Stoney units

In physics, Stoney scale units are units of measurement named after the Irish physicist George Johnstone Stoney, who first proposed them in 1881. They are an example of natural units, i.e. units of measurement designed so that certain fundamental physical constants are normalized to unity. The constants that Stoney units normalize are the following.

• Elementary charge, e;
• Speed of light in a vacuum, c;
• Dielectric vacuum constant, ε;
• Gravitational vacuum "dielectric constant", ε_G;
• Planck constant, h;
• Boltzmann's constant, k_B (or simply k).

Each of these constants can be associated with at least one fundamental physical theory: c with special relativity, ε_G with general relativity and Newtonian gravity, e and ε_E with electrostatics, and k with statistical mechanics and thermodynamics. Stoney units have profound significance for theoretical physics since they simplify several recurring algebraic expressions of physical law by nondimensionalization. They are particularly relevant in research on unified theories such as quantum gravity.

History

Contemporary physics has settled on the Planck scale as the most suitable scale for the unified theory. The Planck scale was however anticipated by George Stoney. James G. O’Hara pointed out in 1974 that Stoney’s derived estimate of the unit of charge, 10⁻²⁰ Ampere (later called the Coulomb), was 1⁄16 of the correct value of the charge of the electron. Stoney’s use of the quantity 10¹⁸ for the number of molecules presented in one cubic millimetre of gas at standard temperature and pressure. Using Avogadro’s number 6.0238×10²³, and the volume of a gram-molecule (at s.t.p.) of 22.4146×10⁶ mm³, we derive, instead of 10¹⁸, the estimate 2.687×10¹⁶. So, the Stoney charge differs from the modern value for the charge of the electron about 1% (if he took the true number of molecules).

Stoney scale and Planck scale are intermediate between microscopic and cosmic processes and it was soon realized that either could be the right scale for a unified theory. The only notable attempt to construct such a theory from the Stoney scale was that of H. Weyl, who associated a gravitational unit of charge with the Stoney length. and who appears to have inspired Dirac’s fascination with LNH. However, Weyl’s dogmatic adherence to the principle of locality reduced his theory to a mathematical construct with some non-physical implications. The Stoney scale thereafter fell into such neglect that it should to be re-discovered by M. Castans and J. Belinchon, and by Ross McPherson

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 gravity by using the Maxwell-like gravitational equations during last decades, became clear that Stoney scale is independent scale of matter. Furthermore, it is the base of the contemporary electrodynamics and gravidynamics (classical and quantum). Due to McDonald first who used Maxwell equations to describe gravity was Oliver Heaviside The point is that in the weak gravitational field the standard theory of gravity could be written in the form of Maxwell equations It is evident that in 19th century there was no SI units, and therefore the first mention of the gravitational constants possibly due to Forward (1961)

In the 1980s Maxwell-like equations were considered in the Wald book of general relativity In the 1990s Kraus first introduced the gravitational characteristic impedance of free space, which was detailed later by Kiefer, and now Raymond Y. Chiao who is developing the ways of experimental determination of the gravitational waves.

Fundamental units of vacuum

Dielectric constant:

$\varepsilon _{E}=\varepsilon _{0}=8.854187817\cdot 10^{-12}\$ F m⁻¹

Magnetic constant:

$\mu _{E}=\mu _{0}={\frac {1}{\varepsilon _{0}c^{2}}}=1.2566370614\cdot 10^{-6}\$ H m⁻¹

Electrodynamic velocity of light:

$c_{E}={\frac {1}{\sqrt {\varepsilon _{E}\mu _{E}}}}=2.99792458\cdot 10^{8}\$ m s⁻¹

Electrodynamic vacuum impedance:

$\rho _{E0}={\sqrt {\frac {\mu _{E}}{\varepsilon _{E}}}}=376.730313461\$ Ohm

Dielectric-like gravitational constant:

$\varepsilon _{G}={\frac {1}{4\pi G}}=1.192708\cdot 10^{9}\$ kg s² m⁻³

Magnetic-like gravitational constant:

$\mu _{G}={\frac {4\pi G}{c^{2}}}=9.328772\cdot 10^{-27}\$ m kg⁻¹

Gravidynamic velocity of light:

$c_{G}={\frac {1}{\sqrt {\varepsilon _{G}\mu _{G}}}}=2.9979246\cdot 10^{8}\$ m s⁻¹

Gravidynamic vacuum impedance:

