Nonstandard physics/Stoney scale

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

$~\alpha_{S} = \frac{m_S^2}{2hc\varepsilon_G}= \frac{e^2}{2hc\,\varepsilon_0} =\alpha,$

where

• $~ m_S = \sqrt{\frac{e^2}{4\pi \varepsilon_0 G}}$ is Stoney mass;
• $~e$ is elementary charge;
• $~\varepsilon_0$ is vacuum permittivity;
• $~G$ is gravitational constant;
• $~h$ is Planck constant;
• $~c$ is speed of light in vacuum;
• $~\varepsilon_G= \frac{1}{4\pi G}$ is electric-like gravitational constant.

The dimensionless parameter $~\alpha_S = \alpha$ 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 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).

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, ? = 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)	$\lambda_S = \frac{h}{m_Sc}$	$1.18860\cdot 10^{-33}$ m
Stoney time	Time (T)	$t_S = \frac{\lambda_S}{c}$	$3.96474\cdot 10^{-42}$ s
Stoney classical radius	Length (L)	$r_{Sc} = \frac{\alpha \lambda_S}{2\pi}$	$1.38045\cdot 10^{-36}$ m
Stoney Schwarzschild radius	Length (L)	$r_{SS} = 2r_{Sc} \$	$2.76090\cdot 10^{-36}$ m
Stoney temperature	Temperature (Θ)	$T_S = \frac{m_S c^2}{k_B}$	$1.21049\cdot 10^{31}$ 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	Approximate SI equivalent
Stoney area	Area (L2)	$\lambda_S^2 = \frac{1}{2\alpha \varepsilon_Gc^3}$	$1.4128\cdot 10^{-66}$ m2
Stoney volume	Volume (L3)	$\lambda_S^3 = \left(\frac{h}{m_Sc}\right)^3$	$1.6792\cdot 10^{-99}$ 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)	$F_S = \frac{W_S}{\lambda_S} = \frac{m_Sc^2}{\lambda_S}$	$1.4061\cdot 10^{41}$N
Stoney power	Power (L2MT −3)	$P_S = \frac{W_S}{t_S} = 2\alpha \varepsilon_Gc^5$	$4.2153\cdot 10^{49}$ 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)	$p_S = \frac{F_S}{\lambda_S^2} = \frac{m_Sc^2}{\lambda_S^3}$	$7.5615\cdot 10^{115}$ 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)	$V_S = \frac{W_S}{q_S} = \frac{m_Sc^2}{e}$	$1.0431\cdot 10^{27}$ V
Stoney electric impedance	Resistance (L2MT −1Q−2)	$R_{ES} = \frac{V_S}{I_S} = \frac{h}{e^2}$	$2.5813\cdot 10^{4}$ ?
Stoney gravitational charge current	Gravitational current (MT −1)	$I_{GS} = \frac{m_S}{t_S} = \frac{m_S^2c^2}{h}$	$4.6902\cdot 10^{32}$ 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}. \$

See also

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

References