On fixing the magnitudes of gravitational constant and strong coupling constant |
|
U. V. S. Seshavatharam 1*, S. Lakshminarayana 2 |
|
1 Honorary faculty, I-SERVE, Alakapuri, Hyderabad-35,Ttelengana, India 2 Dept. of Nuclear Physics, Andhra University, Visakhapatnam-03,AP, India *Corresponding author E-mail: seshavatharam.uvs@gmail.com |
In the earlier published papers the authors suggested
that, “Magnitude of the unified force can be assumed to be equal to the
classical or astrophysical force limit. Strength of any interaction can be defined as the ratio
of the operating force magnitude and the magnitude of
.
If strength
of the Schwarzschild interaction is assumed to be unity, then weak interaction strength seems to be ‘squared
Avogadro number’ times less than the Schwarzschild interaction. The
characteristic atomic force can be represented by
”.
Thinking in this way, atomic gravitational constant can be expressed as
With current atomic
physical constants and with the assumed two new grand unified back ground
numbers
analytically
- value of
can be fixed for 10 digits and can be verified. Inverse of the strong
coupling constant can be considered as the ‘natural logarithm of square root of
ratio of gravitational and electromagnetic force ratio of down quark mass where
the operating gravitational constant is squared Avogadro number times the
gravitational constant’. Finally an attempt is made to fit and understand the
mystery of Up and Down quarks, nuclear stability, and nuclear binding energy.
For medium and heavy atomic nuclides, at the stable mass number, nuclear binding
energy seems to be equal to the sum of rest energy of
up quarks and
down quarks.
Keywords: Gravitational constant; Schwarzschild’s interaction; Astrophysical force limit; Avogadro number; Particle rest masses; Strong interaction; nuclear binding energy; and Electron’s (n2) quantum states.
1. Introduction
From final unification point of view, it is very much
essential to couple the universal gravitational constant with the elementary
physical constants. Then only the essence of unification can be understood. So
far scientists proposed several interesting models (P. A. M. Dirac 1937),
(Witten, Edward 1981), (David Gross 2005), (Abdus Salam 1981), (Salam A. &
Sivaram C 1993), (Recami E 1994), (Dine, Michael 2007), (Roberto Onofrio 2013). In this context, readers may go through the authors
published papers (U. V. S. Seshavatharam & S. Lakshminarayana 2014, 2013,
2011). By introducing two new
back ground unified numbers , in the published paper the authors
expressed their views (U. V. S. Seshavatharam & S. Lakshminarayana 2014) on
final unification and proposed three characteristic relations for connecting,
fitting and verifying the Newtonian gravitational constant in a unified approach
via the Avogadro number
In this paper, the
topics covered and reviewed are: Schwarzschild interaction strength, meaning of
strength of interaction in atomic physics, significance of Avogadro number,
fitting of the gravitational constant, muon an tau rest masses, strong coupling
constant, fine structure ratio, reduced Planck’s constant, rest masses of Up
&
Down quarks, Nucleon rest masses, rms radius of proton, nuclear charge radius, nuclear stability, nuclear binding energy and unified atomic mass unit. Important points can be expressed as follows.
1) In any inverse square law of force, system is sustained only by means of the central attractive force and it is the root cause of revolving body’s angular momentum. If it is confirmed that, revolving body’s angular momentum is discrete, then it is a clear indication of the discrete nature of the central force acting on the revolving body. If one is willing to think in this direction, the historical mystery of Bohr’s discrete atomic structure and discrete angular momentum can be understood.
2) Note that, as per the basic
concepts of final unification, there exists a fundamental unified force from
which all the observed forces emerged. If so, magnitude of the unified force
can be assumed to be equal to the astrophysical force limit. Note that, magnitude of the radial
inward force acting on any black hole surface (U. V. S. Seshavatharam & S.
Lakshminarayana 2014) is of the order of
.
2. The classical limits of force and power
To unify cosmology, quantum mechanics and
the four observed fundamental cosmological interactions certainly a ‘unified
force’ is required. In this connection can
be considered as the classical force or astrophysical
force limit. Similarly
can be considered as the classical power
limit. If it is true that
and
are fundamental physical constants in
physics, then
and
can also be considered as
fundamental compound physical constants. These classical limits are more
powerful than the Uncertainty limit. Without
considering the current notion of black hole physics, Schwarzschild radius of
black hole (Roger Penrose 1996), (Subrahmanyan Chandrasekhar 1983) can be understood
with the characteristic astrophysical limiting force of magnitude
. Note that by considering
the famous
Planck mass can be obtained very easily.
