Algorithms for Empirical Equations in Terms of the Cosmic Microwave Background Temperature ()
1. Introduction
The symbol list is shown in Section 2. Previously, we described Equations (1), (2) and (3) in terms of the cosmic microwave background (CMB) temperature [1]-[5].
(1)
(2)
(3)
We then derived an empirical equation for the fine-structure constant [6].
(4)
(5)
Equations (4) and (5) are related to the transference number [7] [8]. Next, we propose the following values as deviations of the values of 9/2 and π [8] [9].
(6)
(7)
Then,
has units of
. By redefining the Avogadro number and the Faraday constant, these values can be adjusted back to 9/2 and π [9].
(8)
(9)
Every equation can be explained in terms of the Compton length of an electron (λe), the Compton length of a proton (λp) and α [10] Furthermore, every equation can be explained in terms of the Avogadro number and the number of electrons at 1 C [11]. We showed that every equation can be described in terms of the Planck constant [12]. Then, the ratio of the gravitational force to the electric force can be uniquely determined with the assumption of minimum mass. Using the correspondence principle with the thermodynamic principles in solid-state ionics [13], we propose a canonical ensemble to explain the concept of the minimum mass. However, there appear to be dimension mismatch problems. In this report, we present the algorithms for these equations. Then, every dimension mismatch problem can be solved. Quantum mechanics [14] and Gravity [15] have been tried to explain thermodynamically. Our motivation is to use the thermodynamic principles in the area of solid-state ionics discovered by ourselves. However, we can only discuss about the numerical connections in the present.
The remainder of this paper is organized as follows. In Section 2, we present the list of symbols used in our derivations. In Section 3, we propose algorithms to explain these equations in detail. Then, every dimension mismatch problem can be solved. In Section 4, using algorithms, we explain our main equations. In Section 5, our conclusions are described.
2. Symbol List
2.1. MKSA Units1
G: gravitational constant: 6.6743 × 10−11 (m3∙kg−1∙s−2)
(we used the compensated value 6.673778 × 10−11 in this study)
Tc: CMB temperature: 2.72548 (K)
(we used the compensated value 2.726312 K in this study)
k: Boltzmann constant: 1.380649 × 10−23 (J∙K−1)
c: speed of light: 299792458 (m/s)
h: Planck constant: 6.62607015 × 10−34 (J s)
ε0: Electric constant: 8.8541878128 × 10−12 (N⋅m2⋅C−2)
μ0: magnetic constant: 1.25663706212×10−6 (N∙A−2)
e: electric charge of one electron: −1.602176634 × 10−19 (C)
qm: magnetic charge of one magnetic monopole: 4.13566770 × 10−15 (Wb)
(this value is only a theoretical value, qm = h/e)
mp: rest mass of a proton: 1.672621923 × 10−27 (kg)
(We used the compensated value of 1.6726219059 × 10−27 kg to explain 137.0359990841)
me: rest mass of an electron: 9.1093837 × 10−31 (kg)
Rk: von Klitzing constant: 25812.80745 (Ω)
Z0: wave impedance in free space: 376.730313668 (Ω)
α: fine-structure constant: 1/137.035999081
λp: Compton wavelength of a proton: 1.32141 × 10−15 (m)
λe: Compton wavelength of an electron: 2.4263102367 × 10−12 (m)
2.2. Symbol List after Redefinition
(10)
(11)
(12)
(13)
Equation (13) can be rewritten as follows:
(14)
(15)
Equation (15) can be rewritten as follows:
(16)
(17)
(18)
(19)
The Compton wavelength (λ) is as follows:
(20)
This value (λ) should be unchanged since the unit for 1 m is unchanged. However, in Equation (12), Planck’s constant is changed. Therefore, the units for the masses of one electron and one proton need to be redefined.
(21)
(22)
From the dimensional analysis in a previous report [9], the following is obtained:
(23)
To simplify the calculation, GN is defined as follows:
(24)
Now, the value of GN remains unchanged. However, the value of GN_new should change [9] as follows:
(25)
2.3. Symbol List in Terms of the Compton Length of an Electron (λe),
the Compton Length of a Proton (λp) and α
The following equations were proposed in a previous study [10]:
(26)
(27)
(28)
(29)
(30)
(31)
2.4. Symbol List in Terms of the Avogadro Number and the Number
of Electrons in 1 C
The Avogadro number (NA) is 6.02214076 × 1023. This value is related to the following value.
