G: gravitational constant: 6.6743 × 10⁻¹¹ (m³∙kg⁻¹∙s⁻²)

(we used the compensated value 6.673778 × 10⁻¹¹ in this study)

T_c: 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)

q_m: magnetic charge of one magnetic monopole: 4.13566770 × 10⁻¹⁵ (Wb)

(this value is only a theoretical value, q_m = h/e)

m_p: 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)

m_e: 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

λ_p: Compton wavelength of a proton: 1.32141 × 10⁻¹⁵ (m)

λ_e: Compton wavelength of an electron: 2.4263102367 × 10⁻¹² (m)

$e_{new}=e\times \frac{4.48852}{4.5}=1.59809\text{E}-19\left(\text{C}\right)$ (10)

$q_{m\_}{}_{new}=q_m\times \frac{\pi }{3.13201}=4.14832\text{E}-15\left(\text{Wb}\right)$ (11)

$h_{new}=e_{new}\times q_{m\_new}=h\times \frac{4.48852}{4.5}\times \frac{\pi }{3.13201}=6.62938E-34\left(\text{J}\cdot \text{s}\right)$ (12)

$R{k}_{\_}{}_{new}=\frac{q_{m\_new}}{e_{\_new}}=Rk\times \frac{4.5}{4.48852}\times \frac{\pi }{3.13201}=25958.0\left(\Omega \right)$ (13)

Equation (13) can be rewritten as follows:

$R{k}_{new}=4.5\left(\frac{\text{1}}{\text{A}\cdot \text{m}}\right)\times \pi \left(\text{V}\cdot \text{m}\right)\times \frac{m_p}{m_e}=25957.9966027\left(\Omega \right)$ (14)

$Z_{0\_new}=\alpha \times \frac{2h_{new}}{e_{new}{}^2}=2\alpha \times R{k}_{new}=Z_0\times \frac{4.5}{4.48852}\times \frac{\pi }{3.13201}=378.849\left(\Omega \right)$ (15)

Equation (15) can be rewritten as follows:

$Z_{0\_new}=4.5\left(\frac{\text{1}}{\text{A}\cdot \text{m}}\right)\times \pi \left(\text{V}\cdot \text{m}\right)\times 2\alpha \times \frac{m_p}{m_e}=378.8493064\left(\Omega \right)$ (16)

${\mu }_{0\_new}=\frac{Z_{0\_new}}{c}={\mu }_0\times \frac{4.5}{4.48852}\times \frac{\pi }{3.13201}=1.26371\text{E}-06\left(\text{N}\cdot {\text{A}}^{-2}\right)$ (17)

${\epsilon }_{0\_new}=\frac{1}{Z_{0\_new}\times c}={\epsilon }_0\times \frac{4.48852}{4.5}\times \frac{3.13201}{\pi }=8.80466\text{E}-12\left(\text{F}\cdot {\text{m}}^{-1}\right)$ (18)

$c_{\_new}=\frac{1}{\sqrt{{\epsilon }_{0\_new}{\mu }_{0\_new}}}=\frac{1}{\sqrt{{\epsilon }_0{\mu }_0}}=c=299792458\left(\text{m}\cdot {\text{s}}^{-1}\right)$ (19)

The Compton wavelength (λ) is as follows:

$\lambda =\frac{h}{mc}$ (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.

$m_{e\_new}=\frac{4.48852}{4.5}\times \frac{\pi }{3.13201}\times m_e=9.11394\text{E}-31\left(\text{kg}\right)$ (21)

$m_{p\_new}=\frac{4.48852}{4.5}\times \frac{\pi }{3.13201}\times m_p=1.67346\text{E}-27\left(\text{kg}\right)$ (22)

From the dimensional analysis in a previous report [9] , the following is obtained:

$k{T}_{c\_new}=\frac{4.48852}{4.5}\times \frac{\pi }{3.13201}\times k{T}_c=3.7659625\text{E}-23\left(\text{J}\right)$ (23)

To simplify the calculation, G_N is defined as follows:

$G_N=G\times 1\text{kg}\left({\text{m}}^{\text{3}}\cdot {\text{s}}^{-2}\right)=6.673778\text{E}-11\left({\text{m}}^{\text{3}}\cdot {\text{s}}^{-2}\right)$ (24)

Now, the value of G_N remains unchanged. However, the value of G_{N_new} should change [9] as follows:

$G_{N\_new}=G_N\times \frac{4.5}{4.48852}\left(\frac{{\text{m}}^{\text{3}}}{{\text{s}}^{\text{2}}}\right)=G_N\times \frac{e}{e_{new}}=6.69084770\text{E}-11\left(\frac{{\text{m}}^{\text{3}}}{{\text{s}}^{\text{2}}}\right)$ (25)

The following equations were proposed in a previous study [10] :

