyright © 20
20 by author(s) and
Scientific
Research Publishing Inc.
This work is licensed under the Creative
Commons Attribution International
License (CC BY
4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access
R. Burri
DOI:
10.4236/jamp.2020.86086 1136
Journal of Applied Mathematics and Physics
sor equations as solved by the Einstein Field Equation (EFE), in other words, the
contracting space-time places the time variable inside a concept equation:
<minor space ⇔ minor time ⇒ less space → less time>
2. Theoretical Synopsis
Theoretically, the time curvature can be described as: the projection of instanta-
neous time onto an imaginary abscissa shifted in advance towards the future.
However, more concretely we can see it as a “time discrepancy” (∆τ) that ori-
ginates between the event’s instantaneous time and its imaginary projection in
spacetime. These results in a coherent representation of curved spacetime re-
ferred to a generic mass but specified by Planck constants. The topological cha-
racteristic of the representation of a time discrepancy or a spacetime-induced
time warp was developed by placing as a reference theoretical concept: Min-
kowski4 spacetime2 [1]-[7]. The Minkowski4 manifold highlights the position in
Euclidean space of an event recorded in an instant “chronotope” time, however
it does not make possible a connection to an imaginary temporal base. In order
to operate on the imaginary time variable, the study has developed a specific to-
pological manifold applied to the properties of the Hypersphere 3S Sphere
[8]-[12] manifold, which made it possible to represent a complex imaginary time
related to the frame of reference of instantaneous time. The studied manifold
variety, defined as “time-space”, characterizes a pseudo-connection of correla-
tive order between the time referred to the Minkowski space, and the projection
on an imaginary timeframe of reference.
Time-space manifold resolves the correlation between <Minkowski4 space>
and the projection of imaginary timeframe of reference elaborated based on the
higher dimensions’ geometry of the 3-s Sphere hypersphere, running an entan-
glement-type correlation. The time-space variety allows us to specify the imagi-
nary part of the time axis leaving the space coordinates invariant, as shown in
Figure 1 For the sake of simplicity, the graphical representation of the hyper-
sphere is dashed at a semi-circle.
–
Pk
:
chronotope
event
–
t
0:
time
event
–
C
:
pseudo
connection
–
e
0:
Imaginary
time
(
00
et≡
)
The
reference
point
(
t
0)
of
chronotope
time
component
expressed
by
M
4
space
is
correlated
and
in
phase
with
its
imagi-
nary
time
frame
of
reference
component
(
e
0)
when
in
a
theoretical
flat
space-time
condition
.
2Minkowski4 spacetime provides
a logical representation of our Universe considering that there is
no event without a reference to a specific instant in time. Any event of ponderable nature is nece
s-
sarily characterized by a precise and unique set of “space + time” coordinates, called for
this reason
“chronotope”. The representation of the chronotope on a hyperplane or four-dimensional space e
s-
tablishes the uniqueness of that precise spacetime event. The chronotropic coordinates of an event
establish the inviolability defined by the “Pauli exclusion principle”. In fact,
in the same place and at
the same time there can be just one fermion. The chronotope representation on 4-
dimensional space
was the synthesis of the study developed by Hermann Minkowski to provide a coherent geometry to
relativistic theories.
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Figure 1. Correlation time event → projection Imaginary time in flat space (
0
e
) and
curved spacetime (
0
e
−
) In the case of flat space, (
00
et≡
) the two components (real and
imaginary) coincide.
–
B
:
hypersphere
3
-s
Sphere
(
0
e
is
centre
of
gravity
)
–
D
:
discrepancy
time
–
time-space
curvature
(
∆qt
)
–
0
e
−
:
precursive
time
(
00
te
−
⇔
)
Chronotope
event
“
Pk
”
with
time
component
(
t
0),
establishes
a
new
correlation
(
0
e
−
)
becomes
the
new
point
of
hyper-
sphere
centre
of
gravity
.
3. TimeSpace Manifold
The use of the term “timespace”, at variance from spacetime, makes sense be-
cause in the thesis we take the parameter time as the independent function and
space as the dependent function. In fact, the term “chronotope” [13] [14] is
composed, in the order, by: chronos (time) + topos (space) ≡ time space.