(32)
Using the redefined values, the new definition of the Avogadro number (NA_new) is as follows:
(33)
The number of electrons in 1 C (Ne) is as follows:
(34)
Using the redefined values, the new definition of the number of electrons in 1 C (Ne_new) is as follows:
(35)
The following equations were proposed in a previous study [11]:
(36)
(37)
where mp/me (=1836.1526) is not changed after redefinition.
(38)
(39)
(40)
(41)
(42)
2.5. Symbol List for the Advanced Expressions for kTc and GN
Furthermore, we propose the following four equations [11]:
(43)
(44)
In Equations 43 and 44, 2π(1) is dimensionless. For G, there are two equations, as follows:
(45)
(46)
In Equations (45) and (46), 4π(1) and 4.5(1) are dimensionless. Equations (45) and (46) are corrected from a previous report [12]. The details are shown in a later section. There are no dimension mismatch problems.
2.4. Symbol List when the Planck Constant is Changed to 1 Js
When we define the Planck constant as (1 Js), the following equations can be used:
(47)
where hnew(1) (=6.629383E−34) is dimensionless. cgeneral and 1mgeneral are the values for c and 1 m, respectively, after Planck’s constant is changed. Thus, the unit of the meter should be changed. Importantly, Equation (47) does not indicate a change in the light speed.
(48)
(49)
(50)
(51)
(52)
(53)
where 1 Cgeneral, 1 Wbgeneral, 1 kggeneral, egeneral, qm_general, me_general, mp_general, Tc_general and GN_gerneral are the values for 1 C, 1 Wb, 1 kg, e, qm, me, mp, Tc and GN, respectively, when the Planck constant is changed to 1 Js.
The minimum mass (Mmin) is as follows:
(54)
The ratio between the mass of an electron and the minimum mass is as follows:
(55)
The mass ratio of a proton to its minimum mass is as follows:
(56)
For convenience, Equation (1) is rewritten as follows:
(57)
The equation for a fine structure constant is as follows:
(58)
The ratio of the gravitational force to the electric force is as follows:
(59)
Equation (59) is corrected from a previous report [12]. The details are shown in a later section. There are no dimension mismatch problems. From Equation (52),
(60)
Next,
(61)
Using Equations (59), (60) and (61),
(62)
Then, we can explain the ratio of the gravitational force to the electric force. Furthermore, there are no dimension mismatch problems.
3. Methods
In this section, we present the algorithms for these equations. First, we note the final dimension mismatch problem. Next, we propose the first list from the symbol list written in Section 2.2. Using the first list, we can present the second list.
3.1. Solution for the Final Dimension Mismatch Problem
For convenience, Equation (31) is rewritten as follows:
(63)
Therefore,
(64)
For convenience, Equation (31) is rewritten as follows:
(65)
Therefore,
(66)
From Equations (64) and (66),
(67)
In Equation 67, to convert back to the MKSA unit, 4.5 should be changed from 4.5 to 4.8852. Therefore,
(68)
where 4.5(1) is dimensionless. When converting back to the MKSA unit, the unit of 1C should be compensated.
(69)
4.5 in Equations (1) and (2) is dimensionless. Gnew (after redefinition of G) is as follows:
(70)
However, Gnew is not useful rather than GN_new because we must consider the compensation of the unit 1 kg in every equation.
3.2. Algorithms for Our Empirical Equations
3.2.1. Algorithms for Making the First List
Equations (71) and (72) are important for making the first list. Using Equations (26) - (31), the following list can be obtained.
(71)
(72)
(73)
(74)
(75)
(76)
(77)
(78)
(79)
With respect to Equation (78), we find two equations.
(80)
(81)
With respect to Equation (79), we find two equations.
(82)
(83)
3.2.2. The Algorithms for Making the Second List
List 2 can be obtained in this way. In the previous report, in Equation 77, we used the Planck constant h to be 1. Then, cgeneral can be obtained, and every value can be calculated. This means that we can obtain the second list.
(84)
Using Equations (73) and (84),
(85)
Using Equations (74) and (84),
(86)
Using Equations (75) and (84),
(87)
Using Equations (76) and (84),
(88)
Regarding Equations (77) and (84),
(89)
Then, kTc/α is unchanged. The reason is that in Equation (81), ec and mpc2 are unchanged.
(90)
From Equation (82),
(91)
From Equations (53) and (84),
(92)
From Equations (91) and (92), GN/αc is a constant, and GN_general can be obtained. Consequently, we can obtain the second list when c is changed in the first list. These calculated values are the same as the values shown in Section 2.4. This means that when an arbitrary value of the light speed is used, the values of me, mp, e, qm, h, kTc/α and GN/αc in List 1 can be calculated. Another examples are shown in Appendix.