$\begin{array}{l}{m}_{e\_new}{c}^2\times {\left(\frac{m_p}{m_e}+\frac{4}{3}\right)}^2\left(\frac{\text{J}\cdot {\text{m}}^4}{{\text{s}}^2}\right)\\ =\frac{\pi }{4.5}\left(\text{V}\cdot \text{m}\cdot \text{A}\cdot \text{m}=\frac{\text{J}\cdot {\text{m}}^{\text{2}}}{\text{s}}\right)\times {\lambda }_{p}c\left(\frac{{\text{m}}^{\text{2}}}{\text{s}}\right)\\ =2.76564\text{E}-07\left(\frac{\text{J}\cdot {\text{m}}^4}{{\text{s}}^2}\right)=\text{constant}\end{array}$ (26)

$\begin{array}{l}{e}_{new}c\times \left(\frac{m_p}{m_e}+\frac{4}{3}\right)\left(\frac{\text{A}\cdot {\text{m}}^{\text{3}}}{\text{s}}\right)\\ =\frac{1}{4.5}\left(\text{A}\cdot \text{m}\right)\times {\lambda }_{p}c\left(\frac{{\text{m}}^{\text{2}}}{\text{s}}\right)=8.80330\text{E}-08\left(\frac{\text{A}\cdot {\text{m}}^{\text{3}}}{\text{s}}\right)=\text{constant}\end{array}$ (27)

$\begin{array}{l}{m}_{p\_new}{c}^2\times {\left(\frac{m_p}{m_e}+\frac{4}{3}\right)}^2\left(\frac{\text{J}\cdot {\text{m}}^4}{{\text{s}}^2}\right)\\ =\frac{\pi }{4.5}\left(\frac{\text{J}\cdot {\text{m}}^{\text{2}}}{\text{s}}\right)\times {\lambda }_{e}c\left(\frac{{\text{m}}^{\text{2}}}{\text{s}}\right)=5.07814\text{E}-04\left(\frac{\text{J}\cdot {\text{m}}^4}{{\text{s}}^2}\right)=\text{constant}\end{array}$ (28)

$\begin{array}{l}{q}_{m\_new}c\times \left(\frac{m_p}{m_e}+\frac{4}{3}\right)\left(\frac{\text{V}\cdot {\text{m}}^{\text{3}}}{\text{s}}\right)\\ =\pi \left(\text{V}\cdot \text{m}\right)\times {\lambda }_{e}c\left(\frac{{\text{m}}^{\text{2}}}{\text{s}}\right)=2.28516\text{E}-03\left(\frac{\text{V}\cdot {\text{m}}^{\text{3}}}{\text{s}}\right)=\text{constant}\end{array}$ (29)

$\begin{array}{l}k{T}_{c\_new}\times \frac{2\pi }{\alpha }\times {\left(\frac{m_p}{m_e}+\frac{4}{3}\right)}^3\left(\frac{\text{J}\cdot {\text{m}}^{\text{6}}}{{\text{s}}^{\text{3}}}\right)\\ =\frac{\pi }{4.5}\left(\frac{\text{J}\cdot {\text{m}}^{\text{2}}}{\text{s}}\right)\times {\lambda }_{p}c\times {\lambda }_{e}c=2.011697\text{E}-10\left(\frac{\text{J}\cdot {\text{m}}^{\text{6}}}{{\text{s}}^{\text{3}}}\right)=\text{constant}\end{array}$ (30)

$\begin{array}{l}{G}_{N\_new}\left(\frac{{\text{m}}^{\text{3}}}{{\text{s}}^{\text{2}}}\right)\times \left(\frac{m_p}{m_e}+\frac{4}{3}\right)\left(\frac{{\text{m}}^{\text{2}}}{\text{s}}\right)\\ ={\left({\lambda }_{p}c\right)}^2\left(\frac{{\text{m}}^4}{{\text{s}}^2}\right)\times c\left(\frac{\text{m}}{\text{s}}\right)\times \frac{9\alpha }{8\pi }=1.22943\text{E}-07\left(\frac{{\text{m}}^5}{{\text{s}}^3}\right)=\text{constant}\end{array}$ (31)

The Avogadro number (N_A) is 6.02214076 × 10²³. This value is related to the following value.

$N_A=\frac{1\text{g}}{m_p}=5.978637\text{E}+23$ (32)

Using the redefined values, the new definition of the Avogadro number (N_{A_new}) is as follows:

$N_{A\_new}=\frac{1\text{kg}}{m_{p\_new}}=\frac{1{\text{kg}}_{new}}{m_p}=5.975649\text{E}+26\ne \frac{1{\text{kg}}_{new}}{m_{p\_new}}$ (33)

The number of electrons in 1 C (N_e) is as follows:

$N_e=\frac{1\text{C}}{e}=6.241509\text{E}+18$ (34)

Using the redefined values, the new definition of the number of electrons in 1 C (N_{e_new}) is as follows:

$N_{e\_new}=\frac{1{\text{C}}_{new}}{e}=\frac{1\text{C}}{e_{new}}=6.257473\text{E}+18\ne \frac{1{\text{C}}_{new}}{e_{new}}$ (35)

The following equations were proposed in a previous study [11] :

$m_{p\_new}=\frac{1}{N_{A\_new}}$ (36)

$m_{e\_new}=\frac{m_e/m_p}{N_{A\_new}}$ (37)

where m_p/m_e (=1836.1526) is not changed after redefinition.