2.1.
Simple applications of
a) Magnitude of force of
attraction or repulsion between any two charged particles never exceeds.
b) Magnitude of gravitational
force of attraction between any two massive bodies never exceeds.
c) Magnitude of mechanical
force on a revolving/rotating body never exceeds.
d) Magnitude of
electromagnetic force on a revolving body never exceeds.
2.2.
Simple applications of
a) Mechanical power never
exceeds
b) Electromagnetic power
never exceeds
c) Thermal radiation power
never exceeds
d) Gravitational radiation
power never exceeds
3. Understanding the role of in black hole formation and
Planck mass generation
3.1. Schwarzschild radius of a black hole
The four basic physical properties of a rotating black
hole are its mass, size, angular velocity and temperature. Without going deep
into the mathematics of black hole physics in this section an attempt is made
to understand the Schwarzschild radius of a black hole. In all directions, if a
force of magnitude acts on the mass-energy
content of the assumed celestial body it approaches a minimum radius of
in the following way. Origin of the force
may be due to self-weight or internal attraction or external
compression or something else.
(1)
If no force (of zero magnitude) acts on the mass content
of the assumed massive body, its radius
becomes infinity. With reference to the average magnitude of
, the presently believed Schwarzschild
radius can be obtained as
(2)
This proposal is very simple and seems to be different from the existing concepts and may be a unified form of the Newton’s law of gravity, Special theory of relativity and General theory of relativity.
3.2 To derive the Planck mass
So far no theoretical model proposed a derivation for the Planck mass. To derive the Planck mass the following two conditions can be given a chance.
Assuming that gravitational force of attraction between
two Planck particles of mass separated by a
minimum distance (rmin) be,
(3)
With reference to wave mechanics, let
(4)
Here, represents the
wavelength associated with the Planck mass. With these two assumed conditions
Planck mass can be obtained as follows.
(5)
3.3. Understanding the strength of any interaction
From the above relations it is reasonable to say that,
1) If it is true that and
are
fundamental physical constants, then
can be considered as a fundamental
compound constant related to a characteristic limiting force.
2) Black holes are the ultimate state of matter’s geometric structure.
3) Magnitude of the operating force
at the black hole surface is the order of.
4) Gravitational interaction taking place at black holes can be called as ‘Schwarzschild interaction’.
5) Strength of ‘Schwarzschild interaction’ can be assumed to be unity.
6) Strength of any other interaction
can be defined as the ratio of operating force magnitude and the classical or
astrophysical force magnitude.
7) If one is willing to represent the
magnitude of the operating force as a fraction of i.e
,
where
, then
(6)
If is very small,
becomes very large. In this
way,
can be called as the strength of
interaction. Clearly speaking, strength of any interaction is
times less than the ‘Schwarzschild interaction’ and effective
becomes
.
4. Basic concepts and relations final unification
The following concepts and relations can be given a chance in final unification program.
1) With reference to the elementary charge and with mass similar to the Planck mass, a new mass unit can be constructed in the following way.
(7)
It can be called as the Stoney mass (G. J. Stoney,
1881). It is well known that play a vital role in
fundamental physics. With these 3 constants, space-time curvature concepts at a
charged particle surface can be studied. It was first introduced by the physicist
George Johnstone Stoney. He is most famous for
introducing the term ‘electron’ as the
‘fundamental unit quantity of electricity’. In unification program, with
this mass unit and with a suitable proportionality ratio- characteristic mass
of any elementary charge can be generated.
2) Avogadro number is an absolute number and it is having no units like ‘per mole’.
3)
Atomic
interaction strength is times less than the Schwarzschild interaction
and hence atomic gravitational constant can be expressed as:
(8)
4)
Similar to the
classical force limit , in atomic system there exists a characteristic
force of magnitude:
(9)
5) Independent of system of units and without considering the Avogadro number, unified atomic mass unit (P.J. Mohr et al 2010), (B. Andreas et al 2011), (B P Leonard 2007),(K.A. Olive et al 2014) can be fitted as follows.
(10)
Where is the
unified atomic mass unit and
is the average
binding energy per nucleon. If
obtained
magnitude of
Thus it can be suggested that,
accuracy of
depends only on the accurate
‘average binding energy per nucleon’.
