In previous papers, we proposed an empirical equation for the fine-structure constant. Using this equation, we proposed a refined version of our own former empirical equations about the electromagnetic force and gravity in terms of the temperature of the cosmic microwave background. The calculated values of the temperature of the cosmic microwave background ( T c) and the gravitational constant ( G) were 2.726312 K and 6.673778 × 10 -11 m 3 ⋅kg -1 ⋅ s -2, respectively. Then, for the values of the factors 9/2 and π in our equations, we used 4.488519503 and 3.132011447, respectively. However, we could not provide a theoretical explanation for the necessity of these empirical equations. In this paper, using the redefinition method for the UNIT, we show the necessity for our empirical equations.
The symbol list is shown in Section 2. Previously, we discovered Equations (1)-(3) [
G m p 2 h c = 4.5 2 × k T c 1 kg × c 2 (1)
G m p 2 ( e 2 4 π ε 0 ) = 4.5 2 π × m e e × h c × ( C J ⋅ m × 1 kg = 1 V ⋅ m × 1 kg ) (2)
m e c 2 e × ( e 2 4 π ε 0 ) = π × k T c × ( J ⋅ m C = V ⋅ m ) (3)
We attempted to reduce the errors in the previous papers by changing the values of 4.5, π and the temperature of the cosmic microwave background (Tc) [
137.0359991 = 136.0113077 + 1 3 × 13.5 + 1 (4)
13.5 × 136.0113077 = 1836.152654 = m p m e (5)
We thought that Equations (4) and (5) should be related to the transference number [
3.13201 Ω = ( m p m e + 4 3 ) m e c 2 e c (6)
4.48852 = q m c ( m p m e + 4 3 ) m p c 2 (7)
Then, ( m p m e + 4 3 ) have the unit of ( m C ) . We can freely define the UNITs for
1 C, 1 Wb and 1 kg. Therefore, we must show the necessity for Equations (6) and (7) that these values are related to 4.5 and π. Using the redefinition method for the UNIT, we can show the necessity for our empirical equations in this report.
The remainder of the paper is organized as follows. In Section 2, we show the symbol list. In Section 3, we discuss the purpose of this report. In Section 4, we explain the redefinition method for the UNIT. In Section 5, using the redefinition method, we can refine Equations (1)-(3). Furthermore, we propose a fourth empirical equation that relates the gravitational constant and the temperature of the cosmic microwave background. In Section 6, our conclusions are described.
G: gravitational constant: 6.6743 × 10−11 (m3∙kg−1∙s−2)
(we used the compensated value 6.673778 × 10−11 in this report).
Tc: temperature of the cosmic microwave background: 2.72548 (K)
(we used the compensated value 2.726312 K in this report).
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: 1.6726219059 × 10−27 (kg)
(we used the compensated value 1.672621923 × 10−27 kg in this report).
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.
In this section, we show the purpose of this report. For convenience, Equations (5) and (6) are rewritten as follows. The units have been corrected.
3.132011447 ( V ⋅ m ) = ( m p m e + 4 3 ) m e c 2 e c ( m 2 s × J A ⋅ m = J ⋅ m C = V ⋅ m ) (8)
4.488519503 ( 1 A ⋅ m ) = q m c ( m p m e + 4 3 ) m p c 2 ( s m 2 × V ⋅ m J = V J × s m = s C ⋅ m = 1 A ⋅ m ) (9)
Then, ( m p m e + 4 3 ) have the unit of ( m 2 s ) . Next, the deviation from 4.5 and π can be explained as follows.
4.5 π × 3.132011447 4.488519503 = 0.999500154 ≐ 1 (10)
Then, we can freely define the UNITs for 1 C, 1 Wb and 1 kg. It does not seem necessary that these values are related to 4.5 and π. Unfortunately, we could not establish the background theory. Using the redefinition method for the UNIT, we can show the necessity why these values should be related to 4.5 and π.
We redefine the electric charge of one electron as follows.
e new = e × 4.48852 4.5 = 1.59809 E − 19 ( C ) (11)
We redefine the magnetic charge of one magnetic monopole as follows.
q m _ new = q m × π 3.13201 = 4.14832 E − 15 ( Wb ) (12)
Then, we can redefine the Planck constant and von Klitzing constant as follows.
h new = e new × q m _ new = h × 4.48852 4.5 × π 3.13201 = 6.62938 E − 34 (13)
R k _ new = q m _ new e _ new = R k × 4.5 4.48852 × π 3.13201 = 25958.0 ( Ω ) (14)
Then, we can redefine the wave impedance in free space, electric constant and magnetic constant.
