# Physics/Essays/Fedosin/Stoney scale

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In physics, Stoney scale is the fundamental scale, named after the Irish physicist George Johnstone Stoney, who first proposed the ‘’elementary electric charge’’ in 1881. [1] It defines that fine structure constant ${\displaystyle ~\alpha }$ is equal to gravitational coupling constant (and to electric coupling constant) ${\displaystyle ~\alpha _{S}}$ of Stoney scale:

${\displaystyle ~\alpha _{S}={\frac {m_{S}^{2}}{2hc\varepsilon _{g}}}={\frac {e^{2}}{2hc\,\varepsilon _{0}}}=\alpha ,}$

where

• ${\displaystyle ~m_{S}={\sqrt {\frac {e^{2}}{4\pi \varepsilon _{0}G}}}}$ is the Stoney mass;
• ${\displaystyle ~e}$ is the elementary charge;
• ${\displaystyle ~\varepsilon _{0}}$ is the electric constant;
• ${\displaystyle ~G}$ is the gravitational constant;
• ${\displaystyle ~h}$ is the Planck constant;
• ${\displaystyle ~c}$ is the speed of light in vacuum;
• ${\displaystyle ~\varepsilon _{g}}$ is the gravitoelectric gravitational constant.

There is the dimensionless magnetic coupling constant ${\displaystyle ~\beta ={\frac {1}{4\alpha }}}$ that could be named as the Stoney scale force constant since it defines the force interactions (electric, gravitational, etc.) in the Stoney scale.

## History

Contemporary physics has settled on the Planck scale as the most suitable scale for the unified field theory. The Planck scale was however anticipated by George Stoney. [1] James G. O’Hara [2] pointed out in 1974 that Stoney’s derived estimate of the unit of charge, 10-20 Ampere (later called the Coulomb), was ​116 of the correct value of the charge of the electron. Stoney used the quantity 1018 for the number of molecules presented in one cubic millimeter of gas at standard temperature and pressure. Using Avogadro constant 6.0221×1023
, and the volume of a mole (at standard conditions) of 22.711×106
mm3
, we derive, instead of 1018, the estimate 2.652×1016
. So, if Stoney could use the true number of molecules his estimate of the unit of charge was about ​12 of the correct value of the charge.

For a long time the Stoney scale was in the shadow of the Planck scale (something like a "deviation" of it). However, after intensive investigation of gravitation by using the Maxwell-like gravitational equations during last decades, became clear that Stoney scale is independent scale of matter.

## Fundamental units of vacuum

The set of primary vacuum constants is: 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: ${\displaystyle ~\mu _{g}={\frac {4\pi G}{c_{g}^{2}}}}$;

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

Note that all Stoney and Planck units are derivatives from the vacuum constants, therefore the last are more fundamental that units of any scale.

If ${\displaystyle c_{g}=c}$ the above fundamental constants define naturally the following relationship between mass and elementary charge for the Stoney mass:

${\displaystyle m_{S}=e{\sqrt {\frac {\varepsilon _{g}}{\varepsilon _{0}}}}=e{\sqrt {\frac {\mu _{0}}{\mu _{g}}}}=e{\sqrt {\frac {Z_{0}}{\rho _{g}}}}\ }$,

and these constants are the base units of the Stoney scale.

## Primary Stoney units

### Gravitational Stoney units

${\displaystyle m_{S}=e{\sqrt {\frac {\varepsilon _{g}}{\varepsilon _{0}}}}={\sqrt {\alpha }}\ m_{P}=1.85927\cdot 10^{-9}\ }$ kg,

where ${\displaystyle m_{P}\ }$ is the Planck mass.

Stoney gravitational fine structure constant:

${\displaystyle \alpha _{S}={\frac {m_{S}^{2}}{2hc\varepsilon _{g}}}=\alpha =7.29735257\cdot 10^{-3}\ }$.

Stoney fictitious gravitational torsion mass:

${\displaystyle m_{\Omega }={\frac {h}{m_{S}}}=3.563801\cdot 10^{-25}\ }$ J s kg−1.

Stoney scale gravitational torsion coupling constant: [3]

${\displaystyle \beta _{g}={\frac {\varepsilon _{g}hc}{2m_{S}^{2}}}={\frac {h}{2c\mu _{g0}m_{S}^{2}}}={\frac {1}{4\alpha }}=34.258999743\ }$.