4.1.
Semi empirical applications of
There exist two new numbers they
can be called as the ‘primordial unified back ground numbers’. They can also be
called as the ‘back ground analytical numbers’ using by which micro-macro
physical constants can be interlinked qualitatively and quantitatively.
Application-1: Rest masses of electron and proton
Electron rest mass can be expressed in the following way.
(11)
With proton rest
mass can be expressed in the following way.
(12)
Thus,
(13)
Application-2: Rest masses of muon and tau
(14)
Where can be called
as the electron mass index. It can be estimated as:
(15)
With this number,
electron, muon and tau rest masses can be fitted with the semi empirical
relation.
(16)
Where Obtained rest
energies are 0.511 MeV, 105.95 MeV and 1777.4 MeV respectively (K.A. Olive et al 2014). New heavy charged lepton at
may be predicted close to 42262 MeV.
Application-3: The reduced Planck’s constant and the rms radius of proton
From above relations,
(16)
If so, Reduced Planck’s constant can be expressed in the following way.
(17)
Characteristic nuclear radii like rms radius of proton (P.J. Mohr et al 2010), (Geiger H & Marsden. E 1909), (Michael O. Distler et al 2011), (Roberto Onofrio 2013) nuclear charge radius etc can be expressed in the following way.
(18)
If so, it is possible to show that,
(19)
Application-4: To fit and verify the gravitational constant
In astronomy, the only one available characteristic
empirical physical constant is the gravitational constant. Its value has been
measured in the lab only within a range of 1 cm to a few meters. Until one
measures the value of the gravitational constant with microscopic physical
constants, the debate of final unification cannot be stopped up. In this context, G. Rosi et al say (G.
Rosi et al 2014): “There
is no definitive relationship between and
the other fundamental constants, and there is no theoretical prediction for its
value, against which to test experimental results. Improving the precision with
which we know
has not only a pure
metrological interest, but is also important because of the key role that
has in theories of gravitation,
cosmology, particle physics and astrophysics and in geophysical models”. In general, ‘Unification’ means:
a) Understanding the origin of the rest mass of atomic elementary particles.
b) Finding and understanding the critical compositeness of the elementary physical constants.
c) Minimizing the number of elementary physical constants.
d) Merging different branches of physics with possible and suitable physical concepts.
Considering the proposed concepts and relations
accurate values of Gravitational constant (L.L.
Williams 2009), (George T Gillies 1997), (J Stuhler et al 2003), (Terry Quinn 2013), (J. B. Fixler et al 2007), (Brandenburg,
J.E 1992), (Jun Luo and Zhong-Kun Hu 2000), (St. Schlamminger et al 2002) and Avogadro number can be estimated from elementary
atomic physical constants. For the time being (i.e until a perfect model is
developed), if one is willing to consider the revolving electron’s angular
momentum as a compound physical constant and depends on the proton-electron
rest masses, characteristic nuclear charge radius and the proposed discrete
force it paves a path for coupling and interconnecting the
micro-macro elementary physical constants in a consistent manner. Thus it is
possible to couple Avogadro number and Gravitational constant in the following
way.
(20)
(21)
(22)
(23)
Thus relations (20, 21 and 22) can be considered as the
3 characteristic semi empirical unified relations. This assumed value of may not be absolute but can be given
some consideration in unification program for further analysis This entire
procedure depends on the two proposed new numbers
and
needs further research. So far there is no verifying procedure for the measured
or estimated magnitude of
with this
kind of procedure, like other physical constants, value of
can be fixed for 10 digits.
Application-5: Strong coupling constant, Up and Down quarks and nuclear binding energy
Inverse of the strong coupling constant can be fitted as follows (A.V. Manohar and C.T. Sachrajdahttp 2014):
(24)
Now Down quark mass can be expressed as follows (Halzen, F.; Martin, A. D 1984).
(25)
(26)
Ratio of Up and down quark masses can be guessed as follows.
(27)
Thus Up quark mass can be fitted as follows.
(28)
Note that, these proposed Up and Down quark masses are roughly 2.20 times higher than the current estimates and their proposed mass ratio is matching with the current estimates. In a super symmetric approach, neutron and proton mass difference can be expressed as follows.
(29)
Where can be
considered as the super symmetric fermion-boson mass ratio (U. V. S.
Seshavatharam & S. Lakshminarayana 2010, 2011).