Z 0 _ new = α × 2 h new e new 2 = 2 α × R k new = Z 0 × 4.5 4.48852 × π 3.13201 = 378.849 ( Ω ) (15)
μ 0 _ new = Z 0 _ new c = μ 0 × 4.5 4.48852 × π 3.13201 = 1.26371 E − 06 ( N ⋅ A − 2 ) (16)
ε 0 _ new = 1 Z 0 _ new × c = ε 0 × 4.48852 4.5 × 3.13201 π = 8.80466 E − 12 ( F ⋅ m − 1 ) (17)
Next, we must ensure that there are no contradictions.
c _ new = 1 ε 0 _ new μ 0 _ new = 1 ε 0 μ 0 = c = 299792458 ( m ⋅ s − 1 ) (18)
Z 0 _ new = μ 0 _ new ε 0 _ new = μ 0 ε 0 × 4.5 4.48852 × π 3.13201 = Z 0 × 4.5 4.48852 × π 3.13201 = 378.849 ( Ω ) (19)
The value in the speed of light should not be changed because the UNITs for 1 m and 1 s are unchanged. In Equation (19), the value of the impedance in free space is the same as the value in Equation (15).
By our redefinition method for the units of the particles, 1 C and 1 Wb as MKSA units are not fixed. We redefine the electric charge of one electron. Then, the number of electrons in 1 C is changed.
N 1 = 1 C e = 1 1.60218 E − 19 = 6.24151 E + 18 (20)
N 2 = 1 C e new = 1 C 4.48852 4.5 × e = 4.5 4.48852 × 6.24151 E + 18 (21)
Then, the number of magnetic monopoles in 1 Wb is changed.
N 3 = 1 Wb q m = 1 4.13567 E − 15 = 2.41799 E + 14 (22)
N 4 = 1 Wb q m _ new = 1 Wb π 3.13201 × 1 q m = 3.13201 π × 2.41799 E + 14 (23)
The number of electrons in 1 C is related to the Faraday constant and the Avogadro’s number. Therefore, we redefined the Faraday constant and the Avogadro’s number, which will be explained in the later section.
1 C new 1 C = 4.5 4.48852 (24)
1 Wb new 1 Wb = 3.13201 π (25)
However, the relationship between Equations (26) and (27) should hold.
1 J ⋅ s new = 1 C new × 1 Wb new (26)
1 Ω new = 1 Wb new 1 C new (27)
We used A∙m_new and Cnew in the later sections. Then, these units are from the microscopic redefinition. From the macroscopic redefinition, A∙m_new and Cnew should be different values.
The Compton wavelength (λ) is as follows.
λ = h m c (28)
This value should (λ) be unchanged since the UNIT for 1 m is unchanged. However, in Equation (13), the Planck constant is changed. Therefore, the UNIT for the mass of one electron and one proton should be redefined.
m e _ new = 4.48852 4.5 × π 3.13201 × m e = 9.11394 E − 31 ( kg ) (29)
m p _ new = 4.48852 4.5 × π 3.13201 × m p = 1.67346 E − 27 ( kg ) (30)
Next, we must ensure that the following equation is satisfied.
m p _ new m e _ new = 1.67346 E − 27 9.11394 E − 31 = 1836.152654 = m p m e (31)
Equations (8) and (9) can be redefined as follows.
( m p m e + 4 3 ) × m e _ new c 2 e new c = 3.13201 ( V ⋅ m ) × 4.5 4.48852 × 4.48852 4.5 × π 3.13201 = π ( V ⋅ m ) (32)
q m _ new c ( m p m e + 4 3 ) m p _ new c 2 = 4.48852 ( 1 A ⋅ m ) × π 3.13201 × 4.5 4.48852 × 3.13201 π = 4.5 ( 1 A ⋅ m ) (33)
Using Equations (32) and (33), Equations (34) and (35) can be obtained.