$\rho _{G0}={\sqrt {\frac {\mu _{G}}{\varepsilon _{G}}}}=2.7966954\cdot 10^{-18}\$ m² kg⁻¹ s⁻¹

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

The above fundamental constants define naturally the following relationship between mass and electric charge:

$m_{S}=e{\sqrt {\frac {\varepsilon _{G}}{\varepsilon _{E}}}}=e{\sqrt {\frac {\mu _{E}}{\mu _{G}}}}=e{\sqrt {\frac {\rho _{E0}}{\rho _{G0}}}}\$

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

Primary Stoney units

Gravitational Stoney units

Stoney mass:

$m_{S}=e{\sqrt {\frac {\varepsilon _{G}}{\varepsilon _{E}}}}={\sqrt {\alpha }}\,m_{P}=1.85921\cdot 10^{-9}\$ kg,

where $m_{P}\$ is Planck mass. Stoney gravitational fine structure constant:

$\alpha _{GS}={\frac {m_{S}^{2}}{2hc\varepsilon _{G}}}=\alpha =7.29973506\cdot 10^{-3}\$

Stoney "dynamic mass", or gravitational magnetic-like flux:

$\varphi _{GS}={\frac {h}{m_{S}}}=3.56333\cdot 10^{-25}\$ J s kg⁻¹

Stoney scale gravitational magnetic-like fine structure constant

$\beta _{GS}={\frac {\varphi _{GS}^{2}}{2hc\mu _{G}}}={\frac {1}{4\alpha }}=34.259009\$

Stoney gravitational impedance quantum:

$R_{GS}={\frac {\varphi _{GS}}{m_{S}}}={\frac {h}{m_{S}^{2}}}=1.91624\cdot 10^{-16}\$ J s kg⁻¹

Electromagnetic Stoney units

Stoney charge:

$q_{S}=e=1.6021892\cdot 10^{-19}\$ C

Stoney electric fine structure constant:

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

Stoney magnetic charge, or flux:

$\varphi _{MS}={\frac {h}{e}}=\varphi _{0}=4.1357013\cdot 10^{-15}\$ Wb

Stoney scale magnetic fine structure constant

$\beta _{MS}={\frac {\varphi _{MS}^{2}}{2hc\mu _{E}}}={\frac {1}{4\alpha }}=34.259009\$

Stoney electrodynamic impedance quantum:

$R_{ES}={\frac {\varphi _{MS}}{q_{S}}}={\frac {h}{e^{2}}}=25812.815\$ Ohm

is the s.c. von Klitzing constant.

Secondary Stoney scale units

All systems of measurement feature 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, which are 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.)

Used keys in the tables below: L = length, T = time, M = mass, Q = electric charge, Θ = temperature. The values given without uncertainties are exact due to the definitions of the metre and the ampere.