Stoney gravitational impedance quantum:

${\displaystyle R_{g}={\frac {m_{\Omega }}{m_{S}}}={\frac {h}{m_{S}^{2}}}=1.91677\cdot 10^{-16}\ }$ J s kg−2.

### Electromagnetic Stoney units

Stoney charge:

${\displaystyle q_{S}=e=1.602176565\cdot 10^{-19}\ }$ C.

Stoney electric fine structure constant:

${\displaystyle \alpha _{S}={\frac {q_{S}^{2}}{2hc\,\varepsilon _{0}}}=\alpha .\ }$

Stoney fictitious magnetic charge:

${\displaystyle q_{m}={\frac {h}{e}}=4.1356675\cdot 10^{-15}\ }$ Wb.

Stoney scale magnetic coupling constant:

${\displaystyle \beta ={\frac {\varepsilon _{0}hc}{2e^{2}}}={\frac {h}{2c\mu _{0}e^{2}}}={\frac {1}{4\alpha }}=34.258999743\ }$.

Stoney electrodynamic impedance quantum:

${\displaystyle R_{e}={\frac {q_{m}}{e}}={\frac {h}{e^{2}}}=25,812.807449\ }$ Ohm

which appears as the von Klitzing constant.

## Secondary Stoney scale units

All systems of measurement feature is base units: in the International System of Units (SI), for example, the base unit of length is the meter. In the system of Stoney units, the Stoney base unit of length is known simply as the ‘’Stoney length’’, the base unit of time is the ‘’Stoney time’’, and so on. These units are derived from the presented above primary Stoney units, and arranged in Table 1 so as to cancel out the unwanted dimensions, leaving only the dimension appropriate to each unit. (Like all systems of natural units, Stoney units are an instance of dimensional analysis.)

The keys which are used in the Tables below: L = length, T = time, M = mass, Q = electric charge, Θ = temperature.

Table 1: Secondary Stoney units
Name Dimension Expressions SI equivalent [4]
Stoney wavelength Length (L) ${\displaystyle \lambda _{S}={\frac {h}{m_{S}c}}}$ ${\displaystyle 1.18876\cdot 10^{-33}}$ m
Stoney time Time (T) ${\displaystyle t_{S}={\frac {\lambda _{S}}{c}}}$ ${\displaystyle 3.96528\cdot 10^{-42}}$ s
Stoney classical radius Length (L) ${\displaystyle r_{Sc}={\frac {\alpha \lambda _{S}}{2\pi }}}$ ${\displaystyle 1.38064\cdot 10^{-36}}$ m
Stoney Schwarzschild radius Length (L) ${\displaystyle r_{SS}={\frac {2Gm_{S}}{c^{2}}}=2r_{Sc}\ }$ ${\displaystyle 2.76127\cdot 10^{-36}}$ m
Stoney temperature Temperature (Θ) ${\displaystyle T_{S}={\frac {m_{S}c^{2}}{k_{B}}}}$ ${\displaystyle 1.21032\cdot 10^{31}}$ K

## Derived Stoney scale units

In any system of measurement, units for many physical quantities can be derived from base units. Table 2 offers a sample of derived Stoney units, some of which in fact are seldom used. As with the base units, their use is mostly confined to theoretical physics because most of them are too large or too small for empirical or practical use and there are large uncertainties in their values.