( m p m e + 4 3 ) × m e _ new c 2 e new c × q m _ new c ( m p m e + 4 3 ) m p _ new c 2 = m e _ new e new × q m _ new m p _ new = 4.5 × π ( V ⋅ m A ⋅ m = Ω ) (34)
q m _ new c ( m p m e + 4 3 ) m p _ new c 2 ( m p m e + 4 3 ) × m e _ new c 2 e new c = h new c 2 ( m p m e + 4 3 ) 2 × m e _ new c 2 × m p _ new c 2 = 4.5 π ( 1 A ⋅ m × V ⋅ m = s J × m 2 ) (35)
When we define the Avogadro’s number (NA) and the Faraday constant (F) as follows,
N A = 1 g m p = 5.97565 E + 23 ≈ 6.02214076 E + 23 (A)
F = e × N A = 9.57405 E + 04 ≈ 9.6485 E + 04 (B)
The redefined Avogadro’s number(NA) and the refined Faraday constant (F) are as follows,
N A _ new = 1 g m p _ new = 1 g m p × 4.5 4.48852 × 3.13201 π (C)
F new = e new × N A _ new = 1 g m p × 3.13201 π (D)
Next, we can define 1 J freely. Using arbitrary number (a), N5 and N6 can be calculated as follows.
N 5 = a × 1 J ( m p m e + 4 3 ) m e _ new c 2 ( s m 2 ) = a × 6.64398 E + 09 ( s m 2 ) (36)
N 6 = a × 1 J e _ new c ( J A ⋅ m ) = a × 2.08726 E + 10 ( J A ⋅ m ) (37)
N 6 N 5 = 2.08726 E + 10 6.64398 E + 09 ( J A ⋅ m × m 2 s = J ⋅ m C = V ⋅ m ) = π ( V ⋅ m ) (38)
The ratio should be π V∙m and constant. Next,
N 7 = a × 1 J ( m p m e + 4 3 ) m p _ new c 2 ( s m 2 ) = a × 3.61843 E + 06 ( s m 2 ) (39)
N 8 = a × 1 J q m _ new c ( J Wb ⋅ m / s = J ⋅ s Wb ⋅ m = C m ) = a × 8.04095 E + 05 ( C m ) (40)
N 7 N 8 = 3.61843 E + 06 8.04095 E + 05 = 4.5 ( s m 2 × m C = 1 A ⋅ m ) (41)
The ratio should be 4.5 and π are constant. Consequently, we can explain the necessity for the values of 4.5 and π in Equations (8) and (9).
Equations (1)-(3) are our main three equations. Then, the unit (V∙m and 1/A∙m) should be replaced. Alternatively, the redefinition of V∙m and 1/A∙m is needed in the equations.
G m p 2 h c = 4.48852 2 k T c 1 kg × c 2 × ( A ⋅ m ) (42)
G m p 2 ( e 2 4 π ε 0 ) = 4.48852 2 × 3.13201 × m e e × h c × ( A ⋅ m × V ⋅ m × 1 V ⋅ m × 1 kg = A ⋅ m × 1 kg ) (43)
m e c 2 e × ( e 2 4 π ε 0 ) = 3.13201 × k T c × ( 1 V ⋅ m × V ⋅ m = 1 ) (44)
Our fourth equation can be derived as follows. From Equation (43),
G m p 2 ( e 2 4 π ε 0 ) 2 = 4.48852 2 × 3.13201 × m e e × 2 π α × ( A ⋅ m × 1 kg ) (45)
From Equation (44),
e 2 4 π ε 0 = 3.13201 × k T c × e m e c 2 (46)
From Equations (45) and (46),
G m p 2 ( 3.13201 × k T c × e m e c 2 ) 2 = 4.48852 2 × 3.13201 × m e e × 2 π α × ( A ⋅ m × 1 kg ) (47)
Therefore,
G m p 2 ( k T c ) 2 = 4.48852 × 3.13201 × e m e c 4 × π α × ( A ⋅ m × 1 kg ) (48)
Here,
4.48852 × 3.13201 ( Ω ) = q m e × m e m p ( Ω ) (49)
In Equation (48), the unit of Ω is already considered. Therefore,
G m p 2 = q m m p c 4 × π α × ( k T c ) 2 × ( A ⋅ m × 1 kg ) (50)
Equation (50) is our fourth equation.
We define G free from 1 kg (GN) as follows.
G N = G × 1 kg (51)
The UNIT for GN is m3∙s−2. By our redefinition method, the UNITs for 1 m and 1 s are unchanged. However, when GN is a function of C or Wb, GN should be changed.