The “time-space manifold” can be understood as an entanglement-type con-
nection with symbol “<
ee
>” between |Minkowski4 space| and <3-s Sphere
hyperspace>, mathematically it is expressed according to the quaternion algebra.
a
(
x
);
b
(
y
);
c
(
z
), and time component:
()
00
dt e→
; pure quaternion part:
( )
,,
i jk
xyz
( ) ( )
( )
( )
000 0 0 0 0 0 0
,,, ,
i jk
xyz t e t e xyz e e
−
∀ ∈→ ⇒ ∈ ⊂ ⇔
(
)
( )
000 0
;, ,, ,,
k i jk
P xyz xyze
−
∀∈
( )
( )
000 0
;,,
,,,
i jk
xyz xyze
−
( )
,
i jk
x yz
σ
≡
, hyperspace coordinate o (
σ
sigma-point)
The expression can be described according by Hurwitz-Lipschitz [15] [16],
since the components are unitary quantities:
00
0
0
0
000
0 00
00 0
000
i
j
k
et
xx
yy
zz
−
−
−
+
−
{ } { }
{ }
{ }
00 0 0 0
,0,0,0 , 0, ,0,0 , 0,0, ,0 , 0, 0,0,
i jk
et xx yy zz−+ − − −
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{ }
0 00 0 0
,, ,
i jk
e tx xy yz z
−+ − − −
The “precursive time” (
0
e−
) is the variable that quantifies the “time curvature”
effect that is generated between the instant time of the event and the projection
on spacetime.
( )
00 0
te e
−
⇔
The concept of time curvature is expressed by the “time-space” complex ma-
nifold (Figure 2).
The timespace manifold layout is represented by two hemispheres which re-
spectively indicate: (red) the mass relative to the event and (blue) the hy-
per-dimensional component temporally shifted with respect to the inertial frame
reference of the chronotope.
–
Pk
:
chronotope
(
event
)
–
t
0:
real
time
event
–
e
0:
Imaginary
projection
event
time
(
Hypersphere
centre
of
gravity
)
–
0
e−
:
precursive
time
event
–
M
4:
Minkowski
4
space
– 3
-Sph
;
hypersphere
3
-s
Sphere
The spacetime variation induced by a generic mass (red hemisphere) is quan-
tifiable by its time curvature that is generated between the instantaneous real
time (
t
0) and the correlated complex imaginary time: (
00
ee
−
→
).
The manifold “timespace” allows us to depict the concept of precursive time
(
0
,e
τ
−
∆
). The connection points are respectively: the centre of gravity (
Pk
) and
the hypersphere centre of gravity (3-Sph).
3.1. Time Curvature Conceptual Meaning
The proportionality correlation that is determined between the inertial mass re-
ferred to the event and the time curvature that it generates on spacetime allows
us to describe the spacetime variations by operating via quantum field theory
(QFT).The invariance to the reference system as defined by the Lorentz trans-
formations is preserved by the event’s frame of reference (Figure 3).
Calculations of time curvature (∆
τ
) and precursive time (
0
e−
) as referred to a
generic inertial mass.
The theses and concepts from which the main equations were obtained were
described in the previous HDTSS publication, [17]; in this study we make some
improvements in the calculation and further analyze and evaluate the procedure.
We will examine a fundamental correlation that occurs between: <nuclear mass -
weak nuclear force-orbital levels>.
3.2. Time Curvature Equation
The time-curvature (
∆qt
) is calculated in Planck time.
The Equation (1) allows us to determine time-space curvatures (∆
τ
) of a ge-
neric inertial mass:
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Journal of Applied Mathematics and Physics
Figure 2. Timespace manifold depiction: The connection points are respectively:
the centre of gravity the mass event and the centre of gravity the hypersphere
3-Sph (red hemisphere). The “precursive” time discrepancy (∆
τ
) specifies exactly
he inertial mass value (blue hemisphere). Notice that the instantaneous time im-
aginary projection undergoes a shift as a function of the inertial mass.
Figure 3. In this figure, we distinguish the event on the arrow
of time as in the classic scalar representation and on the im-
aginary abscissa the related “conjugate time” (-sec.).
( ) ( ) ( ) ( )
( ) ( )
1
22
2
1joules sec
kk
pk
k
hh
MP tP s
tc M P c h
MPc
τ
−
=→= → ∆
(1)
–
5
p
G
tc
=
Planck
time
–
M
(
Pk
) =
Planck
mass
event
–
c
2 =
Speed
of
Light
Squared
From the Equation (1) we get the value of the precursive time ( 0
e
−) (-seconds)
referred to the instant time inertial frame of the reference event.
3.3. Precursive Time Meaning
2
1joules sec
v
p
h
mc
τ
=
As Figure 4 shows, Equation (1) determines the amount of deflected time in-
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duced by the time curvature of the inertial mass of the event; we can distinguish:
• (
∆τ
);
time
curvature
: (∆
τ
) indicates the deflection effect of the real time
coordinate induced by the time curvature.
• (
v
τ
);
precursive
time
: the precursive time is specifically the amount of time
deflected by the time curvature. We remember that the time deflection effect
is dictated by the equation:
<minor space ⇔ minor time> so <less space → less time>. From a relativistic
point of view, the dynamics (torsion → contraction) of spacetime simultaneous-
ly determines a deflation effect of space (less space) together with a deflection of
time (less time):
|time deflection → space deflation| ⇒ spacetime variation
Consequently, the
time
curvature
becomes the primary cause of the spaceti-
mecurvature according general theory of relativity (Figure 5).