Table 1: Secondary Stoney units

Name	Dimension	Expressions	SI equivalent with uncertainties
Stoney wavelength	Length (L)	$\lambda _{S}={\frac {h}{m_{S}c}}$	1.188 60 × 10−33 m
Stoney time	Time (T)	$t_{S}={\frac {\lambda _{S}}{c}}$	3.964 74 × 10−42 s
Stoney classical radius	Length (L)	$r_{Sc}={\frac {\alpha \lambda _{S}}{2\pi }}$	1.380 45 × 10−36 m
Stoney Schwarzschild radius	Length (L)	$r_{SS}=2r_{Sc}\$	2.760 90 × 10−36 m
Stoney temperature	Temperature (Θ)	$T_{S}={\frac {m_{S}c^{2}}{k_{B}}}$	1.210 49 × 1031 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	Approximate SI equivalent
Stoney area	Area (L2)	$\lambda _{S}^{2}={\frac {1}{2\alpha \varepsilon _{G}c^{3}}}$	$1.4128\cdot 10^{-66}$ m2
Stoney volume	Volume (L3)	$\lambda _{S}^{3}=\left({\frac {h}{m_{S}c}}\right)^{3}$	$1.6792\cdot 10^{-99}$ m3
Stoney momentum	Momentum (LMT −1)	$m_{S}c={\frac {h}{\lambda _{S}}}$	$5.5748\cdot 10^{-1}$ kg m/s
Stoney energy	Energy (L2MT −2)	$W_{S}=m_{S}c^{2}=2\alpha c^{4}\varepsilon _{G}\$	$1.6713\cdot 10^{8}$ J
Stoney force	Force (LMT −2)	$F_{S}={\frac {W_{S}}{\lambda _{S}}}={\frac {m_{S}c^{2}}{\lambda _{S}}}$	$1.4061\cdot 10^{41}$N
Stoney power	Power (L2MT −3)	$P_{S}={\frac {W_{S}}{t_{S}}}=2\alpha \varepsilon _{G}c^{5}$	$4.2153\cdot 10^{49}$ W
Stoney density	Density (L−3M)	$\rho _{Sm}={\frac {m_{S}}{\lambda _{S}^{3}}}$	$1.1074\cdot 10^{90}$ kg/m3
Stoney angular frequency	Frequency (T −1)	$\omega _{S}={\frac {2\pi }{t_{S}}}={\frac {2\pi c}{\lambda _{S}}}$	$1.5848\cdot 10^{42}$ rad s−1
Stoney pressure	Pressure (L−1MT −2)	$p_{S}={\frac {F_{S}}{\lambda _{S}^{2}}}={\frac {m_{S}c^{2}}{\lambda _{S}^{3}}}$	$7.5615\cdot 10^{115}$ Pa
Stoney current	Electric current (QT −1)	$I_{S}={\frac {q_{S}}{t_{S}}}={\frac {ec}{\lambda _{S}}}$	$4.0411\cdot 10^{22}$ A
Stoney voltage	Voltage (L2MT −2Q−1)	$V_{S}={\frac {W_{S}}{q_{S}}}={\frac {m_{S}c^{2}}{e}}$	$1.0431\cdot 10^{27}$ V
Stoney electric impedance	Resistance (L2MT −1Q−2)	$R_{ES}={\frac {V_{S}}{I_{S}}}={\frac {h}{e^{2}}}$	$2.5813\cdot 10^{4}$ Ω
Stoney gravitational charge current	Gravitational current (MT −1)	$I_{GS}={\frac {m_{S}}{t_{S}}}={\frac {m_{S}^{2}c^{2}}{h}}$	$4.6902\cdot 10^{32}$ kg s−1
Stoney gravitational charge voltage	Gravitational voltage (L2T −2)	$V_{GS}={\frac {W_{S}}{m_{S}}}=c^{2}$	$8.9876\cdot 10^{16}$ m2 s−2
Stoney gravitational charge impedance	gravitational impedance (LT −2)	$R_{GS}={\frac {V_{GS}}{I_{GS}}}={\frac {h}{m_{S}^{2}}}$	$1.9162\cdot 10^{-16}$ m s−2
Stoney electric capacitance per unit area	Electric capacitance (L−2M−1T2Q2)	$C_{ES}={\frac {\varepsilon _{E}}{\lambda _{S}}}$	$7.4493\cdot 10^{21}$ F m−2
Stoney electric inductance per unit area	Electric inductance (L2MT −2Q−2)	$L_{ES}={\frac {\mu _{S}}{\lambda _{S}}}$	$1.0572\cdot 10^{27}$ H m−2
Stoney gravity capacitance per unit area	Gravitational capacitance (L−4MT2 )	$C_{GS}={\frac {\varepsilon _{G}}{\lambda _{S}}}$	$1.0035\cdot 10^{42}$ m−4 kg s2
Stoney gravity inductance per unit area	Gravitational inductance (M−1)	$L_{GS}={\frac {\mu _{G}}{\lambda _{S}}}$	$7.8485\cdot 10^{6}$ kg−1
Stoney particle radius	Length (L)	$r_{S}={\frac {\lambda _{S}}{2\pi {\sqrt {2}}}}$	$1.3376\cdot 10^{-34}$ m
Stoney particle area	Area (L2)	$S_{S}=4\pi r_{S}^{2}={\frac {\lambda _{S}^{2}}{2\pi }}$	$2.2485\cdot 10^{-67}$ m 2

Stoney scale forces

Stoney scale static forces

Electric Stoney scale force:

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

where $\alpha _{SE}={\frac {e^{2}}{2hc\varepsilon _{E}}}=\alpha \$ is the electric fine structure constant. Gravity Stoney scale force:

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

where $\alpha _{SG}={\frac {m_{S}^{2}}{2hc\varepsilon _{G}}}=\alpha \$ is the gravity fine structure constant. Mixed (charge-mass interaction) Stoney force:

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

where ${\sqrt {\alpha _{SG}\alpha _{SE}}}=\alpha \$ is the mixed fine structure constant.