$e_{new}=\frac{1}{N_{e\_new}}$ (38)

$q_{m\_new}=\frac{4.5\pi \times m_p/m_e}{N_{e\_new}}=4.148319\text{E}-15$ (39)

$h_{new}=\frac{4.5\pi \times m_p/m_e}{{\left(N_{e\_new}\right)}^2}=6.62938382\text{E}-34$ (40)

$k{T}_{c\_new}=\frac{4.5\times c^3\times \alpha }{2\pi \times N_{e\_new}\times N_{A\_new}}=3.7659625\text{E}-23$ (41)

$G_{N\_new}=\frac{{4.5}^3\times m_p/m_e\times N_{A\_new}\times c^2\times \alpha }{4\times N_{e\_new}^3}=6.6908477\text{E}-11$ (42)

Furthermore, we propose the following four equations [11] :

$k{T}_{c\_new}\left(\text{J}\right)=\frac{\alpha }{2\pi \left(1\right)}\times \frac{1}{\pi }\left(\frac{\text{1}}{\text{V}\cdot \text{m}}\right)\times q_{m\_new}c\times m_{e\_new}{c}^2=3.76596254\text{E}-23$ (43)

$k{T}_{c\_new}\left(\text{J}\right)=\frac{\alpha }{2\pi \left(1\right)}\times 4.5\left(\frac{\text{1}}{\text{A}\cdot \text{m}}\right)\times e_{new}c\times m_{p\_new}{c}^2=3.76596254\text{E}-23$ (44)

In Equations 43 and 44, 2π(1) is dimensionless. For G, there are two equations, as follows:

$\begin{array}{c}{G}_{N\_new}\left(\frac{{\text{m}}^{\text{3}}}{{\text{s}}^{\text{2}}}\right)=\alpha c\frac{4.5\left(1\right)}{4\pi \left(1\right)}\times \left(4.5\times e_{new}c\right)\times e_{new}c\times \frac{q_{m\_new}c}{m_{p\_new}{c}^2}\\ =6.69084770\text{E}-11\left(\frac{{\text{m}}^{\text{3}}}{{\text{s}}^{\text{2}}}\right)\end{array}$ (45)

$\begin{array}{c}{G}_{N\_new}\left(\frac{{\text{m}}^{\text{3}}}{{\text{s}}^{\text{2}}}\right)=\alpha c\frac{4.5\left(1\right)}{4\pi \left(1\right)}\times {\left(4.5\times e_{new}c\right)}^2\times e_{new}c\times \frac{\pi \left(\text{V}\cdot \text{m}\right)}{m_{e\_new}{c}^2}\\ =6.69084770\text{E}-11\left(\frac{{\text{m}}^{\text{3}}}{{\text{s}}^{\text{2}}}\right)\end{array}$ (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.

When we define the Planck constant as (1 Js), the following equations can be used:

$\begin{array}{c}{c}_{genaral}\left(\frac{{\text{m}}_{general}}{\text{s}}\right)=c\times \sqrt{h_{new}\left(1\right)}=299792458\times \sqrt{6.62938\text{E}-34}\\ =7.71893\text{E}-09\left(\frac{{\text{m}}_{general}}{\text{s}}\right)\end{array}$ (47)

where h_new(1) (=6.629383E−34) is dimensionless. c_general and 1m_general 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.

$e_{general}=\sqrt{\frac{1\left(\text{J}\cdot \text{s}\right)}{4.5\pi \times m_p/m_e}}=6.20675231\text{E}-03\left({\text{C}}_{general}\right)$ (48)

$q_{m\_general}=\sqrt{1\left(\text{J}\cdot \text{s}\right)\times 4.5\pi \times m_p/m_e}=1.61114855\text{E}+02\left({\text{Wb}}_{general}\right)$ (49)

$m_{e\_general}=\sqrt{\frac{1\left(\text{J}\cdot \text{s}\right)\times \pi \times m_e/m_p}{c_{general}^2\times 4.5}}\times {\left(\frac{m_p}{m_e}+\frac{4}{3}\right)}^{-1}=1.37477924\text{E}+03\left({\text{kg}}_{general}\right)$ (50)

$m_{p\_general}=\sqrt{\frac{1\left(\text{J}\cdot \text{s}\right)\times \pi \times m_p/m_e}{c_{general}^2\times 4.5}}\times {\left(\frac{m_p}{m_e}+\frac{4}{3}\right)}^{-1}=2.52430455\text{E}+06\left({\text{kg}}_{general}\right)$ (51)

$\frac{k{T}_{c\_general}}{\alpha \times c_{general}^2}=\frac{1\left(\text{J}\cdot \text{s}\right)}{2\pi \left(1\right)}\times {\left(\frac{m_p}{m_e}+\frac{4}{3}\right)}^{-1}=8.66155955\text{E}-05\left({\text{kg}}_{general}\right)$ (52)

