Physics/Essays/Anonymous/Weak natural scale

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In physics, Weak Natural scale is the fundamental scale of matter, named after

It defines the Weak Natural coupling constant:

${\displaystyle \alpha _{WN}={\frac {m_{\nu }^{2}}{2hc\,\varepsilon _{G}}},\ }$

where

• ${\displaystyle \!m_{\nu }}$ is the upper limit for electron neutrino mass;
• ${\displaystyle \!h}$ is the Planck constant;
• ${\displaystyle \!c}$ is the speed of light in vacuum;
• ${\displaystyle \!\varepsilon _{G}}$ is the gravitational electric-like constant. Thus, the Natural scale defines the force interaction dimensionless parameter, something like gravitational fine structure constant.

History

Usually, the Weak Natural scale now considered for definition of the weak interaction force and has not appropriate attention that should be fo the scale of matter. However, the real strength of forces is determined by the scale only, but not the metter type: charge or mass.

Fundamental units of vacuum

Dielectric constant [1]:

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

Magnetic constant:

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

Electrodynamic velocity of light:

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

Electrodynamic vacuum impedance:

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

Dielectric-like gravitational constant:

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

Magnetic-like gravitational constant:

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

Gravidynamic velocity of light:

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

Gravidynamic vacuum impedance:

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

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

Weak interaction Natural scale units

The weak scale of Natural units is based on the electron 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:

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

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

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

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

Table 5: Base weak Natural scale units

Name	Dimension	Expressions	SI equivalent with uncertainties [1]
Neutrino mass	Mass (M)	${\displaystyle m_{WN}={\sqrt {\alpha _{W}}}m_{N}\ }$	${\displaystyle 1.2127370\cdot 10^{-35}}$ kg
Neutrino wavelength	Length (L)	${\displaystyle \lambda _{WN}={\frac {h}{m_{WN}c}}}$	${\displaystyle 1.8822505\cdot 10^{-7}}$ m
Weak interaction force constant	Dimensionless	${\displaystyle \alpha _{W}=\left({\frac {\alpha }{2}}\right)^{4}}$	${\displaystyle 1.7723168\cdot 10^{-10}}$
Weak gravity force constant	Dimensionless	${\displaystyle \alpha _{WN}=\alpha _{W}\cdot \alpha _{GN}}$	${\displaystyle 3.10472\cdot 10^{-55}}$
Weak Natural "dynamic mass"	Dynamic mass (L2T −1)	${\displaystyle \varphi _{WGN}={\frac {h}{m_{WN}}}\ }$	${\displaystyle 5.46373\cdot 10^{0}}$ m2 s−1
Weak Natural "dynamic mass" force constant	Dimensionless	${\displaystyle \beta _{GN}={\frac {\varphi _{WGN}^{2}}{2hc\mu _{G}}}={\frac {1}{4\alpha _{WN}}}\ }$	${\displaystyle 3.10472\cdot 10^{53}}$
Weak Natural time	Time (T)	${\displaystyle t_{\text{WN}}={\frac {\lambda _{\text{WN}}}{c}}}$	${\displaystyle 6.07922\cdot 10^{-16}}$ 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:

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

that is the same as in the weak natural scale.

The weak primordial mass will be:

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

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

The weak primordial wavelength is:

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

The weak primordial time is:

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

Work function and Universe scale

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

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

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

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

where

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

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

Considering the Universe bubble as the minimal energy scale:

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

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

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

from which the Universe length parameter could be derived:

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

which value is consistent with the 15 billion years.

Weak Planck scale and Solar planatery system

Planet resonator characteristics

Geometrical parameters of any planetary object determine the following resonance frequency:

${\displaystyle \omega _{p}={\frac {c}{R_{p}}}=4.7055793\cdot 10^{1}rad/s\ }$,

here is used the Earth as an example. This resonance frequency could be connected with the "minimal mass":

${\displaystyle m_{pmin}={\frac {\hbar \omega _{p}}{c^{2}}}=5.521382\cdot 10^{-50}kg\ }$,

where ${\displaystyle \hbar }$ is the reduced Planck constant, and ${\displaystyle c}$ is the speed of light.