It is important to note that (
v
τ
) expressed in Planck time (joule/sec) coincides
with the amount of energy of the inertial mass:
( )
( )
energy inertial mass
v
Mi
ττ
⇒∆ ≡≡
(2)
The equation states that the inertial mass is induced by the time curvature;
more specifically we can state that the inertial mass is exactly the value the time
curvature.
Figure 4. Equation (1) specifies the time curvature generated by the
event in spacetime. The dynamic is centripetal, because it precursive,
whereby involves from the future (
0
e
−
) toward the time event (
0
t
).
Figure 5. Depiction as “spacetime curvature” and “spacetime deflation”. The depiction of
spacetime in twist dynamics allows us to act on hyper-dimensional coordinate working
with quantum correlations.
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Two other fundamental parameters are acquired from Equation (1):
1) (
−s
) time deflection in second (Planck time in precursive second).
2) (
σ
) space deflation. It coincides with the spacetime curvature of the GR
(Figure 5).
Equation (1) shows that spacetime variations can be interpreted as classically
described by GR through quantum physics.
Equation (1) also confirms that:
<
For
each
event
with
[
mass
> 0]
there
is
an
imaginary
time
component
proportional
to
the
value
of
the
inertial
mass
>.
Equation (2) extends the concept of complex variable mass in which the iner-
tial frame of reference is a function of the time-space curvature (∆
τ
), the equality
(
mi
≡ ∆
τ
) indicates that:
Theinertial mass in the absence of interacting gravitational forces is a physical
quantity determined by the time curvature.
Note:
Conceptually, the hypothesis of time curvature deals with two important in-
explicable issues that cannot be demonstrated through the classic depiction of
space-curved GR:
1) It unifies general relativity and quantum field theory.
2) It unifies inertial and gravitational mass phenomena.
3.4. Time Space Quadrant as a Hyper-Dimensional Manifold
The representation in TimeSpace quadrant allows us to get a photograph of the
event in instant time, as a “snapshot” of the imaginary timespace curvature
(
ϖ
σ
). The coordinates
( )
,,
i jk
xyz
can be resolved as “hyperdimensional” since
the complementary reference base (
00
ee
−
→
) is placed on an abscissa dimen-
sionally shifted with respect to the event (instantaneous time) (Figure 6).
Figure 6. Representation in quadrant depiction; this allows us to have a comprehensive picture of the
coordinates as referred to the event.
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• Coordinates
( )
,
,
i jk
xyz
define the “curvature” of the deflated space.
• Sigma point (
ϖ
σ
) which indicates the point of deflated edge space (is the
concrete physical space even if calculated with imaginary coordinates).
• (
0
e
−
) is a deflection Imaginary time coordinate assuming the local obser-
vables reference frame, induced by the inertial mass (
Pk
) on time space (time
curvature) (
0
e
−
) is positioned in advance (precursive) with respect to (
0
t
),
because <less space → less time>.
3.5. Time Curvarture Coordinates
From Timespace manifold:
( )( )
( )
000 0 0
, ,, ,,
k i jk
P xyz t xyze
−
∀∈ ⊆
()
( )
0 0 00 0
;34,,
,,, ,
i jk
xyzt s xyze
M
−
;
( )( ) ( )
( )
0 000 0
,, ,;,
k k i jk
Pt x yz Pe xyz
−
′
;
Summing up, each event is characterized by “seven” coordinates:
( )( ) ( )( )
0 000 0
, ;,
kk
Pt xyz Pe
ϖ
σ
−
′
real coordinates referring to the chronotropic position of the M4 event:
( )( )
000 0
,: ,
k
P xyz t
and four 3S hyperdimensional coordinates:
( )
()
0
,:,
i jk
xyz e
ϖ
σ
−
where, (
t
0) is the event time of inertial frame of reference.
3.6. Calculation of Inertial Mass as a Function of Spacetime
From the analyses carried out in this study, we can assume that matter and
spacetime are in fact a single entity consisting of two joint parts, real partand
spacetime the complex hyperdimensional joint part, the interaction <inertial
mass ⇔ spacetime> assumes the concept of a bivalent set. Ref. timespace mani-
fold (Figure 2).
From Equation (1), we can infer correlation (3) that introduces a concept of
mass expressed in Planck parameters, directly related to the spacetime interac-
tion properties.
It is a new expression of mass calculation as a function of the precursive time,
that is, time-space curvatures, a new concept of mass that we distinguish with
the name “mass-matter” (
ψ
).