$\begin{array}{c}{G}_{N\_general}=\frac{\alpha c_{general}^3}{m_p/m_e}\times \frac{{4.5}^2}{4{\pi }^2}\times 1\left(\text{J}\cdot \text{s}\right)\times \left(\frac{m_p}{m_e}+\frac{4}{3}\right)\\ =1.72273202\text{E}-27\left(\frac{{\text{m}}_{general}^3}{{\text{s}}^2}\right)\end{array}$ (53)

where 1 C_general, 1 Wb_general, 1 kg_general, e_general, q_{m_general}, m_{e_general}, m_{p_general}, T_{c_general} and G_{N_gerneral} are the values for 1 C, 1 Wb, 1 kg, e, q_m, m_e, m_p, T_c and G_N, respectively, when the Planck constant is changed to 1 Js.

The minimum mass (M_min) is as follows:

$M_{\min }\left({\text{kg}}_{general}\right)=\frac{k{T}_c}{\alpha \times c^2}=\frac{1\left(\text{J}\cdot \text{s}\right)}{2\pi }\times {\left(\frac{m_p}{m_e}+\frac{4}{3}\right)}^{-1}$ (54)

The ratio between the mass of an electron and the minimum mass is as follows:

$m_{e\_general}\times \frac{\alpha c_{general}^2}{k{T}_{c\_general}}=2\pi \left(1\right)\times \frac{\pi }{q_{m\_general}{c}_{general}}=1.587219\text{E}+07$ (55)

The mass ratio of a proton to its minimum mass is as follows:

$m_{p\_general}\times \frac{\alpha c_{general}^2}{k{T}_{c\_general}}=\frac{2\pi \left(1\right)}{4.5\times e_{general}{c}_{general}}=2.914376\text{E}+10$ (56)

For convenience, Equation (1) is rewritten as follows:

$\frac{G{m}_p^2}{hc}=\frac{4.5}{2}\times \frac{k{T}_c}{1\text{kg}\times c^2}$ (57)

The equation for a fine structure constant is as follows:

$\frac{e^2}{4\pi {\epsilon }_0}=\frac{hc}{2\pi }\times \alpha$ (58)

The ratio of the gravitational force to the electric force is as follows:

$\frac{G{m}_p^2}{\left(\frac{e^2}{4\pi {\epsilon }_0}\right)}=4.5\left(1\right)\times \pi \left(1\right)\times \frac{\frac{k{T}_{c\_general}}{\alpha \times c_{general}^2}\left({\text{kg}}_{general}\right)}{1\text{kg}}$ (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),

$\frac{k{T}_{c\_general}}{\alpha \times c_{general}^2}\text{=}8.66155955\text{E}-05\left({\text{kg}}_{general}\right)$ (60)

Next,

$\frac{{\text{kg}}_{general}}{1\text{kg}}=\frac{m_{p\_new}}{m_{p\_general}}=\frac{1.6734583781\text{E}-27}{2.52430455\text{E}+06}=6.629384\text{E}-34=h\left(1\right)$ (61)

Using Equations (59), (60) and (61),

$\begin{array}{c}\frac{G{m}_p^2}{\left(\frac{e^2}{4\pi {\epsilon }_0}\right)}=4.5\left(1\right)\times \pi \left(1\right)\times 8.66155955\text{E}-05\times 6.629384\text{E}-34\\ =8.11767475\text{E}-37\end{array}$ (62)

Then, we can explain the ratio of the gravitational force to the electric force. Furthermore, there are no dimension mismatch problems.

For convenience, Equation (31) is rewritten as follows:

$\begin{array}{l}{G}_{N\_new}\left(\frac{{\text{m}}^{\text{3}}}{{\text{s}}^{\text{2}}}\right)\times \left(\frac{m_p}{m_e}+\frac{4}{3}\right)\left(\frac{{\text{m}}^{\text{2}}}{\text{s}}\right)\\ ={\left({\lambda }_{p}c\right)}^2\left(\frac{{\text{m}}^4}{{\text{s}}^2}\right)\times c\left(\frac{\text{m}}{\text{s}}\right)\times \frac{9\alpha }{8\pi }=1.22943\text{E}-07\left(\frac{{\text{m}}^5}{{\text{s}}^3}\right)\end{array}$ (63)

Therefore,

$G_{N\_new}\left(\frac{{\text{m}}^{\text{3}}}{{\text{s}}^{\text{2}}}\right)=\alpha c\frac{4.5}{4\pi \left(1\right)}\times {\left(\frac{h}{m_p}\right)}^2\times {\left(\frac{m_p}{m_e}+\frac{4}{3}\right)}^{-1}=6.6908477\text{E}-11$ (64)

For convenience, Equation (31) is rewritten as follows:

$e_{new}c\times \left(\frac{m_p}{m_e}+\frac{4}{3}\right)\left(\frac{\text{A}\cdot {\text{m}}^{\text{3}}}{\text{s}}\right)=\frac{1}{4.5}\left(\text{A}\cdot \text{m}\right)\times {\lambda }_{p}c\left(\frac{{\text{m}}^{\text{2}}}{\text{s}}\right)=8.80330\text{E}-08\left(\frac{\text{A}\cdot {\text{m}}^{\text{3}}}{\text{s}}\right)$ (65)