With Up and Down quark masses nuclear binding energy (Chowdhury, P.R. et al 2005), (W.D. Myers & W.J. Swiatecki 1994), (G. Audi & A.H. Wapstra 1993) can be fitted as follows.
Step-1: To fit the stable mass number of
(30)
Step-2: To fit the nuclear binding energy at stable
mass number of
(31)
Where,
See table-1 for the estimated nuclear binding energy near to the stable mass number. Considering even-odd corrections on the estimated stable mass number and
With further research, data accuracy can be improved. From the data it is very clear to say that:
1) At the stable mass number, nuclear
binding energy seems to be equal to the sum of rest energy of up quarks and
down quarks.
2) As per the quark theory proton constitutes two Up quarks and one Down quark. Hence it can be guessed that, near to stable mass number, nuclear binding energy seems to depend only on the proton number.
Step-3: To fit the nuclear binding energy above and
below the stable mass number of
(32)
Table 1: To fit the nuclear binding energy
near to stable mass number of
Proton number |
Estimated stable mass number |
Estimated value of |
Binding energy in MeV (near to stable mass number) |
2 |
4 |
0.6368 |
25.3 |
3 |
6 |
0.6813 |
40.5 |
4 |
8 |
0.7148 |
56.7 |
5 |
10 |
0.7418 |
73.5 |
6 |
12 |
0.7647 |
91.0 |
7 |
14 |
0.7846 |
108.9 |
8 |
16 |
0.8023 |
127.3 |
9 |
18 |
0.8182 |
146.0 |
10 |
21 |
0.8327 |
165.1 |
11 |
23 |
0.8460 |
184.5 |
12 |
25 |
0.8584 |
204.2 |
13 |
27 |
0.8699 |
224.2 |
14 |
29 |
0.8807 |
244.5 |
15 |
31 |
0.8909 |
265.0 |
16 |
34 |
0.9005 |
285.7 |
17 |
36 |
0.9097 |
306.6 |
18 |
38 |
0.9184 |
327.8 |
19 |
40 |
0.9267 |
349.1 |
20 |
42 |
0.9347 |
370.6 |
21 |
45 |
0.9423 |
392.4 |
22 |
47 |
0.9496 |
414.2 |
23 |
49 |
0.9567 |
436.3 |
24 |
52 |
0.9635 |
458.5 |
25 |
54 |
0.9701 |
480.9 |
26 |
56 |
0.9764 |
503.4 |
27 |
58 |
0.9826 |
526.0 |
28 |
61 |
0.9886 |
548.8 |
29 |
63 |
0.9944 |
571.8 |
30 |
66 |
1.0000 |
594.8 |
31 |
68 |
1.0000 |
614.7 |
32 |
70 |
1.0000 |
634.5 |
33 |
73 |
1.0000 |
654.3 |
34 |
75 |
1.0000 |
674.2 |
35 |
78 |
1.0000 |
694.0 |
36 |
80 |
1.0000 |
713.8 |
37 |
82 |
1.0000 |
733.6 |
38 |
85 |
1.0000 |
753.5 |
39 |
87 |
1.0000 |
773.3 |
40 |
90 |
1.0000 |
793.1 |
41 |
92 |
1.0000 |
812.9 |
42 |
95 |
1.0000 |
832.8 |
43 |
97 |
1.0000 |
852.6 |
44 |
100 |
1.0000 |
872.4 |
45 |
102 |
1.0000 |
892.3 |
46 |
105 |
1.0000 |
912.1 |
47 |
108 |
1.0000 |
931.9 |
48 |
110 |
1.0000 |
951.7 |
49 |
113 |
1.0000 |
971.6 |
50 |
115 |
1.0000 |
991.4 |
51 |
118 |
1.0000 |
1011.2 |
52 |
121 |
1.0000 |
1031.1 |
53 |
123 |
1.0000 |
1050.9 |
54 |
126 |
1.0000 |
1070.7 |
55 |
129 |
1.0000 |
1090.5 |
56 |
131 |
1.0000 |
1110.4 |
57 |
134 |
1.0000 |
1130.2 |
58 |
137 |
1.0000 |
1150.0 |
59 |
139 |
1.0000 |
1169.9 |
60 |
142 |
1.0000 |
1189.7 |
61 |
145 |
1.0000 |
1209.5 |
62 |
148 |
1.0000 |
1229.3 |
63 |
150 |
1.0000 |
1249.2 |
64 |
153 |
1.0000 |
1269.0 |
65 |
156 |
1.0000 |
1288.8 |
66 |
159 |
1.0000 |
1308.7 |
67 |
162 |
1.0000 |
1328.5 |
68 |
164 |
1.0000 |
1348.3 |
69 |
167 |
1.0000 |
1368.1 |
70 |
170 |
1.0000 |
1388.0 |
71 |
173 |
1.0000 |
1407.8 |
72 |
176 |
1.0000 |
1427.6 |
73 |
179 |
1.0000 |
1447.4 |
74 |
182 |
1.0000 |
1467.3 |
75 |
184 |
1.0000 |
1487.1 |
76 |
187 |
1.0000 |
1506.9 |
77 |
190 |
1.0000 |
1526.8 |
78 |
193 |
1.0000 |
1546.6 |
79 |
196 |