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

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

Stoney scale dynamic forces

Magnetic Stoney scale force:

$F_{S}(\varphi _{MS}\cdot \varphi _{MS})={\frac {1}{4\pi \mu _{E}}}\cdot {\frac {\varphi _{MS}^{2}}{r^{2}}}={\frac {\beta _{SE}\hbar c}{r^{2}}},\$

where $\beta _{SE}={\frac {\varphi _{MS}^{2}}{2hc\mu _{E}}}=\beta \$ is the magnetic fine structure constant. Gravitational magnetic-like force:

$F_{S}(\varphi _{GS}\cdot \varphi _{GS})={\frac {1}{4\pi \mu _{G}}}\cdot {\frac {\varphi _{GS}^{2}}{r^{2}}}={\frac {\beta _{SG}\hbar c}{r^{2}}},\$

where $\beta _{SG}={\frac {\varphi _{GS}^{2}}{2hc\mu _{G}}}=\beta \$ is the magnetic-like gravitational fine structure constant. Mixed dynamic (charge-mass interaction) gorce:

$F_{S}(\varphi _{MS}\cdot \varphi _{GS})={\frac {1}{4\pi {\sqrt {\mu _{G}\mu _{E}}}}}\cdot {\frac {\varphi _{MS}\cdot \varphi _{GS}}{r^{2}}}={\sqrt {\beta _{G}\beta _{E}}}{\frac {\hbar c}{r^{2}}}={\frac {\beta \hbar c}{r^{2}}},\$

where ${\sqrt {\beta _{SG}\beta _{SE}}}=\beta ={\frac {1}{4\alpha }}.\$

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

$F_{S}(\varphi _{MS}\cdot \varphi _{MS})=F_{S}(\varphi _{GS}\cdot \varphi _{GS})=F_{S}(\varphi _{MS}\cdot \varphi _{GS})={\frac {\beta \hbar c}{r^{2}}}.\$

Planck scale units

For the sake of completeness in the Table 3 presented the main Planck units in the form consistent with above tables for Stoney scale.

Table 3: Base Planck units

Name	Dimension	Expressions	SI equivalent with uncertainties	Other equivalent
Planck mass	Mass (M)	$m_{\text{P}}={\sqrt {\frac {\hbar c}{G}}}={\sqrt {2hc\varepsilon _{G}}}\$	2.176 44(11) × 10−8 kg	1.220 862(61)× 1019 GeV/c2
Planck wavelength	Length (L)	$\lambda _{\text{P}}={\frac {h}{m_{P}c}}$	1.013 56 × 10−34 m
Planck gravity fine structure constant	Dimensionless	$\alpha _{GP}={\frac {m_{P}^{2}}{2hc\varepsilon _{G}}}$	1
Planck "dynamic mass"	Dynamic mass (L2T −1)	$\varphi _{GP}={\frac {h}{m_{P}}}\$	3.043 96 × 10−26 m2 s−1
Planck "dynamic mass" fine structure constant	Dimensionless	$\beta _{GP}={\frac {\varphi _{P}^{2}}{2hc\mu _{G}}}$	1/4
Planck time	Time (T)	$t_{\text{P}}={\frac {\lambda _{\text{P}}}{c}}$	3.386 86 × 10−43 s
Planck charge	Electric charge (Q)	$q_{\text{P}}=m_{\text{P}}{\sqrt {4\pi \varepsilon _{0}G}}={\frac {e}{\sqrt {\alpha }}}$	1.875 545 870(47) × 10−18 C	11.706 237 6398(40) e
Planck electric fine structure constant	Dimensionless	$\alpha _{EP}={\frac {q_{P}^{2}}{2hc\varepsilon _{E}}}={\frac {\alpha }{\alpha }}$	1
Planck "magnetic charge"	magnetic charge (L2MT −1Q−1)	$\varphi _{MP}={\frac {h}{q_{P}}}={\sqrt {\alpha }}\varphi _{0}\$	3.532 90 × 10−16 Wb
Planck "magnetic charge" fine structure constant	Dimensionless	$\beta _{MP}={\frac {\varphi _{MP}^{2}}{2hc\mu _{E}}}$	1/4
Planck gravity impedance quantum	Gravitational impedance (L2M−1T −1)	$R_{GP}={\frac {\varphi _{GP}}{m_{P}}}={\frac {h}{m_{P}^{2}}}\$	1.398 35 × 10−18 m2 kg−1 s−1
Planck electromagnetic impedance quantum	Electrical impedance (L2M−1T −1Q−2)	$R_{EP}={\frac {\varphi _{MP}}{q_{P}}}=\alpha {\frac {h}{e^{2}}}\$	1.883 65 × 102 Ω