Therefore,

$4.5e_{new}c=\frac{h}{m_p}\times {\left(\frac{m_p}{m_e}+\frac{4}{3}\right)}^{-1}$ (66)

From Equations (64) and (66),

$G_{N\_new}\left(\frac{{\text{m}}^{\text{3}}}{{\text{s}}^{\text{2}}}\right)=\alpha c\frac{4.5}{4\pi \left(1\right)}\times \left(\frac{h}{m_p}\right)\times 4.5e_{new}c=6.6908477\text{E}-11$ (67)

In Equation 67, to convert back to the MKSA unit, 4.5 should be changed from 4.5 to 4.8852. Therefore,

$G_N\left(\frac{{\text{m}}^3}{{\text{s}}^{\text{2}}}\right)=\alpha c\frac{4.5\left(1\right)}{4\pi \left(1\right)}\times \left(\frac{h}{m_p}\right)\times 4.48852e_{new}c=6.6737778665\text{E}-11$ (68)

where 4.5(1) is dimensionless. When converting back to the MKSA unit, the unit of 1C should be compensated.

$G_N\left(\frac{{\text{m}}^{\text{3}}}{{\text{s}}^{\text{2}}}\right)=\alpha c\frac{4.5\left(1\right)}{4\pi \left(1\right)}\times \frac{e_{new}}{e}\times \left(\frac{h}{m_p}\right)\times 4.5e_{new}c=6.6737778665\text{E}-11$ (69)

4.5 in Equations (1) and (2) is dimensionless. G_new (after redefinition of G) is as follows:

$\begin{array}{c}{G}_{new}\left(\frac{{\text{m}}^{\text{3}}}{\text{kg}\cdot {\text{s}}^{\text{2}}}\right)=1\text{kg}\times \frac{G_{N\_new}}{1{\text{kg}}_{new}}=\frac{1\text{kg}}{1{\text{kg}}_{new}}\times \frac{e}{e_{new}}\times G\\ =\frac{\pi }{3.13201}\times G=6.69419377\text{E}-11\end{array}$ (70)

However, G_new is not useful rather than G_{N_new} because we must consider the compensation of the unit 1 kg in every equation.

Equations (71) and (72) are important for making the first list. Using Equations (26) - (31), the following list can be obtained.

$\frac{h}{m_p}=\frac{h_{new}}{m_{p\_new}}=3.9614871\text{E}-07=\text{constant}\left(\text{experimental result}\right)$ (71)

$\frac{h}{m_e}=\frac{h_{new}}{m_{e\_new}}=7.2738951\text{E}-04=\text{constant}\left(\text{experimental result}\right)$ (72)

$m_{e\_new}{c}^2\left(\text{J}\right)=\frac{\pi }{4.5}\times \frac{h}{m_p}\times {\left(\frac{m_p}{m_e}+\frac{4}{3}\right)}^{-2}\left(\text{J}\right)=8.19120012\text{E}-14\left(\text{J}\right)$ (73)

$e_{new}c\left(\text{A}\cdot \text{m}\right)=\frac{1}{4.5}\times \frac{h}{m_p}\times {\left(\frac{m_p}{m_e}+\frac{4}{3}\right)}^{-1}\left(\text{A}\cdot \text{m}\right)=4.79095067\text{E}-11\left(\text{A}\cdot \text{m}\right)$ (74)

$m_{p\_new}{c}^2\left(\text{J}\right)=\frac{\pi }{4.5}\times \frac{h}{m_e}\times {\left(\frac{m_p}{m_e}+\frac{4}{3}\right)}^{-2}\left(\text{J}\right)=1.50402938\text{E}-10\left(\text{J}\right)$ (75)

$q_{m\_new}c\left(\text{V}\cdot \text{m}\right)=\pi \times \frac{h}{m_e}\times {\left(\frac{m_p}{m_e}+\frac{4}{3}\right)}^{-1}\left(\text{V}\cdot \text{m}\right)=1.24363481\text{E}-06\left(\text{V}\cdot \text{m}\right)$ (76)

$\begin{array}{c}{h}_{new}{c}^2\left(\frac{\text{J}\cdot {\text{m}}^{\text{2}}}{\text{s}}\right)=\frac{\pi }{4.5}\times \frac{h}{m_p}\times \frac{h}{m_e}\times {\left(\frac{m_p}{m_e}+\frac{4}{3}\right)}^{-2}\left(\frac{\text{J}\cdot {\text{m}}^{\text{2}}}{\text{s}}\right)\\ =5.9581930\text{E}-17\left(\frac{\text{J}\cdot {\text{m}}^{\text{2}}}{\text{s}}\right)\end{array}$ (77)

