User:Lemmiwinks2/Stoney scale units

Stoney scale units

 * main stoney scale units

Fundamental units
Boltzmann's constant, kB (or simply k) = 1. Dielectric constant :
 * $$\epsilon_E = \epsilon_0 = 8.854187817\cdot 10^{-12} \ $$ F m-1

Magnetic constant:
 * $$\mu_E = \mu_0 = \frac{1}{\epsilon_0 c^2} = 1.2566370614\cdot 10^{-6} \ $$ H m-1

Electrodynamic velocity of light:
 * $$c_E = \frac{1}{\sqrt{\epsilon_E\mu_E}} = 2.99792458\cdot 10^8 \ $$ m s-1

Electrodynamic vacuum impedance:
 * $$\rho_{E0} = \sqrt{\frac{\mu_E}{\epsilon_E}} = 376.730313461 \ $$ Ohm

Dielectric-like gravitational constant:
 * $$\epsilon_G = \frac{1}{4\pi G} = 1.192708\cdot 10^9 \ $$ 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:
 * $$c_G = \frac{1}{\sqrt{\epsilon_G\mu_G}} = 2.9979246\cdot 10^8 \ $$ m s-1

Gravidynamic vacuum impedance:
 * $$\rho_{G0} = \sqrt{\frac{\mu_G}{\epsilon_G}} = 2.7966954\cdot 10^{-18} \ $$ m2 kg-1 s-1

The above fundamental constants define naturally the following relationship between mass and electric charge:
 * $$m_S = e\sqrt{\frac{\epsilon_G}{\epsilon_E}} = e\sqrt{\frac{\mu_E}{\mu_G}} = e\sqrt{\frac{\rho_{E0}}{\rho_{G0}}} \ $$

And since:
 * $$\alpha =\ \frac{e^2}{\hbar c \ 4 \pi \varepsilon_0}\ =\ \frac{e^2 c \mu_0}{2 h} = \frac{k_\mathrm{e} e^2}{\hbar c} = \frac{1}{137.035999}$$

Since the Bohr radius equals:
 * $$a_0 = \frac{4 \pi \epsilon_0 \hbar^2}{m_e e^2} = \frac{\hbar}{m_e\,c\,\alpha}$$

And the Bohr scale velocity equals:
 * $$v_B = \omega_B\cdot a_B = \frac{\hbar}{m_0a_B} = \alpha c $$(velocity of electron in Bohr atom)

Therefore the above fundamental constants also define the reduced Planck's constant as:
 * $$ \hbar = \frac{1}{\alpha} = a_0 m_e\,c\,\alpha$$(ℏ = twice the angular momentum of hydrogen electron = energy * time)

Since the Rydberg constant equals:
 * $$R_\infty = \frac{\alpha^2 m_e c}{4 \pi \hbar} = \frac{\alpha^2}{2 \lambda_e} \ $$

Therefore the Frequency of the hydrogen electron equals:
 * $$2 cR_H = \frac{ m_e e^4}{4 \epsilon_0^2 h^3} = \frac{\alpha^2 m_e c^2}{2 \pi \hbar} $$(freq * wavelength = velocity)

The Bohr magneton is defined in SI units by
 * $$\mu_\mathrm{B} = {{e \hbar} \over {2 m_\mathrm{e}}}$$

If the electron is visualized as a classical charged particle literally rotating about an axis with angular momentum $$\vec{L}$$, its magnetic dipole moment $$\vec{\mu}$$ is given by:
 * $$\vec{\mu} = \frac{-e}{2m_e}\, \vec{L}$$

Here the charge is $$-e$$ where $$e$$ is the elementary charge. The mass is the electron rest mass $$m_e$$. Note that the angular momentum $$\vec{L}$$ in this equation may be the spin angular momentum, the orbital angular momentum, or the total angular momentum. It turns out the classical result is off by a proportional factor for the spin magnetic moment. As a result, the classical result is corrected by multiplying it with a correction factor.
 * $$\vec{\mu} = g \, \frac{-e}{2m_e}\, \vec{L}$$

The dimensionless correction factor g is known as the g-factor. Finally, it is customary to express the magnetic moment in terms of the Planck constant and the Bohr magneton:
 * $$\vec{\mu} = -g \mu_B \frac{\vec{L}}{\hbar} $$

Here $$\mu_{B}$$ is the Bohr magneton and $$\hbar\,$$ is the reduced Planck constant.

The Bohr radius including the effect of reduced mass can be given by the following equation:
 * $$ \ a_0^* \ = \frac{\lambda_p + \lambda_e}{2\pi\alpha}$$,

the Classical electron radius equals:
 * $$r_e = \frac{\alpha \lambda_e}{2\pi} = \alpha^2 a_0$$(lambda = Compton wavelength)

Rydberg energy(equal to 1/2 Hartree):
 * $$R_y = h c R_\infty = \frac{h c \alpha^2}{2 \lambda_e} = \frac{h f_C \alpha^2}{2} = \frac{\hbar \omega_C}{2} \alpha^2 \ $$ (binding energy of the hydrogen electron)

Gravitational units
Stoney mass:
 * $$m_S = e\sqrt{\frac{\epsilon_G}{\epsilon_E}} = \sqrt{\alpha} m_P = 1.85921\cdot 10^{-9}  \ $$ kg,

where $$m_P \ $$ is Planck mass. Stoney gravitational fine structure constant:
 * $$\alpha_{GS} = \frac{m_S^2}{2hc\epsilon_G} = \alpha = 7.29973506\cdot 10^{-3} \ $$

Stoney "dynamic mass", or gravitational magnetic-like flux:
 * $$\phi_{GS} = \frac{h}{m_S} = 3.56333\cdot 10^{-25} \ $$ J s kg-1