1.0000 |
1566.4 |
80 |
199 |
1.0000 |
1586.2 |
81 |
202 |
1.0000 |
1606.1 |
82 |
205 |
1.0000 |
1625.9 |
83 |
208 |
1.0000 |
1645.7 |
84 |
211 |
1.0000 |
1665.6 |
85 |
214 |
1.0000 |
1685.4 |
86 |
217 |
1.0000 |
1705.2 |
87 |
220 |
1.0000 |
1725.0 |
88 |
223 |
1.0000 |
1744.9 |
89 |
227 |
1.0000 |
1764.7 |
90 |
230 |
1.0000 |
1784.5 |
91 |
233 |
1.0000 |
1804.4 |
92 |
236 |
1.0000 |
1824.2 |
93 |
239 |
1.0000 |
1844.0 |
94 |
242 |
1.0000 |
1863.8 |
95 |
245 |
1.0000 |
1883.7 |
96 |
248 |
1.0000 |
1903.5 |
97 |
252 |
1.0000 |
1923.3 |
98 |
255 |
1.0000 |
1943.1 |
99 |
258 |
1.0000 |
1963.0 |
100 |
261 |
1.0000 |
1982.8 |
5. To understand the discrete behavior and the total energy of electron in hydrogen atom
Step-1: To understand the discrete behavior
From Bohr’s theory of
Hydrogen atom (N.Bohr 1913) maximum number of electrons that can be
accommodated in any principal quantum shell are this proposal can be
reinterpreted as follows: In Hydrogen atom, in
principal quantum shell, electron
can exist in
different quantum states. It
can be understood as follows. Guess that currently believed s- shell is the
basic unit of all quantum shells and it constitutes a maximum of 2 numbers of
electrons. With reference to the current concept of
electrons,
there can exit
number of s-shells.
If one s-shell represents on quantum state, then with reference to
number of s-shells, one can expect
number of different quantum states with
different energy levels.
Step-2: To understand the
potential energy of different states
Let potential energy of electron at any one quantum state be:
(33)
Where is the
distance between electron and
proton corresponding to
quantum state.
Potential energy of possible
quantum states can be:
(34)
Based
on the Virial theorem (Celso L. Ladera et
al 2010) in a central force field,
quantitatively kinetic energy is half the potential energy. Following this
idea, total kinetic energy of electron for quantum
state can be:
(35)
Thus, total energy of
electron for quantum states can be:
(36)
If so, potential energy of electron at any one quantum state can be:
(37)
Kinetic energy of electron at any one quantum state can be:
(38)
Total energy of electron at any one quantum state can be:
(39)
Step-3: To understand the emitted photon energy
With reference to the jumping nature of electron from one quantum state to another quantum sate, emitted photon energy can be:
(40)
Where
6. Conclusion
So far no model succeeded in coupling and understanding
the unified concepts of gravity, electromagnetic and strong interactions. Based
on the proposed concepts and accurate relations and with further research and
analysis, different models of final unification can be developed with different
proportionality ratios and finally a unified model can be standardized. The absolute magnitude of can be fixed and uncertainty in its current recommended
magnitude can be minimized
Acknowledgements
The first author is indebted to professor K. V. Krishna Murthy, Chairman, Institute of Scientific Research on Vedas (I-SERVE), Hyderabad, India and Shri K. V. R. S. Murthy, former scientist IICT (CSIR) Govt. of India, Director, Research and Development, I-SERVE, for their valuable guidance and great support in developing this subject.
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