$\frac{k{T}_{c\_new}}{\alpha }\left(\text{J}\right)=\frac{1}{2\pi \left(1\right)}\times \frac{\pi }{4.5}\times \frac{h}{m_p}\times \frac{h}{m_e}\times {\left(\frac{m_p}{m_e}+\frac{4}{3}\right)}^{-3}\left(\text{J}\right)=5.1607244\text{E}-21\left(\text{J}\right)$ (78)

$G_{N\_new}\left(\frac{{\text{m}}^{\text{3}}}{{\text{s}}^{\text{2}}}\right)=\alpha c\frac{4.5\left(1\right)}{4\pi \left(1\right)}\times {\left(\frac{h}{m_p}\right)}^2\times {\left(\frac{m_p}{m_e}+\frac{4}{3}\right)}^{-1}\left(\frac{{\text{m}}^{\text{3}}}{{\text{s}}^{\text{2}}}\right)=6.6908477\text{E}-11$ (79)

With respect to Equation (78), we find two equations.

$\frac{k{T}_{c\_new}}{\alpha }\left(\text{J}\right)=\frac{1}{2\pi \left(1\right)}\times \frac{q_{m\_new}c}{\pi }\times m_{e\_new}{c}^2\left(\text{J}\right)=3.76596254\text{E}-23\left(\text{J}\right)$ (80)

$\frac{k{T}_{c\_new}}{\alpha }\left(\text{J}\right)=\frac{1}{2\pi \left(1\right)}\times 4.5\times e_{new}c\times m_{p\_new}{c}^2\left(\text{J}\right)=3.76596254\text{E}-23\left(\text{J}\right)$ (81)

With respect to Equation (79), we find two equations.

$G_{N\_new}\left(\frac{{\text{m}}^{\text{3}}}{{\text{s}}^{\text{2}}}\right)=\alpha c\frac{4.5\left(1\right)}{4\pi \left(1\right)}\times \left(4.5\times e_{new}c\right)\times \frac{h_{new}{c}^2}{m_{p\_new}{c}^2}=6.69084770\text{E}-11\left(\frac{{\text{m}}^{\text{3}}}{{\text{s}}^{\text{2}}}\right)$ (82)

$\begin{array}{c}{G}_{N\_new}\left(\frac{{\text{m}}^{\text{3}}}{{\text{s}}^{\text{2}}}\right)=\alpha c\frac{4.5\left(1\right)}{4\pi \left(1\right)}\times {\left(4.5\times e_{new}c\right)}^2\times e_{new}c\times \frac{\pi \left(\text{V}\cdot \text{m}\right)}{m_{e\_new}{c}^2}\\ =6.69084770\text{E}-11\left(\frac{{\text{m}}^{\text{3}}}{{\text{s}}^{\text{2}}}\right)\end{array}$ (83)

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, c_general can be obtained, and every value can be calculated. This means that we can obtain the second list.

$c_{genaral}=7.71893\text{E}-09\left(\frac{{\text{m}}_{general}}{\text{s}}\right)$ (84)

Using Equations (73) and (84),

$m_{e\_general}=\frac{8.19120012\text{E}-14}{{\left(7.71893\text{E}-09\right)}^2}\left({\text{kg}}_{general}\right)=1.37477924\text{E}+03\left({\text{kg}}_{general}\right)$ (85)

Using Equations (74) and (84),

$e_{general}=\frac{4.79095067\text{E}-11}{7.71893\text{E}-09}\left({\text{C}}_{general}\right)=6.20675231\text{E}-03\left({\text{C}}_{general}\right)$ (86)

Using Equations (75) and (84),

$m_{p\_general}=\frac{1.50402938\text{E}-10}{{\left(7.71893\text{E}-09\right)}^2}\left({\text{kg}}_{general}\right)=2.52430455\text{E}+06\left({\text{kg}}_{general}\right)$ (87)

Using Equations (76) and (84),

$q_{m\_general}=\frac{1.24363481\text{E}-06}{7.71893\text{E}-09}\left({\text{Wb}}_{general}\right)=1.61114855\text{E}+02\left({\text{Wb}}_{general}\right)$ (88)

Regarding Equations (77) and (84),

$\frac{k{T}_{c\_general}}{\alpha c^2}=\frac{5.16072439\text{E}-21}{{\left(7.71893\text{E}-09\right)}^2}\left({\text{kg}}_{general}\right)=8.66155955\text{E}-05\left({\text{kg}}_{general}\right)$ (89)

Then, kT_c/α is unchanged. The reason is that in Equation (81), ec and m_pc² are unchanged.