Equation (3): calculation of inertial mass according to the precursive time
equation:
2
1
pv F
vp
p
m
K
h
mc
τ
τ ψπ
= →=
Γ
(3)
–
ψ
:
mass-matter
(
inertial
mass
as
function
oftime
curvature
)
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–
mp
:
chronotope
mass
(
Pk
)
in
Planckmass
–
v
τ
:
precursive
time
–
p
Γ
:
quantum
specific
Heat
capacity
:
–
K
:
Boltzmann
Constant
–
F
π
:
quantum
compensation
spacetime
, (
F
)
Fibonacci
coefficient
[18] [19]
[20] [21].
The equation of the calculation of the inertial mass as a function of the time-
space curvature implies a direct correlation to Quantum Heat capacity (
p
Γ
):
Q. Heat capacity:
2
v
p
C
K
τ
Γ=
(4)
The equation expressed with (
p
Γ
) is calculated as Planck temperature re-
ferred to (
v
τ
).
Equation (3) highlights that:
( )
p
f
υ
ψτ
Γ→
this leads to asserting that (
p
Γ
)is a state property of matter.
From the calculation of Q.Heat capacity it is pointed out that is a characteris-
tic of the mass as a function of the time curvature; in fact the correlation pro-
vides us with a reading key to the interpretation of the rebus of absolute zero:
( )
0K
p
ψ
→Γ >
From the correlations expressed, we can say that: a generic mass
0mi >
has
a heat state of Q. Heat capacity determined by its time curvature. This assump-
tion implies that it cannot degrade below the Absolute Zero (0 K).
3.7. Quantum Compensation Spacetime (πF)
(
F
) Fibonacci coefficient
Let’s highlight the remarkable coefficient (
πF
) obtained from theoretical
processing of spacetime twist dynamics: quantum compensation spacetime
(q.c.s.). This coefficient allows us to obtain aprecise measure of the value of the
inertial mass as a function of the precursive time. The representation of space-
time calculated as a function of the Fibonacci exponential faithfully describes the
variation of spacetime in twist (torsion) dynamics with an accuracy of the time
curvatures of (10−15), as shown below in Figure 8 and Figure 9 with regard to
nuclear masses. The constant of “quantum compensation spacetime”, describes
the development layout of the twist-contraction dynamics of spacetime (Figure
7).
Given the coordinates: (
()
0e
yi
;
( )
0e
xi
) the graphic elaboration model draws a
centripetal cycloid where the space-time (eigenvectors) contracts towards the
centre of gravity of the chronotope. The four segments (
F
) represent equipoten-
tial space-time areas with a different frame of reference.
It is important to underline that the different areas or equipotential regions
(
F
1,
F
2,
F
3,
F
4) are also iso-temporal, this being in accordance with Equations (1)
and (2). The summation of Fibonacci’s hyperdimensional regions: complex time
+ complex space, is equivalent to the value of the inertial mass as referred to the
chronotope: (
n
F
):
1234nn nn
vvvv
ττττ ψ
+++ =
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Figure 7. Left image: computer processing of space-time torsion contraction, Coordinates
(Yj and Xi) draw a centripetal cycloid with a Fibonacci progression (
πF
) (
Spiral sinks
complex eigenvalues graphics processing
). Right image: comparison with a spiral gener-
ated by an electron in a bubble chamber –
Photo
:
Harvard project
,
Elementary Particles
[22].
The hypothesis under study seems to draw an important meaning as a physi-
cal consequence, that is: mass and spacetime are not two different interacting
entities as we would interpret classically, but we can state that:
<inertial mass and spacetime are constituents of a single entity in which pre-
cursive time is the mediator>.
3.8. Energy Equation
The energy equation
ψ
function of time curvature, becomes a concept of
mass peculiarly related to spacetime interaction. As a result of this, the equation:
(
2
E mc=
), can be reexamined in:
3
Ec
ψ
=
(5)
Equation (5) which confirms the thesis, in fact it can take over from Einstein’s
equation.
23
E mc c
ψ
= ≡
From the exposed correlations [matter → precursive time → energy], we can
support the principle stating that: [matter ⇔ spacetime]
uniquely
phenomena
.
4. Unified General Relativity ⇔ Quantum Field Theory
In this unpublished second part of the publication we submit the working hypo-
thesis to the test of coherence applied to the nuclear masses. The theses that are
here developed involve an atypical research approach, as general relativity (GR)
and quantum field theory (QFT):
4.1. Correlation: Timespace Curvature ⇔ Nuclearenergy
Figure 8 shows an example of application of Equations (1) (3)-(5) to the nuclear
masses of the following atomic elements.
Figure 8 confirms that the correlations are consistent, in fact the value of the
energy calculated in GeV is correct.
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The value of the energy GeV confirms: [
23
E mc c
ψ
= ≡
]
– Column (∆
τ
) J/sec indicates the precursive time (time curvature) calculated
from Equation (1).
– Column (Ψ) indicates “mass-matter” calculated from Equation (3).
– The energy calculation (GeV) is obtained from Equation (5).
Figure 8 confirms the thesis that:
1) We can specify spacetime in quantum physics.