Table 2: Derived Stoney units

Name Dimensions Expression SI equivalent
Stoney area Area (L2) ${\displaystyle \lambda _{S}^{2}={\frac {h}{2\alpha \varepsilon _{g}c^{3}}}}$ ${\displaystyle 1.4127\cdot 10^{-66}}$ m2
Stoney volume Volume (L3) ${\displaystyle \lambda _{S}^{3}=\left({\frac {h}{m_{S}c}}\right)^{3}}$ ${\displaystyle 1.6794\cdot 10^{-99}}$ m3
Stoney momentum Momentum (LMT −1) ${\displaystyle m_{S}c={\frac {h}{\lambda _{S}}}}$ ${\displaystyle 5.5739\cdot 10^{-1}}$ kg m/s
Stoney energy Energy (L2MT −2) ${\displaystyle W_{S}=m_{S}c^{2}\ }$ ${\displaystyle 1.6710\cdot 10^{8}}$ J
Stoney force Force (LMT −2) ${\displaystyle F_{S}={\frac {W_{S}}{\lambda _{S}}}={\frac {m_{S}c^{2}}{\lambda _{S}}}}$ ${\displaystyle 1.4057\cdot 10^{41}}$ N
Stoney power Power (L2MT −3) ${\displaystyle P_{S}={\frac {m_{S}c^{2}}{t_{S}}}}$ ${\displaystyle 4.2141\cdot 10^{49}}$ W
Stoney density Density (L−3M) ${\displaystyle \rho _{Sm}={\frac {m_{S}}{\lambda _{S}^{3}}}}$ ${\displaystyle 1.1071\cdot 10^{90}}$ kg/m3
Stoney angular frequency Frequency (T −1) ${\displaystyle \omega _{S}={\frac {2\pi }{t_{S}}}={\frac {2\pi c}{\lambda _{S}}}}$ ${\displaystyle 1.5846\cdot 10^{42}}$ rad s−1
Stoney pressure Pressure (L−1MT −2) ${\displaystyle p_{S}={\frac {F_{S}}{\lambda _{S}^{2}}}={\frac {m_{S}c^{2}}{\lambda _{S}^{3}}}}$ ${\displaystyle 9.9504\cdot 10^{106}}$ Pa
Stoney current Electric current (QT −1) ${\displaystyle I_{S}={\frac {q_{S}}{t_{S}}}={\frac {ec}{\lambda _{S}}}}$ ${\displaystyle 4.0405\cdot 10^{22}}$ A
Stoney voltage Voltage (L2MT −2Q−1) ${\displaystyle V_{S}={\frac {W_{S}}{q_{S}}}={\frac {m_{S}c^{2}}{e}}}$ ${\displaystyle 1.0429\cdot 10^{27}}$ V
Stoney electric impedance Resistance (L2MT −1Q−2) ${\displaystyle R_{ES}={\frac {V_{S}}{I_{S}}}={\frac {h}{e^{2}}}}$ ${\displaystyle 2.5813\cdot 10^{4}}$ Ohm
Stoney gravitational current Gravitational current (MT −1) ${\displaystyle I_{GS}={\frac {m_{S}}{t_{S}}}={\frac {m_{S}^{2}c^{2}}{h}}}$ ${\displaystyle 4.6889\cdot 10^{32}}$ kg s−1
Stoney gravitational potential Gravitational potential (L2T −2) ${\displaystyle V_{GS}={\frac {W_{S}}{m_{S}}}=c^{2}}$ ${\displaystyle 8.9875\cdot 10^{16}}$ m2 s−2
Stoney gravitational impedance Gravitational impedance (L2M −1T −1) ${\displaystyle R_{GS}={\frac {V_{GS}}{I_{GS}}}={\frac {h}{m_{S}^{2}}}}$ ${\displaystyle 1.9168\cdot 10^{-16}}$ m2 kg−1 s−1
Stoney electric capacitance per unit area Electric capacitance (L−4M−1T2Q2) ${\displaystyle C_{ES}={\frac {e}{V_{S}\lambda _{S}^{2}}}={\frac {2\varepsilon _{0}\alpha }{\lambda _{S}}}}$ ${\displaystyle 1.0870\cdot 10^{20}}$ F m−2
Stoney electric inductance per unit area Electric inductance (L2MQ−2) ${\displaystyle L_{ES}=R_{ES}t_{S}={\frac {\mu _{0}}{2\alpha \lambda _{S}}}}$ ${\displaystyle 7.2430\cdot 10^{28}}$ H m−2
Stoney gravitational capacitance per unit area Gravitational capacitance (L−4MT2 ) ${\displaystyle C_{GS}={\frac {m_{s}}{V_{GS}\lambda _{S}^{2}}}={\frac {2\varepsilon _{g}\alpha }{\lambda _{S}}}}$ ${\displaystyle 1.4643\cdot 10^{40}}$ m−4 kg s2
Stoney gravitational inductance per unit area Gravitational inductance (L2M−1) ${\displaystyle L_{GS}=R_{GS}t_{S}={\frac {\mu _{g}}{2\alpha \lambda _{S}}}}$ ${\displaystyle 5.3769\cdot 10^{8}}$ m2 kg−1
Stoney particle radius Length (L) ${\displaystyle r_{S}={\frac {\lambda _{S}}{2\pi {\sqrt {2}}}}}$ ${\displaystyle 1.3378\cdot 10^{-34}}$ m
Stoney particle area Area (L2) ${\displaystyle S_{S}=4\pi r_{S}^{2}={\frac {\lambda _{S}^{2}}{2\pi }}}$ ${\displaystyle 2.2491\cdot 10^{-67}}$ m2