$\frac{k{T}_{c\_new}}{\alpha }\left(\text{J}\right)=8.66155955\text{E}-05\times {\left(7.71893324\text{E}-09\right)}^2=5.16072439\text{E}-21$ (90)

From Equation (82),

$\frac{G_{N\_new}}{\alpha c}\left(\frac{{\text{m}}^{\text{2}}}{\text{s}}\right)=\frac{6.69084770\text{E}-11}{\alpha \times 299792458}\left(\frac{{\text{m}}^{\text{2}}}{\text{s}}\right)=3.05840582\text{E}-17\left(\frac{{\text{m}}^{\text{2}}}{\text{s}}\right)$ (91)

From Equations (53) and (84),

$\frac{G_{N\_general}}{\alpha c_{genaral}}\left(\frac{{\text{m}}^{\text{2}}}{\text{s}}\right)=\frac{1.72273202\text{E}-27}{\alpha \times 7.71893\text{E}-09}\left(\frac{{\text{m}}^{\text{2}}}{\text{s}}\right)=3.05840582\text{E}-17\left(\frac{{\text{m}}^{\text{2}}}{\text{s}}\right)$ (92)

From Equations (91) and (92), G_N/αc is a constant, and G_{N_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 m_e, m_p, e, q_m, h, kT_c/α and G_N/αc in List 1 can be calculated. Another examples are shown in Appendix.

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.

$\left(\frac{m_p}{m_e}+\frac{4}{3}\right)=\frac{q_{m}c}{4.5m_p{c}^2}=\frac{\pi ec}{m_e{c}^2}=1837.485988$ (93)

$2\pi \left(1\right)\times \frac{\pi }{q_{m\_new}c}=1.587219\text{E}+07$ (94)

$\frac{2\pi \left(1\right)}{4.5\times e_{new}c}=2.914376\text{E}+10$ (95)

$\frac{k{T}_{c\_new}}{\alpha }\left(\text{J}\right)=\frac{1}{2\pi \left(1\right)}\times \frac{q_{m\_new}c}{\pi }\times m_{e\_new}{c}^2\left(\text{J}\right)=3.76596254\text{E}-23\left(\text{J}\right)$ (96)

$\frac{G_{N\_new}}{\alpha c}\left(\frac{{\text{m}}^{\text{2}}}{\text{s}}\right)=\frac{4.5\left(1\right)}{4\pi \left(1\right)}\times \left(4.5\times e_{new}c\right)\times \frac{h_{new}{c}^2}{m_{p\_new}{c}^2}=3.05840582\text{E}-17\left(\frac{\text{m}}{\text{s}}\right)$ (97)

We note that another list is possible.

$\left(\frac{m_p}{m_e}+\frac{4}{3}\right)=\frac{\pi ec}{m_e{c}^2}$ (98)

Therefore,

$e\pi =c\times \left(m_p+\frac{4}{3}{m}_e\right)$ (99)

Therefore,

$m_p=\frac{e\pi }{c}-\frac{4}{3}{m}_e$ (100)

Here,

$\left(\frac{m_p}{m_e}+\frac{4}{3}\right)=\frac{q_{m}c}{4.5m_p{c}^2}$ (101)

S0,

$\frac{q_m}{4.5}=m_{p}c\times \left(\frac{m_p}{m_e}+\frac{4}{3}\right)$ (102)

From Equations (100) and (102),

$\frac{q_m}{4.5}=c\times \left(\frac{e\pi }{c}-\frac{4}{3}{m}_e\right)\left(\frac{\frac{e\pi }{c}-\frac{4}{3}{m}_e}{m_e}+\frac{4}{3}\right)=\left(e\pi -\frac{4}{3}{m}_{e}c\right)\frac{e\pi }{m_{e}c}=\frac{{\left(e\pi \right)}^2}{m_{e}c}-\frac{4}{3}e\pi$ (103)

Therefore,

$m_e=\frac{e\pi }{c}{\left(\frac{q_m}{4.5e\pi }+\frac{4}{3}\right)}^{-1}$ (104)

Therefore,

$m_p=\frac{e\pi }{c}\times \left(1-\frac{4}{3}{\left(\frac{Rk}{4.5\pi }+\frac{4}{3}\right)}^{-1}\right)$ (105)

Therefore,

$\frac{m_p}{m_e}=\frac{\frac{e\pi }{c}\times \left(1-\frac{4}{3}{\left(\frac{Rk}{4.5\pi }+\frac{4}{3}\right)}^{-1}\right)}{\frac{e\pi }{c}\times {\left(\frac{Rk}{4.5\pi }+\frac{4}{3}\right)}^{-1}}=\frac{Rk}{4.5\pi }=1836.15265426$ (106)

When we use $\left(\frac{m_p}{m_e}+\frac{5}{3}\right)$ instead of $\left(\frac{m_p}{m_e}+\frac{4}{3}\right)$,

$\frac{m_p}{m_e}=\frac{\frac{e\pi }{c}\times \left(1-\frac{5}{3}{\left(\frac{Rk}{4.5\pi }+\frac{5}{3}\right)}^{-1}\right)}{\frac{e\pi }{c}\times {\left(\frac{Rk}{4.5\pi }+\frac{5}{3}\right)}^{-1}}=\frac{Rk}{4.5\pi }=1836.15265426$ (107)

Consequently, another first list is possible. However, the calculated values of G and kT_c are much different from the observed values.

From this section onward, we ensure that our second list is correct in our three equations. Strictly speaking, m_e should be written as m_{e_general}. However, we omit the subscript “general” to avoid unnecessary notational complexity.