2) Spacetime can also be calculated for atomic nuclei.
In Figure 9, it is possible to quantify the spacetime variation even for nuclear
masses. We apply Equations (1) (3)-(5) to the nuclear masses of the following
atomic elements:
4.2. Correlation: Timespace Curvature ⇔ Electron Energy Levels
The column (picometers) is calculated:
•
( )
2
: orbital radius
v
qt Ptc
τ
⇒∆=
• nuclear mass
3
c
ψ
= GeV
Figure 9 correlates time curvatures (
τ
∆
)and the distance (
ϖ
σ
) in picome-
ters.
We can note that time curvatures generated by the nucleus’ mass determines
the distance of the electronic orbitals’ levels (
ϖ
σ
).
Figure 8. This figure shows how the time curvatures equation can determine the energy
value of the relevant respective nuclear. The calculated values are correct and confirm
our thesis.
Figure 9. This figure calculates the time curvature (Δ
τ
) generated by the
mass of the nucleus. The space gap in picometers is exactly the electron dis-
tance from the nucleus according to its energy level.
elements un.atomic mass (u) Mass (Kg) planck mass ∆τ (jou le s /se c) ψ mas s -ma t te r ψC 3 GeV
Hydrogen 1,0079400E+00 1,6726200E-27 7,6900000E-20 3,2289743E+15 5,57933E-36 1,50329E-10 0,9382824
Li 6,9400000E+00 1,1500000E-26 5,2950000E-19 8,4729453E+15 3,8414E-35 1,03503E-09 6,460129
Na 2,2989769E+01 3,8175409E-26 1,7540000E-18 1,5421131E+16 1,27249E-34 3,42859E-09 21,39956
K3,9098300E+01 6,4924300E-26 2,9830000E-18 2,0110740E+16 2,1641E-34 5,83094E-09 36,39386
Rb 8,5467800E+01 1,4192000E-25 6,5210000E-18 2,9734355E+16 4,73083E-34 1,27468E-08 79,55901
Cs 1,3290545E+02 2,2069470E-25 1,0140000E-17 3,7078350E+16 7,35633E-34 1,98209E-08 123,7124
Fr 2,2300000E+02 3,7030000E-25 1,7010000E-17 4,8023474E+16 1,23404E-33 3,32499E-08 207,5293
elements
∆τ (joules/sec) (PlanckTime) Pt m*s ec
(
σω)
Picometers GeV
Hydrogen 3,22897E+15 1,741E-28 1,56473E-11 15,64733 0,9382824
Li 8,47295E+15 4,568E-28 4,10551E-11 41,05513656 6,460129
Na 1,54211E+16 8,314E-28 7,47225E-11 74,72250556 21,39956
K2,01107E+16 1,084E-27 9,74251E-11 97,42506138 36,39386
Rb 2,97344E+16 1,603E-27 1,4407E-10 144,0704552 79,55901
Cs 3,70783E+16 1,999E-27 1,79661E-10 179,6611602 123,7124
Fr 4,80235E+16 2,589E-27 2,32688E-10 232,6877158 207,5293
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5. The Electromagnetic, Weak, and Strong Interactions
5.1. Correlation: Nuclear Mass ⇔ Nuclear Electronic Orbital
Levels
In the following diagram we have highlighted the correlation ratio determined
between the orbitals
i.e.
Na -sodium (2, 8, 1) and the respective distance from
the nucleus. The energy contained in orbitals (2, 8.1) is directly related to the
distance between the nucleus and the orbital radius.
– Values in red refer to the energies ((∆
τ
) J/sec))of the different orbital levels.
– Values in blue refer to the positions (Pm) in picometers attributed to orbitals.
The orbital takes on an energy value as a function of the distance determined
from the nucleus.
From the Equation (1) (
v
τ
) it is possible to calculate the levels of the orbitals
and the number of electrons between orbitals according to the distance from the
nucleus.
Figure 10 shows the position of the three sodium levels and the overall energy
equal to the number of electrons on that level:
5.2. Correlation:
p
τ
∆ →Γ
Exothermic ↔ Endothermic Q Value
The movement of electrons between orbital levels in emission or absorption
form is indeed a variation of the orbital radius, the correlation is:
( )
3
v
c
ϖ
τσψ
→→
Each level is characterized by a space position coordinate (
n
µ
σ
) (Figure 11)
and a corresponding amount of energy as a function of precursive time (
v
τ
).