G N _ n e w ≠ G N (52)
From Equations (42) and (51),
G N m p 2 h c = 4.4885 2 k T c c 2 × ( A ⋅ m ) (53)
Therefore,
G N m p _ new 2 × ( 4.5 4.48852 × 3.13201 π ) 2 h new × 4.5 4.48852 × 3.13201 π c = 4.48852 2 k T c c 2 × A ⋅ m (54)
Therefore,
G N m p _ new 2 h new c = 4.5 2 × ( 4.48852 4.5 ) 2 × π 3.13201 × k T c c 2 × ( A ⋅ m ) (55)
Next, using GN_new and kTc_new, we proposed the following equation, which is the same formula as Equation (1). Then, the unit of 4.5 is 1/A∙m in Equation (9).
G N _ new m p _ new 2 h new c = 4.5 2 × k T c _ new c 2 × ( A ⋅ m ) new (56)
Therefore,
G N _ new G N = ( 4.5 4.48852 ) 2 × 3.13201 π × k T c _ new k T × ( A ⋅ m ) new A ⋅ m (57)
Using GN, Equation (43) is rewritten as follows.
G N m p 2 ( e 2 4 π ε 0 ) = 4.48852 2 × 3.13201 × m e e × h c × ( A ⋅ m ) (58)
Therefore,
G N m p _ new 2 = m e _ new e new h new c 4 π ε 0 _ new × 4.48852 2 × 3.13201 × 4.5 4.48852 × 4.48852 4.5 × 3.13201 π × ( A ⋅ m ) (59)
Using GN_new and A∙mnew, we propose the following equation, which is the same formula as Equation (2).
G N _ new m p _ new 2 ( e 2 4 π ε 0 ) new = 4.5 2 × π × m e _ new e _ new × h _ new c × ( A ⋅ m ) new (60)
Therefore,
G N _ new G N = 4.5 4.48852 × ( A ⋅ m ) new A ⋅ m (61)
Equation (44) is rewritten as follows.
m e c 2 e × ( e 2 4 π ε 0 ) = 3.13201 × k T c (62)
Therefore,
m e _ new c 2 e new × e new 2 4 π ε 0 _ new × 4.5 4.48852 × 3.13201 π = π × k T c (63)
Next, using kTc_new, we propose the following Equation, which is the same formula as Equation (3).
m e _ new c 2 e new × e new 2 4 π ε 0 _ new = π × k T c _ new (64)
Therefore,
k T c _ new k T c = 4.48852 4.5 × π 3.13201 (65)
Using GN, Equation (50) is rewritten as follows.
G N m p 2 = q m m p c 4 × π α × ( k T c ) 2 × ( A ⋅ m ) (66)
Therefore,
G N = q m _ new m p _ new 3 × π α × ( k T c ) 2 c 4 × ( 4.48852 4.5 ) 3 × ( π 3.13201 ) 2 × ( A ⋅ m ) (67)
Next, using GN_new, kTc_new and A∙mnew, we propose the following equation.
G N _ new = q m _ new m p _ new 3 × π α × ( k T c _ new ) 2 c 4 × ( A ⋅ m ) new (68)
Therefore,
G N _ n e w G N = ( 4.5 4.48852 ) 3 × ( 3.13201 π ) × ( k T c _ new k T c ) 2 × ( A ⋅ m ) new A ⋅ m (69)
Equations (57), (61), (65) and (69) are rewritten as follows.
G N _ new G N = ( 4.5 4.48852 ) 2 × 3.13201 π × k T c _ new k T × ( A ⋅ m ) new A ⋅ m (70)
G N _ new G N = 4.5 4.48852 × ( A ⋅ m ) new A ⋅ m (71)
k T c _ new k T c = 4.48852 4.5 × π 3.13201 (72)
G N _ new G N = ( 4.5 4.48852 ) 3 × ( 3.13201 π ) × ( k T c _ new k T c ) 2 × ( A ⋅ m ) new A ⋅ m (73)
In Equation (72), kTc is redefined and determined. Next,
( A ⋅ m ) new A ⋅ m = ( C × m / s ) new C × m / s = C new C = 4.48852 4.5 (74)
From Equations (71) and (74),
G N _ new G N = 1 (75)
Then, Equations (70) and (73) can be explained. In Equation (72), redefinition of kTc,is from redefinition of the mass of the particle as follows:
k T c _ new k T = 4.48852 4.5 × π 3.13201 = m e _ new m e (76)
However, the unit Am remains in the three equations. We proposed the following unit. We will explain the unit (GN/Am) in a future report, which maybe different from the standard approach for G and thermodynamics [
G N / A m _ new G N / A m = ( G × 1 kg A ⋅ m ) new G × 1 kg A ⋅ m = ( G N A ⋅ m ) new G N A ⋅ m = 4.5 4.48852 (77)
Equation (56) is rewritten as follows:
G N / A m _ new m p _ new 2 h new c = 4.5 2 × k T c _ new c 2 = 9.4279464618 E − 40 (78)
Equation (60) is rewritten as follows:
G N / A m _ new m p _ new 2 ( e 2 4 π ε 0 ) new = 4.5 2 × π × m e _ new e _ new × h _ new c = 8.1176747490 E − 37 (79)
Equation (64) is rewritten as follows:
m e _ new c 2 e new × e new 2 4 π ε 0 _ new = π × k T c _ new = 1.18311202 E − 22 (80)
Equation (68) is rewritten as follows:
G N / A m _ new = q m _ new m p _ new 3 × π α × ( k T c _ new ) 2 c 4 = 6.6908477020 E − 11 (81)
Consequently, our four equations can be redefined successfully.