4. Results
4.1. Unchanged Values
After the first list is obtained, the following values are unchanged when c is changed to obtain the second list. These values can be obtained directly via the first list; thus, when c is changed, the values of the equations cannot be changed.
(93)
(94)
(95)
(96)
(97)
4.2. Another Possible List
We note that another list is possible.
(98)
Therefore,
(99)
Therefore,
(100)
Here,
(101)
S0,
(102)
From Equations (100) and (102),
(103)
Therefore,
(104)
Therefore,
(105)
Therefore,
(106)
When we use
instead of
,
(107)
Consequently, another first list is possible. However, the calculated values of G and kTc are much different from the observed values.
4.3. Explanation of Our Main Three Equations
From this section onward, we ensure that our second list is correct in our three equations. Strictly speaking, me should be written as me_general. However, we omit the subscript “general” to avoid unnecessary notational complexity.
4.3.1. Explanation of Our First Equation
For convenience, Equation (1) is rewritten as follows:
(108)
Therefore, the following can be applied:
(109)
where 4.5(1) is dimensionless. Therefore,
(110)
,
,
and
are constant when c is changed. Therefore, our second list is correct.
4.3.2. Explanation of Our Second Equation
For convenience, Equation (2) is rewritten as follows:
(111)
Therefore, the following can be obtained:
(112)
The equation for a fine structure constant is as follows:
(113)
Therefore,
(114)
Therefore,
(115)
When c is changed,
,
,
,
, and ec are constant. Therefore, our second list is correct.
4.3.3. Explanation of Our Third Equation
For convenience, Equation (3) is rewritten as follows:
(116)
The equation for a fine structure constant is as follows:
(117)
Therefore,
(118)
Therefore,
(119)
Then,
,
,
, and ec are constant when c is changed. Therefore, our second list is correct.
4.4. Mathematical Proof for the Ratio of the Gravitational Force to
the Electric Force
Equation (2) is rewritten as follows:
(120)
The fine-structure constant is defined as follows:
(121)
Therefore, the following can be obtained:
(122)
Therefore,
(123)
Therefore,
(124)
Consequently, the ratio of the gravitational force to the electric force is unchanged in the second list.
4.5. The Theoretical Meaning of the Second List
The theoretical meaning of the second list is as follows. Einstein discovered the following equation:
(125)
When we know the unit of work (J) and when we do not know the unit of mass (kg), after the light speed (m/s) is measured, the unit of mass (kg) is uniquely determined.
5. Conclusions
First, we report the final dimension mismatch problem. The value 4.5 in Equations (1) and (2) is dimensionless. When converting back to the MKSA unit, the unit of 1 C should be compensated. The correct equation is as follows:
(126)
The main point is that the dimensionless value of 4.5(1) is not related to the unit (1/Am). Thus, there are no dimension mismatch problems. Furthermore, in this report, we describe the algorithms used to explain these equations in detail. Using Equations (26) - (31), the following first list can be obtained.
(127)
(128)
(129)
(130)
(131)
(132)
(133)
Next, using the arbitrary value of c, me, mp, e, qm, h,
and
are determined as the second list. When these values are used, our three main equations are correct. Furthermore, the ratio of the gravitational force to the electric force is unchanged in the second list.
(134)
The theoretical meaning is highly related to the thermodynamic principles in solid-state ionics, which will be published in future reports.
Appendix A
In this appendix, using the arbitrary value of the light speed, we show another examples for the second list.
(A1)
where carbitrary and 1 marbitrary are the values for c and 1 m when the arbitrary value of light speed is used, respectively,
(A2)
(A3)
(A4)
(A5)
(A6)
(A7)
(A8)
where 1 Carbitrary, 1 Wbarbitrary, 1 kgarbitrary, earbitrary, qm_arbitrary, me_arbitrary, mp_arbitrary, Tc_arbitrary and GN_arbitrary are the values for 1 C, 1 Wb, 1 kg, e, qm, me, mp, Tc and GN, respectively, when the arbitrary value of light speed is used.
From Equations 50 and A6,
(A9)
Equations A2 equals to Equations A9. From Equations 48 and A6,
(A10)
Equations A3 equals to Equations A10. From Equations 49 and A6,
(A11)
Equations A3 equals to Equations A10. From Equations 51 and A6,
(A12)
Equations A5 equals to Equations A12. From Equations 52 and A6,
(A13)
Equations A7 equals to Equations A13. From Equations 53 and A6,
(A14)
Equations A8 equals to Equations A14. Consequently, we can show other examples.
NOTES
1These values were obtained from Wikipedia.