The complexity of the system is due to the double correlation that specifies the
coordinate set (
+
) for each Orbital level (
K
,
L
,
M
,
N
):
-
( )
( )
3
:,,
k
ki kj kk v k
K xyz c
τψ
Σ⇒
-
( )
( )
3
:,,
l
li lj lk v l
L xyz c
τψ
Σ⇒
-
()
( )
3
:,,m
mi mj mk v m
M xyz c
τψ
Σ⇒
-
( )
( )
3
:,,
n
ni nj nk v n
N xyz c
τψ
Σ⇒
Nucleus:
– chronotope nucleus position (nucleus coordinate):
( )( )
000 0
,: ,
k
P xyz t
– Nucleus energy:
( )
3
qc
τ
∆
– Atom cross sections:
()
(
)
0
,:,
i jk
xyz e
ϖ
σ
−
– orbital (N) stretch dynamic: (
12
2
1NN
e
n
ττ
= −
)
( )
4
2
22
0
8
Np
meh
ττ
ε
→ ∆ −Γ
→ orbital contraction (Figure 12).
( )
4
1
22
0
8
Np
meh
ττ
ε
→ ∆ +Γ
→ orbital stretch (Figure 12).
In summary, each orbital consists of a space position (
n
µ
σ
) (hyperspace posi-
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Journal of Applied Mathematics and Physics
tioning) and a time quantum level (
v
τ
) which determines its energy (Figure 11).
The correlation is extremely significant, as it places the nucleus mass as the cause
of the energies distributed over the orbital levels (Figure 12 general diagram).
Figure 10.
Orbital sodium levels obtained from a calculation model from Equation (1).
Figure 11. The correlation ratio between nucleus mass
and orbital radius defines the electroweak force.
Figure 12. General diagram: The dynamic of the <exothermic ⇔ endothermic> reaction
can be very similarly imagined to a (stretching dynamic).
Slide show
2019
LANR/CF
Colloquium at MIT
,
March
23-24, 2019.
Cambridge MA
02139 (
USA
).
τ
υ
proton
τ
υ
electron
3,2279244E+15 7,53426E+13 Lev H
(1st gap) Gap1
Lev 2 Gap 2 Lev 3
σ
ϖ
15,6383 Pm 0,3651
ne 2 8 1
21,3996
GeV | MeV 0,5104 1,0208 3,0625 4,0833 5,1041 0,5104 14,2915
74,7225 Pm
0,3651 0,7301 2,1899 2,9205 3,6507 0,3650 10,2213
Nucleus
orbitals levels
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6. Theoretical Fallout
6.1. Extending the Standard Model in Hyper-Dimensional
Mechanics
As shown on the chart in Figure 11, the electrons are emitted from quantum
time precursive orbital, with respect to the nucleus chronotypal position, (
the
frame
of
reference
of
the
nucleus
is
delayed
compared
electronicorbitals
), it fol-
lows that the electron does not have a specific position “x” and momentum “p”,
but is in a weak interactions’ regime in the latency energy form of.
The deductive conclusions suggest an understanding of the structure of a
four-dimensional atom much more complex than the one presented by the cur-
rent “standard model”.
The concepts presented so far are coherent with the standard model, Bohr
model representation models hitherto analyzed, however the electroweak inte-
raction dynamics defined by the new variable suggest a mathematical structure
which goes under the name the Hopf fibration [23] [24] [25].
Hopf’s fibrational dynamics configuration allows the standard model to be
extended to hyperspace correlations between nucleus and electronic orbitals. Fi-
brational mechanics allow a consistent representation of the theory and allow to
calculate the energies of the electronic orbitals (Figure 13 referred to Figure 10)
The fibration dynamics explain the reason why there is no conflict between
the negative charge of the electron and the positive charge of the nucleus, the
system to continue in balance because it is the nucleus itself the one generating
the spacetime variations that give rise to the electron charge of the electro-weak
interaction.
– fibration positioning (
132
sss→→
).
– energy levels (
132
ψψψ
→→
).
The proposed atom fibration depiction dynamics does not modify the energy
levels ascertained by the standard model, it varies the rationality of the origin
causality of the Electroweak interactions.
As regards Quantum Chromodynamics (QCD) theory of the strong interac-
tion between quarks and gluons, a review the interpretations in this new key of
reading necessarily arises.
The hypothesis studying a “hyperdimensional atom” explains the reason for
the incomprehensible conflict “The Wave-particle duality relation” and confirms
Louis de Broglie’s Research on Quantum Theory hypothesized that all matter
can be represented as a de Broglie wave in the manner of light.
6.2. Heisenberg Uncertainty Principle, Why?
The uncertainty principle is basically evincible through the electron levels in fi-
brational dynamic, since it is generally not possible to predict the value of a
quantity with absolute certainty. In fact, any measurement test induces the pa-
rameters (
X
position and
P
momentum) change in (
ω
σ
) of the fibers, just as
demonstrated by Heisenberg’s uncertainty principle.
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Journal of Applied Mathematics and Physics
Figure 13. Preliminary development of orbital levels in terms of Hopf’s fibration.
( )
xn
fS
χυ
τ
∆∆→
( )
pn
Px
fS
ω
σ
∆∆→
6.3. How Is Gravitational Force Generated and Why Is It Always
Attractive?