From Equation (8),
3.132011447 ( V ⋅ m ) = ( m p m e + 4 3 ) m e c 2 e c ( m 2 s × J A ⋅ m = J ⋅ m C = V ⋅ m ) (82)
From Equation (44),
m e c 2 × e c 4 π ε 0 c = 3.132011447 ( V ⋅ m ) × k T c (83)
In Equation (83), there are no dimension mismatch problems. From Equations (81) and (82),
m e c 2 × e c 4 π ε 0 c = ( m p m e + 4 3 ) m e c 2 e c × k T c (84)
Therefore,
k T c e 2 c 4 π ε 0 = 1 ( m p m e + 4 3 ) ( s m 2 ) = 1 1837.485988 ( s m 2 ) (85)
Therefore,
( k T c e 2 c 4 π ε 0 ) new 1 1837.485988 ( s m 2 ) = 1 ( m p m e + 4 3 ) ( s m 2 ) (86)
Consequently, Equation (85) can be successfully redefined. Therefore,
( k T c h c 2 ) new = 1 6.3206454 E − 07 = 1 ( m p m e + 4 3 ) × α 2 π (87)
Furthermore, Equation (87) can also be redefined.
In this report, using the redefinition method for the UNIT, we showed the necessity for our empirical equations. The redefinition method was explained in detail. Then, we proposed four empirical equations.
G N / A m _ new m p _ new 2 h new c = 4.5 2 ( 1 A ⋅ m ) × k T c _ new c 2 = 9.4279464618 E − 40 (88)
where GN/Am = G × 1 kg/A∙m
G N / A m _ new m p _ new 2 ( e 2 4 π ε 0 ) new = 4.5 2 × π ( 1 A ⋅ m × V ⋅ m ) × m e _ new e _ new × h _ new c = 8.1176747490 E − 37 (89)
m e _ new c 2 e new × e new 2 4 π ε 0 _ new = π ( V ⋅ m ) × k T c _ new = 1.18311202 E − 22 (90)
G N / A m _ new = q m _ new m p _ new 3 × π ( V ⋅ m ) α × ( k T c _ new ) 2 c 4 = 6.6908477020 E − 11 (91)
Furthermore, we derived four important equations.
m e _ new e n e w × q m _ new m p _ new = 4.5 × π ( V ⋅ m A ⋅ m = Ω ) (92)
h new c 2 ( m p m e + 4 3 ) 2 × m e _ new c 2 × m p _ new c 2 = 4.5 π ( 1 A ⋅ m × V ⋅ m = s J × m 2 ) (93)
k T c e 2 c 4 π ε 0 = 1 1837.485988 ( s 2 m ) = 1 ( m p m e + 4 3 ) ( s 2 m ) (94)
( k T c h c 2 ) new = 1 6.3206454 E − 07 = 1 ( m p m e + 4 3 ) × α 2 π ( s 2 m ) (95)
About numerical connections considering the units, our redefinition is perfect, in which there are not any dimension mismatch problems. About the unit (GN/Am), we will explain in the future report.
The author declares no conflicts of interest regarding the publication of this paper.
Miyashita, T. (2023) Explanation of the Necessity of the Empirical Equations That Relate the Gravitational Constant and the Temperature of the CMB. Journal of Modern Physics, 14, 432-444. https://doi.org/10.4236/jmp.2023.144024