In physics, Natural scale is the scale, which connects with the mass of electron. The method of constructing of the scale is the same as for Planck scale and Stoney scale. Usually, the Natural scale now considered for definition of the gravitational force between electrons and has not appropriate attention that should be for the scale of matter.

The natural gravitational coupling constant is:

${\displaystyle \alpha _{N}={\frac {m_{e}^{2}}{2hc\,\varepsilon _{g}}}=1.7517846\cdot 10^{-45},\ }$

where

• ${\displaystyle \!m_{e}}$ is the electron mass;
• ${\displaystyle \!h}$ is the Planck constant;
• ${\displaystyle \!c}$ is the speed of light in vacuum;
• ${\displaystyle ~\varepsilon _{g}={\frac {1}{4\pi G}}}$ is the gravitoelectric gravitational constant;
• ${\displaystyle \!G}$ is the gravitational constant.

Fundamental units of vacuum

The set of primary vacuum constants is: ^[1] the speed of light ${\displaystyle ~c}$; the electric constant ${\displaystyle ~\varepsilon _{0}}$; the speed of gravity ${\displaystyle ~c_{g}}$ (usually equated to the speed of light); the gravitational constant ${\displaystyle ~G}$.

The set of secondary vacuum constants is: The vacuum permeability: ${\displaystyle \mu _{0}={\frac {1}{\varepsilon _{0}c^{2}}}\ }$;

The electromagnetic impedance of free space:

${\displaystyle Z_{0}=\mu _{0}c={\sqrt {\frac {\mu _{0}}{\varepsilon _{0}}}}={\frac {1}{\varepsilon _{0}c}}}$;

The gravitoelectric gravitational constant: ${\displaystyle ~\varepsilon _{g}={\frac {1}{4\pi G}}}$;

The gravitomagnetic gravitational constant: ${\displaystyle ~\mu _{g}={\frac {4\pi G}{c_{g}^{2}}}}$;

The gravitational characteristic impedance of free space:

${\displaystyle ~\rho _{g}={\sqrt {\frac {\mu _{g}}{\varepsilon _{g}}}}={\frac {4\pi G}{c_{g}}}.}$

Natural scale units based on electron mass

For the sake of completeness in the Table 1 the main Natural scale units in the form consistent with Tables for the Stoney scale and vacuum constants are presented.

Table 1: Base Natural scale units

Name	Dimension	Expressions	SI equivalent
Electron mass	Mass (M)	${\displaystyle m_{e}\ }$	${\displaystyle 9.10938215(45)\cdot 10^{-31}}$ kg
Electron wavelength	Length (L)	${\displaystyle \lambda _{N}={\frac {h}{m_{e}c}}}$	${\displaystyle 2.4263102175(33)\cdot 10^{-12}}$ m
Natural gravitational coupling constant	Dimensionless	${\displaystyle \alpha _{N}={\frac {m_{e}^{2}}{2hc\varepsilon _{g}}}}$	${\displaystyle 1.7517846\cdot 10^{-45}}$
Natural fictitious gravitational torsion mass	Dynamic mass (L2T −1)	${\displaystyle m_{\Omega }={\frac {h}{m_{e}}}\ }$	${\displaystyle 7.27339\cdot 10^{-4}}$ m2 s−1
Natural torsion coupling constant	Dimensionless	${\displaystyle \beta _{N}={\frac {m_{\Omega }^{2}}{2hc\mu _{g}}}={\frac {1}{4\alpha _{N}}}\ }$	${\displaystyle 1.42757\cdot 10^{44}}$
Natural charge	Electric charge (Q)	${\displaystyle q_{N}={\sqrt {2hc\varepsilon _{0}\alpha _{N}}}=e{\sqrt {\frac {\alpha _{N}}{\alpha }}}=e\left({\frac {m_{e}}{m_{S}}}\right)}$	${\displaystyle 7.84854579\cdot 10^{-41}}$ C
Natural electric coupling constant	Dimensionless	${\displaystyle \alpha _{N}={\frac {q_{N}^{2}}{2hc\varepsilon _{0}}}\ }$	${\displaystyle 1.7517846\cdot 10^{-45}}$
Natural fictitious magnetic charge	Magnetic charge (L2MT −1Q−1)	${\displaystyle q_{m}={\frac {h}{q_{N}}}={\frac {h}{e}}{\sqrt {\frac {\alpha }{\alpha _{N}}}}\ }$	{${\displaystyle 8.44229\cdot 10^{6}}$ Wb
Natural magnetic coupling constant	Dimensionless	${\displaystyle \beta _{N}={\frac {q_{m}^{2}}{2hc\mu _{0}}}={\frac {1}{4\alpha _{N}}}}$	${\displaystyle 1.42757\cdot 10^{44}}$
Natural gravitational impedance quantum	Gravitational impedance (L2M−1T −1)	${\displaystyle R_{N}={\frac {m_{\Omega }}{m_{e}}}={\frac {h}{m_{e}^{2}}}\ }$	${\displaystyle 7.98492\cdot 10^{26}}$ m2 kg−1 s−1
Natural electromagnetic impedance quantum	Electrical impedance (L2M−1T −1Q−2)	${\displaystyle R_{e}={\frac {q_{m}}{q_{N}}}={\frac {h}{e^{2}}}{\frac {\alpha }{\alpha _{N}}}\ }$	${\displaystyle 1.07562\cdot 10^{47}}$ Ohm.

The difference of Natural coupling constant ${\displaystyle \alpha _{N}\ }$ and fine structure constant ${\displaystyle \alpha \ }$ is so high, that the Natural scale now is the base for the Dirac large numbers and different numerology approaches. ^[2]

Actually, the relationship between fine structure constant and Natural coupling constant yields the Dirac number:

${\displaystyle \xi _{D}={\frac {\alpha }{\alpha _{N}}}=\left({\frac {e}{m_{e}}}\right)^{2}{\frac {\varepsilon _{g}}{\varepsilon _{0}}}=\left({\frac {m_{S}}{m_{e}}}\right)^{2}=4.167\cdot 10^{42},\ }$

where ${\displaystyle m_{S}\ }$ is the Stoney mass.

See also

• Planck scale
• Stoney scale
• Selfconsistent gravitational constants
• Selfconsistent electromagnetic constants
• Maxwell-like gravitational equations
• Quantum Gravitational Resonator

References