As could be seen from the table, the main difference between Stoney and Planck units - the fine structure constants. For example, the wave vacuum impedance in the Planck scale will be:

$\rho _{EP}={\sqrt {\frac {\mu _{E}}{\varepsilon _{E}}}}=2\alpha \cdot {\frac {h}{e^{2}}}=2\cdot {\frac {h}{q_{P}^{2}}}.\$

This is due to the difference in fine structure constants. Actually, the relationship between "static" and "dynamic" forces in the Planck scale is:

${\sqrt {\frac {F_{Pst}}{F_{Pdyn}}}}={\sqrt {\frac {\alpha _{P}}{\beta _{P}}}}=2\alpha _{P}=2,\$

but in the Stoney scale it will be:

${\sqrt {\frac {F_{Sst}}{F_{Sdyn}}}}={\sqrt {\frac {\alpha _{S}}{\beta _{S}}}}=2\alpha _{S}=2\alpha .\$

Natural scale units based on electron mass

For the sake of completeness in the Table 4 presented the main Natural scale units based on electron mass in the form consistent with above tables for Stoney scale.

Table 4: Base Natural scale units

Name	Dimension	Expressions	SI equivalent with uncertainties	Other equivalent
Electron mass	Mass (M)	$m_{N}=m_{0}\$	9.109382 15(45) × 10−31 kg	5.109989 10(13) × 10−1 MeV
Electron wavelength	Length (L)	$\lambda _{N}=\lambda _{0}={\frac {h}{m_{0}c}}$	2.42631021 75(33) × 10−12 m
Natural gravity fine structure constant	Dimensionless	$\alpha _{GN}={\frac {m_{0}^{2}}{2hc\varepsilon _{G}}}$	1.751 12 × 10−45
Natural "dynamic mass"	Dynamic mass (L2T −1)	$\varphi _{GN}={\frac {h}{m_{0}}}\$	7.273 39 × 10−4 m2 s−1
Natural "dynamic mass" fine structure constant	Dimensionless	$\beta _{GN}={\frac {\varphi _{GN}^{2}}{2hc\mu _{G}}}={\frac {1}{4\alpha _{GN}}}\$	1.427 57 × 10+44
Natural charge	Electric charge (Q)	$q_{N}={\sqrt {2hc\varepsilon _{E}\alpha _{GN}}}=e{\sqrt {\frac {\alpha _{GN}}{\alpha }}}=e\left({\frac {m_{0}}{m_{S}}}\right)$	7.848 545 79 × 10−41 C
Natural electric fine structure constant	Dimensionless	$\alpha _{EN}={\frac {q_{N}^{2}}{2hc\varepsilon _{E}}}=\alpha _{GN}\$	1.751 12 × 10−45
Natural "magnetic charge"	magnetic charge (L2MT −1Q−1)	$\varphi _{MN}={\frac {h}{q_{N}}}={\frac {h}{e}}{\sqrt {\frac {\alpha }{\alpha _{GN}}}}\$	8.442 29 × 106 Wb
Natural "magnetic charge" fine structure constant	Dimensionless	$\beta _{MN}={\frac {\varphi _{MN}^{2}}{2hc\mu _{E}}}={\frac {1}{4\alpha _{GN}}}$	1.427 57 × 1044
Natural gravity impedance quantum	Gravitational impedance (L2M−1T −1)	$R_{GN}={\frac {\varphi _{GN}}{m_{0}}}={\frac {h}{m_{0}^{2}}}\$	7.984 92 × 1026 m2 kg−1 s−1
Natural electromagnetic impedance quantum	Electrical impedance (L2M−1T −1Q−2)	$R_{EN}={\frac {\varphi _{MN}}{q_{N}}}={\frac {h}{e^{2}}}{\frac {\alpha }{\alpha _{GN}}}\$	1.075 62 × 1047 Ω

Note that, the Natural scale has different values for the fine structure constants (as the Planck scale does). However, this difference is so high, that this scale now is the base for the LNH and different numerology approaches . Actually, the relationship between Stoney and Natural fine structure constants yields the s.c. Dirack number:

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

Weak interaction Natural scale units

The weak scale of Natural units is based on the neutrino mass. As is known, neutrinos are generated during the annihilation process, which is going through intermidiate positronium atom. The effective mass of the positronoum atom is:

$m_{Bp}={\frac {m_{e}m_{p}}{m_{e}+m_{p}}}=0.5m_{N},\$

where $m_{e},m_{p}=m_{N}$ are electron and positron mass respectively. The energy scale for the positronium atom is:

$W_{Bp}={\frac {\hbar ^{2}}{2m_{Bp}a_{Bp}^{2}}}=({\frac {\alpha }{2}})^{2}\cdot m_{N}c^{2}=m_{WN}c^{2},\$

where $a_{Bp}=2a_{B}={\frac {\lambda _{N}}{\pi \alpha }}$ is the length scale for positronium, and $m_{WN}={\sqrt {\alpha _{W}}}\cdot m_{N}$ is the upper value for the neutrino mass, and $\alpha _{W}=({\frac {\alpha }{2}})^{4}=1.7723168\cdot 10^{-10}$ is the weak interaction force constant (or weak fine structure constant).

Table 4: Base weak Natural scale units

Name	Dimension	Expressions	SI equivalent with uncertainties	Other equivalent
Neutrino mass	Mass (M)	$m_{WN}={\sqrt {\alpha _{W}}}m_{N}\$	1.21273 70 × 10−35 kg
Neutrino wavelength	Length (L)	$\lambda _{WN}={\frac {h}{m_{WN}c}}$	1.88225 05 × 10−7 m
Weak interaction force constant	Dimensionless	$\alpha _{W}=({\frac {\alpha }{2}})^{4}$	1.77231 68 × 10−10
Weak gravity force constant	Dimensionless	$\alpha _{WN}=\alpha _{W}\cdot \alpha _{GN}$	3.1047 2 × 10−55
Weak Natural "dynamic mass"	Dynamic mass (L2T −1)	$\varphi _{WGN}={\frac {h}{m_{WN}}}\$	5.4637 3 × 10−0 m2 s−1
Weak Natural "dynamic mass" force constant	Dimensionless	$\beta _{GN}={\frac {\varphi _{WGN}^{2}}{2hc\mu _{G}}}={\frac {1}{4\alpha _{WN}}}\$	3.1047 2 × 10+53
Weak Natural time	Time (T)	$t_{\text{WN}}={\frac {\lambda _{\text{WN}}}{c}}$	6.0792 2 × 10−16 s

Weak Planck scale units

The primordial level of matter has two standard scales: Planck (defines the Planck mass) and Stoney (defines the Stoney mass). However, it has the third primordial scale that could be named as the weak interaction scale, which has the following force constant:

$\alpha _{W}=({\frac {\alpha }{2}})^{4},\$

that is the same as in the weak natural scale.

The weak primordial mass will be:

$m_{WP}={\sqrt {\alpha _{W}}}\cdot m_{P}=2.897473\cdot 10^{-13}\$kg,

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

The weak primordial wavelength is:

$\lambda _{WP}={\frac {h}{cm_{WP}}}=7.62809\cdot 10^{-30}\$m

The weak primordial time is:

$t_{WP}={\frac {h}{c^{2}m_{WP}}}=2.544458\cdot 10^{-38}\$s

Work function and Universe scale

The standard definition of the work function in the strength field is:

$A_{\lambda }=F_{\lambda }\cdot \lambda ={\frac {\hbar c}{\lambda }}={\frac {m_{lambda}c^{2}}{2\pi }}.\$

So, the complex weak displacement work in the weak natural force will be:

$A_{NWW}=F_{WN}\cdot \lambda _{W}=\alpha _{W}\alpha _{N}\cdot {\frac {\hbar c}{\lambda _{W}}},\$

where

$F_{WN}={\frac {m_{WN}^{2}}{2hc\epsilon _{G}}}=\alpha _{W}\alpha _{N}\cdot {\frac {\hbar c}{r^{2}}}\$

is the weak natural force, and $\lambda _{W}$ is the weak Planck wavelength.

Considering the Universe bubble as the minimal energy scale:

$W_{U}=h\nu _{U}={\frac {\hbar c}{\lambda _{U}}},\$

where $\lambda _{U}$ is the Universe wavelength, and equating the above energies, we derive the following fundamental relationship:

${\frac {\lambda _{W}}{\alpha _{W}\alpha _{N}}}={\frac {\lambda _{U}}{2\pi }},\$

from which the Universe length parameter could be derived:

$\lambda _{U}={\frac {2\pi \lambda _{W}}{\alpha _{W}\alpha _{N}}}=1.5437\cdot 10^{26}\$m

which value is consistent with the 15 billion years.

See also

• Dimensional analysis
• Stoney mass
• Natural units
• Physical constants
• Planck scale

External links

• Stoney Scale and Large Number Coincidences