For convenience, Equation (1) is rewritten as follows:

$\frac{G{m}_p^2}{hc}=\frac{4.5}{2}\times \frac{k{T}_c}{1\text{kg}\times c^2}$ (108)

Therefore, the following can be applied:

$\frac{G_N{m}_p^2}{hc}=\frac{4.5\left(1\right)}{2}\times \frac{k{T}_c}{c^2}$ (109)

where 4.5(1) is dimensionless. Therefore,

$\frac{G_N}{\alpha c}\frac{{\left(m_p{c}^2\right)}^2}{h{c}^2}=\frac{4.5\left(1\right)}{2}\times \frac{k{T}_c}{\alpha }=1.16116299\text{E}-20$ (110)

$\frac{G_N}{\alpha c}$, $m_p{c}^2$, $h{c}^2$ and $\frac{k{T}_c}{\alpha }$ are constant when c is changed. Therefore, our second list is correct.

For convenience, Equation (2) is rewritten as follows:

$\frac{G{m}_p^2}{\left(\frac{e^2}{4\pi {\epsilon }_0}\right)}=\frac{4.5\left(1\right)}{2\pi }\times \frac{m_e}{e}\times hc$ (111)

Therefore, the following can be obtained:

$\frac{G_N{m}_p^2}{hc}=\frac{4.5\left(1\right)}{2\pi }\times \frac{m_e}{e}\times \frac{e^2}{4\pi {\epsilon }_0}$ (112)

The equation for a fine structure constant is as follows:

$\frac{e^2}{4\pi {\epsilon }_0}=\frac{hc}{2\pi \left(1\right)}\times \alpha$ (113)

Therefore,

$\frac{G_N{m}_p^2}{hc}=\frac{4.5\left(1\right)}{2\pi }\times \frac{m_e}{e}\times \frac{hc}{2\pi \left(1\right)}\times \alpha$ (114)

Therefore,

$\frac{G_N}{\alpha c}\frac{{\left(m_p{c}^2\right)}^2}{h{c}^2}=\frac{4.5\left(1\right)}{2\pi \left(1\right)}\times \frac{m_e{c}^2}{ec}\times \frac{h{c}^2}{2\pi }=1.16116299\text{E}-20$ (115)

When c is changed, $\frac{G_N}{\alpha c}$, $m_p{c}^2$, $h{c}^2$, $m_e{c}^2$, and ec are constant. Therefore, our second list is correct.

For convenience, Equation (3) is rewritten as follows:

$\frac{m_e{c}^2}{e}\times \frac{e^2}{4\pi {\epsilon }_0}=\pi \times k{T}_c$ (116)

The equation for a fine structure constant is as follows:

$\frac{e^2}{4\pi {\epsilon }_0}=\frac{hc}{2\pi \left(1\right)}\times \alpha$ (117)

Therefore,

$\frac{m_e{c}^2}{e}\times \frac{hc}{2\pi \left(1\right)}\times \alpha =\pi \times k{T}_c$ (118)

Therefore,

$\frac{m_e{c}^2}{ec}\times \frac{h{c}^2}{2\pi \left(1\right)}=\pi \times \frac{k{T}_c}{\alpha }$ (119)

Then, $\frac{kT}{\alpha }$, $m_e{c}^2$, $h{c}^2$, and ec are constant when c is changed. Therefore, our second list is correct.

Equation (2) is rewritten as follows:

$\frac{G{m}_p^2}{hc}=\frac{4.5\left(1\right)}{2}\times \frac{k{T}_c}{1\text{kg}\times c^2}$ (120)

The fine-structure constant is defined as follows:

$\frac{e^2}{4\pi {\epsilon }_0}=\frac{\alpha hc}{2\pi }=2.30823131\text{E}-28\left(\text{J}\cdot \text{m}\right)$ (121)

Therefore, the following can be obtained:

$\frac{G{m}_p^2}{\left(\frac{e^2}{4\pi {\epsilon }_0}\right)}=4.5\left(1\right)\times \pi \left(1\right)\times \frac{\frac{k{T}_{c\_general}}{\alpha \times c_{general}^2}\left({\text{kg}}_{general}\right)}{1\text{kg}}$ (122)

Therefore,

$\begin{array}{c}\frac{{\text{kg}}_{general}}{1\text{kg}}\times \frac{1}{{\left(c_{general}\right)}^2}=\frac{m_{p\_new}}{m_{p\_general}}\times \frac{1}{{\left(c_{general}\right)}^2}=\frac{m_{p\_new}}{m_{p\_new}\times c^2}\\ =\frac{1}{c^2}=1.11265\text{E}-17\end{array}$ (123)

Therefore,

$\frac{G{m}_p^2}{\left(\frac{e^2}{4\pi {\epsilon }_0}\right)}=4.5\left(1\right)\times \pi \left(1\right)\times \frac{k{T}_c}{\alpha }\times 1.11265\text{E}-17=8.11767475\text{E}-37$ (124)

Consequently, the ratio of the gravitational force to the electric force is unchanged in the second list.

The theoretical meaning of the second list is as follows. Einstein discovered the following equation:

$E=m{c}^2$ (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.