Stoney scale gravitational magnetic-like fine structure constant
 * $$\beta_{GS} = \frac{\phi_{GS}^2}{2hc\mu_G} = \frac{1}{4\alpha} = 34.259009 \ $$

Stoney gravitational impedance quantum:
 * $$R_{GS} = \frac{\phi_{GS}}{m_S} = \frac{h}{m_S^2} = 1.91624\cdot 10^{-16} \ $$ J s kg-1

Electrical units
Stoney charge:
 * $$q_S = e = 1.6021892\cdot 10^{-19} \ $$ C

Stoney electric fine structure constant: (the speed of the electron in a Bohr atom)
 * $$\alpha_{ES} = \frac{q_S^2}{2hc\epsilon_E} = \alpha. \ $$

Stoney magnetic charge, or flux:
 * $$\phi_{MS} = \frac{h}{e} = \phi_0 = 4.1357013\cdot 10^{-15} \ $$ Wb

Stoney scale magnetic fine structure constant
 * $$\beta_{MS} = \frac{\phi_{MS}^2}{2hc\mu_E} = \frac{1}{4\alpha} = 34.259009 \ $$

Stoney electrodynamic impedance quantum:
 * $$R_{ES} = \frac{\phi_{MS}}{q_S} = \frac{h}{e^2} = 25812.815 \ $$ Ohm

is the s.c. von Klitzing constant.

Stoney scale static forces
Electric Stoney scale force:
 * $$F_S(q_S\cdot q_S) = \frac{1}{4\pi \epsilon_E}\cdot \frac{e^2}{r^2} = \frac{\alpha_{SE}\hbar c}{r^2}, \ $$

Gravity Stoney scale force:
 * $$F_S(m_S\cdot m_S) = \frac{1}{4\pi \epsilon_G}\cdot \frac{m_S^2}{r^2} = \frac{\alpha_{SG}\hbar c}{r^2}, \ $$

Mixed (charge-mass interaction) Stoney force:
 * $$F_S(m_S\cdot q_S) = \frac{1}{4\pi \sqrt{\epsilon_G\epsilon_E}}\cdot \frac{m_S\cdot e}{r^2} = \sqrt{\alpha_G\alpha_E}\frac{\hbar c}{r^2} = \frac{\alpha \hbar c}{r^2}, \ $$

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:
 * $$F_S(\phi_{MS}\cdot \phi_{MS}) = \frac{1}{4\pi \mu_E}\cdot \frac{\phi_{MS}^2}{r^2} = \frac{\beta_{SE}\hbar c}{r^2}, \ $$

Gravitational magnetic-like force:
 * $$F_S(\phi_{GS}\cdot \phi_{GS}) = \frac{1}{4\pi \mu_G}\cdot \frac{\phi_{GS}^2}{r^2} = \frac{\beta_{SG}\hbar c}{r^2}, \ $$

Mixed dynamic (charge-mass interaction) gorce:
 * $$F_S(\phi_{MS}\cdot \phi_{GS}) = \frac{1}{4\pi \sqrt{\mu_G\mu_E}}\cdot \frac{\phi_{MS}\cdot \phi_{GS}}{r^2} =  \sqrt{\beta_G\beta_E}\frac{\hbar c}{r^2} = \frac{\beta \hbar c}{r^2}, \ $$

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:
 * $$F_S(\phi_{MS}\cdot \phi_{MS}) = F_S(\phi_{GS}\cdot \phi_{GS}) = F_S(\phi_{MS}\cdot \phi_{GS}) = \frac{\beta \hbar c}{r^2}. \ $$

Derived Stoney scale 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, &epsilon;;
 * Gravitational vacuum "dielectric constant", &epsilon;G;
 * Planck constant, h;
 * Boltzmann's constant, kB (or simply k).

Each of these constants can be associated with at least one fundamental physical theory: c with special relativity, &epsilon;G with general relativity and Newtonian gravity, e and &epsilon;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 $1/undefined$ 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 $6.024$, and the volume of a gram-molecule (at s.t.p.) of $22.415 mm^{3}$, we derive, instead of 1018, the estimate $2.687$. 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&minus;1

Magnetic constant:
 * $$\mu_E = \mu_0 = \frac{1}{\varepsilon_0 c^2} = 1.2566370614\cdot 10^{-6} \ $$ H m&minus;1

Electrodynamic velocity of light:
 * $$c_E = \frac{1}{\sqrt{\varepsilon_E\mu_E}} = 2.99792458\cdot 10^8 \ $$ m s&minus;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&minus;3

Magnetic-like gravitational constant:
 * $$\mu_G = \frac{4\pi G}{c^2} = 9.328772\cdot 10^{-27} \ $$ m kg&minus;1

Gravidynamic velocity of light:
 * $$c_G = \frac{1}{\sqrt{\varepsilon_G\mu_G}} = 2.9979246\cdot 10^8 \ $$ m s&minus;1

Gravidynamic vacuum impedance:
 * $$\rho_{G0} = \sqrt{\frac{\mu_G}{\varepsilon_G}} = 2.7966954\cdot 10^{-18} \ $$ m2 kg&minus;1 s&minus;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.

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&minus;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&minus;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.

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

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.

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.

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 intermediate positronium atom. The effective mass of the positronoum atom is:


 * $$m_{Bp} = \frac{m_em_p}{m_e + m_p} = 0.5m_N, \ $$

where $$m_e, m_p = m_N $$ are electron and positron mass respectively. The energy scale for the positronium atom is:


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

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 $$\alpha_W = \left(\frac{\alpha}{2}\right)^4 = 1.7723168\cdot 10^{-10} $$ is the weak interaction force constant (or weak fine structure constant).

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