We have explained the concept of time curvarture as an effect of “temporal dis-
crepancy” originating between the chronotropic mass referred to the event and
the spacetime indicated in Planck time defined as precursive time.
The Equations (1) and (2) indicate that the mass and time curvature are pro-
portional. This leads us to infer that time curvature is primarily responsible for
gravitational mass.
In fact, Equation (6) confirms the thesis:
1
2
m
gG gG
r
r
ψ
τ
= ⇒= ∆
(6)
–
g
:
gravity.
–
r
:
radius.
–
G
:
gravitational
constant.
–
τ
∆
:
time-space
curvature.
The reason why the inertial mass and gravitational mass are related is because
both forces originate from the same causal event.
The direct correlation exists between inertial and gravitational mass, so the
gravitational mass value acquires an objective value defined Gravitation as a
<specific attraction degree (
A
0)>.
0q
Ag
τ
≡∆ ≡
Time curvature defines the value of the specific attraction capacity degree of a
body (
A
0).
The two masses attract each other
( )
01 02
g FA FA= −
so the result is the dif-
ference between the two sections of time curvature (Interferential of the Fibo-
nacci areas).
Brief note about it.
In gravitational interactions, areas (
F
) are consistently added into quaternion
matrix as described according to Hurwitz-Lipschitz, since the components are
unitary quantities.
τ
υ
proton
τ
υ
electron
3,2279244E+15 7,53426E+13 Lev H
(1st gap) Gap1
Lev 2 Gap 2 Lev 3
σ
ϖ
15,6383 Pm 0,3651
ne 2 8 1
21,3996
GeV | MeV 0,5104 1,0208 3,0625 4,0833 5,1041 0,5104 14,2915
74,7225 Pm
0,3651 0,7301 2,1899 2,9205 3,6507 0,3650 10,2213
Nucleus
orbitals levels
fibration po sitioning
S1
(picometers)
1,09518 S2 6,20557 S3 10,22131 17,522
energy fibration (MeV)
ψ1 13,781 ψ2 10,719 ψ3 5,615 30,114
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However, in this study we limit the exposure to the gravitational mass equa-
tion or gravitational attraction force, understood as a specific attraction capacity
of a body as a function of the time curvature (space deflated effect). About the
interactions between bodies, that is, between gravitational masses, the theory ex-
tends the concept to the principle of interference, resulting from the mutual ac-
tion of the precursive areas (
q
e
−
) (Figure 14).
7. Discussion
Compared to the first publication of the theory, we collected reviews from many
colleagues who highlighted the difficulty in understanding exactly the sense of
time curvature and precursive time. The difficulty lies in acquiring a new know-
ledge of “binary” matter including of a hyper-dimensional part. As explained,
the study basically demonstrates that matter and spacetime are two constituents
of a single entity. The same principle is applicable on a nuclear scale by estab-
lishing a correlation at the base of the electroweak force, between the nucleus
and the electronic orbitals, so even the Higgs boson, can be considered as the
minimum time discrepancy amount that can give rise to a ponderable unit of
mass through the LHC.
Some colleagues asked us if the theory went towards a GUT hypothesis or if
ToE, given that in the same work it is exposed to gravitational force and elec-
troweak force. However, the fact that the publication exposes aspects of general
relativistic physics, and nuclear physics in the same study, confirms the hypo-
thesis of a Theory of ToE Unification. This made it difficult even to characterize
the physics area for the publication of this study.
As a final point, we can ask ourselves whether there are any scientific clues
leading to acceptance of this hypothesis.
Regarding this difficult question we are sure that the theoretical research was
born from the need to identify a scientific hypothesis consistent with the theo-
ries and phenomenologies gained, to provide an interpretation of the inexplica-
bility/conflictuality of the current physics. To this aim, we can say that until now
neither evidence nor clues have emerged, that exclude it from a valid research
hypothesis.
Figure 14. (
q
e−
):
precursive time events
.
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8. Conclusions
In this study we have made a subsequent deepening of the theoretical hypothesis
of temporal curvature as the basic phenomenon of the interaction between mat-
ter and spacetime. The study summarized the electroweak interaction between
the nucleus and orbital levels, applying the same equations describing spacetime
and gravitation.
The study pointed out the following demonstrations:
1) CORRELATION: TIMESPACE CURVATURE ⇔ NUCLEAR ENERGY
2) CORRELATION: TIMESPACE CURVATURE ⇔ ELECTRON ENERGY
LEVELS
3) CORRELATION: NUCLEAR MASS ⇔ NUCLEAR ELECTRON ORBITAL
ENERGY LEVELS
4) CORRELATION: ∆
τ
→ Γ
p
Exothermic
↔
endothermic Q Value
Mathematical analyses have been demonstrated with Equations: (1) (3)-(6).