Considering that gravitational resonator has its oscillations on the surface, therefore it is interesting to determine the minimal surface radius connected with the "minimal mass":

${\displaystyle r_{pSmin}=R_{p}\cdot {\sqrt {\frac {m_{pmin}}{M_{p}}}}=6.123873\cdot 10^{-31}m\ }$.

The relationship between minimal radius and the Weak Planck length is:

${\displaystyle {\frac {r_{pmin}}{l_{WP}}}=1.98248\ }$,

where ${\displaystyle l_{WP}={\frac {l_{P}}{\sqrt {\alpha _{WP}}}}=1.21405\cdot 10^{-30}m\ }$ is the Weak Planck length. Thus, considering the Solar Planetary System, all objects as gravitational resonators, we will have the small surface area about the Weak Planck scale, where the minimal resonator energy quant ${\displaystyle \hbar \omega _{p}}$ replaced.

The full sets of the planetary data are presented in the Table 2.

Table 2: Solar planatery system

Object	Radius, m	Mass, kg	Minimal Mass, kg	Minimal Radius, m	${\displaystyle l_{W}/r_{min}}$
Sun	${\displaystyle 6.96\cdot 10^{8}}$	${\displaystyle 1.989\cdot 10^{30}}$	${\displaystyle 5.054\cdot 10^{-52}}$	${\displaystyle 1.1095\cdot 10^{-32}}$	${\displaystyle 109.4}$
Jupiter	${\displaystyle 7.13\cdot 10^{7}}$	${\displaystyle 1.899\cdot 10^{27}}$	${\displaystyle 4.934\cdot 10^{-51}}$	${\displaystyle 1.149\cdot 10^{-31}}$	${\displaystyle 10.56}$
Saturn	${\displaystyle 6.01\cdot 10^{7}}$	${\displaystyle 5.686\cdot 10^{26}}$	${\displaystyle 5.853\cdot 10^{-51}}$	${\displaystyle 1.928\cdot 10^{-31}}$	${\displaystyle 6.3}$
Neptun	${\displaystyle 2.51\cdot 10^{7}}$	${\displaystyle 1.03\cdot 10^{26}}$	${\displaystyle 1.402\cdot 10^{-50}}$	${\displaystyle 2.928\cdot 10^{-31}}$	${\displaystyle 4.15}$
Uran	${\displaystyle 2.45\cdot 10^{7}}$	${\displaystyle 8.689\cdot 10^{25}}$	${\displaystyle 1.436\cdot 10^{-50}}$	${\displaystyle 3.149\cdot 10^{-31}}$	${\displaystyle 3.86}$
Earth	${\displaystyle 6.371\cdot 10^{6}}$	${\displaystyle 5.976\cdot 10^{24}}$	${\displaystyle 5.521\cdot 10^{-50}}$	${\displaystyle 6.124\cdot 10^{-31}}$	${\displaystyle 1.98}$
Venus	${\displaystyle 6.07\cdot 10^{6}}$	${\displaystyle 4.87\cdot 10^{24}}$	${\displaystyle 5.795\cdot 10^{-50}}$	${\displaystyle 6.622\cdot 10^{-31}}$	${\displaystyle 1.83}$
Mars	${\displaystyle 3.395\cdot 10^{6}}$	${\displaystyle 6.424\cdot 10^{23}}$	${\displaystyle 1.036\cdot 10^{-49}}$	${\displaystyle 1.364\cdot 10^{-30}}$	${\displaystyle 0.89}$
Mercury	${\displaystyle 2.425\cdot 10^{6}}$	${\displaystyle 3.311\cdot 10^{23}}$	${\displaystyle 1.451\cdot 10^{-49}}$	${\displaystyle 2.185\cdot 10^{-30}}$	${\displaystyle 0.756}$

See also

• Planck scale
• Stoney scale
• Natural scale

References

• 1. Latest (2006) values of the constants