## Stoney scale forces

### Stoney scale static forces

Electric Stoney scale force:

${\displaystyle F_{S}(q_{S},q_{S})={\frac {1}{4\pi \varepsilon _{0}}}\cdot {\frac {e^{2}}{r^{2}}}={\frac {\alpha \hbar c}{r^{2}}}.\ }$

Gravity Stoney scale force:

${\displaystyle F_{S}(m_{S},m_{S})={\frac {1}{4\pi \varepsilon _{g}}}\cdot {\frac {m_{S}^{2}}{r^{2}}}={\frac {\alpha _{S}\hbar c}{r^{2}}},\ }$

where ${\displaystyle \alpha _{S}={\frac {m_{S}^{2}}{2hc\varepsilon _{g}}}=\alpha \ }$ is the gravitational fine structure constant.

Mixed (charge-mass interaction) Stoney force:

${\displaystyle F_{S}(m_{S},q_{S})={\frac {1}{4\pi {\sqrt {\varepsilon _{g}\varepsilon _{0}}}}}\cdot {\frac {m_{S}\cdot e}{r^{2}}}={\sqrt {\alpha _{S}\alpha }}{\frac {\hbar c}{r^{2}}}={\frac {\alpha \hbar c}{r^{2}}}.\ }$

So, at the Stoney scale we have the equality of all static forces which describes interactions between charges and masses:

${\displaystyle F_{S}(q_{S},q_{S})=F_{S}(m_{S},m_{S})=F_{S}(m_{S},q_{S})={\frac {\alpha \hbar c}{r^{2}}}.\ }$

### Stoney scale dynamic forces

Magnetic Stoney scale force:

${\displaystyle F_{S}(q_{m},q_{m})={\frac {1}{4\pi \mu _{0}}}\cdot {\frac {q_{m}^{2}}{r^{2}}}={\frac {\beta \hbar c}{r^{2}}},\ }$

where ${\displaystyle q_{m}={\frac {h}{e}}\ }$ is the fictitious elementary magnetic charge, ${\displaystyle \beta ={\frac {h}{2c\mu _{0}e^{2}}}\ }$ is the magnetic coupling constant.

Gravitational torsion force:

${\displaystyle F_{S}(m_{\Omega },m_{\Omega })={\frac {1}{4\pi \mu _{g0}}}\cdot {\frac {m_{\Omega }^{2}}{r^{2}}}={\frac {\beta _{g}\hbar c}{r^{2}}},\ }$

where ${\displaystyle m_{\Omega }={\frac {h}{m_{S}}}\ }$ is the fictitious gravitational torsion mass, ${\displaystyle \beta _{g}={\frac {\varepsilon _{g}hc}{2m_{S}^{2}}}={\frac {h}{2c\mu _{g0}m_{S}^{2}}}=\beta \ }$ is the gravitational torsion coupling constant for the gravitational torsion mass ${\displaystyle m_{\Omega }\ }$.

Mixed dynamic (magnetic - torsion interaction) force:

${\displaystyle F_{S}(q_{m},m_{\Omega })={\frac {1}{4\pi {\sqrt {\mu _{g0}\mu _{0}}}}}\cdot {\frac {q_{m}\cdot m_{\Omega }}{r^{2}}}={\sqrt {\beta _{g}\beta }}{\frac {\hbar c}{r^{2}}}={\frac {\beta \hbar c}{r^{2}}}.\ }$

So, at the Stoney scale we have the equality of all dynamic forces which describes interactions between dynamic charges and masses:

${\displaystyle F_{S}(q_{m},q_{m})=F_{S}(m_{\Omega },m_{\Omega })=F_{S}(q_{m},m_{\Omega })={\frac {\beta \hbar c}{r^{2}}}.\ }$