From Equation (1) we calculate:
1) Time curvature (Planck time)
2) Precursive time (joule/second) inertial mass
3) Timespacedeflation (
σ
) (second *
c
2) gravitational mass
4) Nuclear Electronic orbital (Planck time *
c
2) - Electroweak interaction.
From Equation (3) we calculate:
Mass-matter or “hypermatter”: calculation of the mass as a function of the
induced space-time curvature.
Equation (3) modifies the concept of mass intended as a scalar quantity, rep-
laces the concept of quantitative aggregate as a derivative of a function compo-
site. The symbol (
ψ
) indicates the complex mass.
The equation introduces two new relationships:
1) Quantum specific heat capacity (QSHC): Planck temperature as a function
of time curvature.
2) QCS (
πF
) coefficient
The processing of the graphical representation models has characterized the
QCS (
πF
) coefficient, allowing to reproduce the centripetal dynamics moment of
the twisting effect of spacetime.
From Equation (5) we calculate:
energy equation:
23
E mc c
ψ
= ≡
The ability to calculate the energy (
2
E mc=
) through the hypermatter calcu-
lation equation (
ψ
), confirms consistency.
The study has been carried out according to a pragmatic theoretical de-
velopment:
<Idea → hypothesis → thesis → theory>
The following stages were developed: conceptual analysis; topological and
mathematical conceptual connections; analysis of phenomenological coherence
with classical theories.
Key concepts elaborated:
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• Timespace correlative manifold variety
• Quaternion algebra; hyperbolic geometry - hypersphere - Minkowski space-
time
• Quadrant Complex depiction (complex time + complex space)
• Quantum compensation spacetime (
πF
); (
F
) Fibonacci coefficient
• Hopf fibration mathematical structure
The study was subject to “Theoretical Stress Test”:
• Inconsistencies
• Theoretical inconsistencies
• Phenomenological inconsistencies
• Calculation inconsistencies
Theoretical Definitions
In the study unpublished definitions were suggested to make exposure unders-
tandable. However, the study proposes them as investigational indicators, sug-
gested to the scientific community for all changes that the research will consider
necessary:
Time
curvature
: It is understood as the discrepancy timethat occurs between
inertial frame of reference and spacetime. The theoretical research that has been
carried out has shown that the spacetime curvature is a phenomenon resulting
from the complex interaction between the three exclusive ingredients that make
up our universe: matter-space-time. The notion makes it possible to combine
and unify quantum physics with classical and relativistic theories.
Precursive
time
: defines the value of the time curvature in quantities of Planck
time or (-sec) of. It is the variable that quantifies the temporal curvature that the
mass induces on spacetime. It can be considered as the physical mediator be-
tween matter and spacetime.
Space
deflation
-
Time
deflection
: In the twist representation, the contraction
of space (deflation) is related to a reduction of time (deflection). In summary, it
takes less time to travel less space.
Chronotope
: Indicates the spatial position of an event at a specific instant time.
The chronotope establishes a condition of exclusivity, as there can be no overlap
of events having the same space and time coordinates
Mass-matter
: The equation with symbol (Ψ), enshrines a concept of complex
mass.
Timespace
manifold
: The topological variety of relationships between Min-
kowski space and the projection of the complex time variable. The variety was
developed to represent in a comprehensible form the points of conjunction be-
tween the real parameters of the event and its imaginary projection. The connec-
tion is hypothesized according to the principle of entanglement correlation.
Timespace
Quadrant
: Representation chart of all real and complex compo-
nents of the event at the time instant.
Q
.
Heat
capacity
: Planck temperature related to time curvature. It is a para-
meter for the calculation of mass-matter. It expresses a thermal characteristic of
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Journal of Applied Mathematics and Physics
matter (
mi
> 0) as a function of spacetime.
Q
.
c
.
s
.
Quantum
compensation
spacetime
(
πF
): coefficient for the calculation
of the mass-matter. It determines the angular momentum of spacetime.
(
A
0)
specific
attraction
degree
: The study shows that inertial and gravitational
mass derive from the same cause described as time curvature. It follows that a
body’s gravitational attraction force can be simply resolvedas a specific attrac-
tion degree.
Acknowledgements
I wish to thank all my trusted and intangible collaborators for all the support
that they have poured into me during these years of study, starting with: Her-
mann Minkowski who despite his extraordinary synthesis of spacetime in the
science of a thousand inexplicables is still somewhat unrecognized. To Albert
who is always willing to whisper advice to me, to David Bohm whom I consider
my putative father. I thank all the other collaborators for what they have given
me their knowledge: Max Planck, W. R. Hamilton, Ludwig Boltzmann, J. H.
Poincaré, J. Krishnamurti, Max Born. Last but not least to Giordano Bruno who
feed the flame of my knowledge.
License
The publication is licensed under Academic Free License (“AFL”) v. 3.0 a Crea-
tive Commons Attribution 4.0 International License.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this
paper.
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