Einstein Probabilistic Units/Why is gravity so weak?

= Why is gravity so weak? = This resource includes primary and/or secondary research.

Einstein Probabilistic Units provides new insights into this elusive yet straightforward question. We will focus our inquiry on the gravitational and electrostatic forces between two electrons

If we take any two electrons and locate them anywhere within the universe, the relative strength of the gravitational force to the electrostatic force is 2.4 x 10-43.$$\frac{F_{e\;gravity}}{F_{e\;electrostatic}}= \frac{G\;m_e^2}{k_e\;q_e^2}= \frac{5.5\times 10^{-71} Nm^2}{2.3\times 10^{-28}Nm^2}= 2.4\times 10^{-43}$$

To help frame this problem we will use dimensional balancing. Then a generalized model using Einstein Probabilistic Units is adapted to provide additional insight into the nature of gravity and it's coupling constants.

Dimensional Balancing
Dimensional Balancing is an extension of dimensional analysis. In dimensional balancing, the exponential dimensional profile on the right-hand side of the equation is always equal to the exponential dimensional profile on the left-hand side of the equation. Applying dimensional balancing to our inquiry reveals that force and charge share an exponential dimensional profile of D2 while mass and energy have an exponential dimensional profile of D3. Some examples of dimensional balancing are;


 * $${Force}=m\times \frac{l}{t^2} = D_m^3 \times {D_l^1}\times {D_t^{-2}}=D_{Force}^2$$
 * $$Charge=m^{1/2}\times\;l^{3/2}\times t^{-1}=D_m^{3/2}\times D_l^{3/2}\times D_t^{-1}=D_{Charge}^2$$
 * $${Energy}=F\times l= D_F^2 \times {D_l^1}=D_{E}^3$$

We can hypothesize that the relative weakness of the gravitational to the electrostatic force is due to the difference in the above exponential dimensional profiles for charge D2  and mass D3.

In contrast, each dimensional D3 mass must undergo a dimensional reduction of D-1 to develop into a D2 gravitational force. In Newton's universal gravitational force equation, for two masses results in the Newton gravitational constant having a dimensional profile of D-2.

Dimensionally, a D2 charge can transform directly into D2 electrostatic force. In Coulomb's electrostatic force equation, for two charges, the Coulomb constant has a dimensional profile of D0.


 * $${Coulomb\;Constant}\; k_e=\frac{m\;l^3}{C^2\;t^2} =\frac{D^3_m\;D^3_l}{D^4_C\;D^2_t}=D_{k_e}^{0}$$

For a pair of electrons the electrostatic force is;


 * $$F_=k_e\;\frac{q_e^2}{r^2}=D^{0}_{k_e}\;\frac{D^4_{q_e}}{D^2_r}=D^2_{F_e\;electrostatic}$$


 * $${Newtons\;Gravitational\;Constant}\; G=\frac{l^3}{m}{\left(\frac{1}{t^2} \right)}=\frac{D^3_l}{D^3_m}\;{\left(\frac{1}{D^2_t}\right)}=D_G^{-2}$$

For a pair of electron the gravitational force is;


 * $$F_=G\;\frac{m_e^2}{r^2}=D^{-2}_{G}\;\frac{D^6_{m_e}}{D^2_r}=D^2_{F_e\;gravity}$$

This supports the observation that the charge to force process dose not require a dimensional transformation.

Einstein’s Gravitational Coefficient
In this section we will generalize Einstein's B coefficient for light to apply to all physical process, relationships and laws. This will provide us additional insight into the gravitational frequency squared dimensional reduction discussed above.

Einstein's B coefficient is given by and has a frequency squared variable;


 * $$B=\frac{l}{m}=\frac{c^3}{h\;v^2}$$

In Einstein's field equations, Einstein’s gravitational constant kappa is proportional to Einstein's Bk gravitational coefficient, (at the Planck scale this is also equal to BP or the B0 zero-point background radiation field).


 * $$B_\kappa=\frac{\kappa}{8 \pi}=\frac{G}{c^2}=B_P= \frac{l_P}{m_P}=7.4\times 10^{-28} \frac{m}{kg}$$

All values of Bx are frequency squared scaled versions of Einstein's Bk gravitational coefficient;


 * $$B_x=B_{\kappa} \times \frac{v^2_{\kappa} }{v^2_x}$$

Einstein’s Be coefficient for the electron is;


 * $$B_e=\frac{r_e}{m_e}=\frac{2.8\times 10^{-15}\;m}{9.1\times 10^{-31}\;kg}= 3.1\times 10^{15} \frac{m}{kg}$$

Comparing the ratio of Bk to Be,


 * $$\frac{B_\kappa}{B_e}=2.4\times 10^{-43}$$

This ratio is exactly equal to the relative strength of the gravitational to the electrostatic force of two electrons. As the B coefficient is proportional to the inverse frequency squared the following holds:


 * $$\frac{G\;{m^2_e}}{k_e\;q_e^2}=\frac{B_\kappa}{B_e}=\frac{\upsilon_{Be}^2}{\upsilon_{B\kappa}^2}=2.4\times 10^{-43}$$

From the above we can also conclude the weakness of the gravitational force is related to a frequency squared dimensional reduction needed to transform mass into force.

Einstein’s Coefficients - Force
Forces can be defined by using Einstein's B coefficient.


 * $$F_B=\frac{c^2}{B}=\frac{h}{c}\;\upsilon^2=m\times a$$

Expressing Newton's law of universal gravitation using Einstein's B coefficient


 * $$F_=G\;\frac{m^2}{r^2}=\frac{G}{B^2}$$

For an electron these forces are,


 * $$F_{B_e}=\frac{c^2}{B_e}=\sqrt{F_P\;F_{BG_e}}=2.9\times 10^1\;N$$


 * $$F_{BG_e}=\frac{G}{B_e^2}=\frac{F_{B_e}^2}{F_{B_P}}=7.0\times 10^{-42}\;N$$


 * $$F_P=\frac{c^4}{G}=\frac{c^2}{B_{\kappa}}=\frac{c^2}{B_P}=\frac{F_{B_e}^2}{F_{{BG}_e}}=1.2\times 10^{44}\;N$$

Thus illustrating the frequency squared reduction.


 * $$\frac{F_{BG_e}}{F_{B_e}}=\frac{G\times B_e}{c^2\times B_e^2}=\frac{G\times\hbar}{c^5}\;\upsilon^2_{B_e}={Constant}\times \;\upsilon^2_{B_e}=2.4\times 10^{-43}$$

We can extend the gravitational to the electrostatic force relationship;


 * $$\frac{G\;{m^2_e}}{k_e\;q_e^2}=\frac{B_\kappa}{B_e}=\frac{\upsilon_{Be}^2}{\upsilon_{P}^2}=\frac{F_{BG_e}}{F_{B_e}}=\frac{F_{B_e}}{F_P}=2.4\times 10^{-43} $$

Einstein B coefficient oscillator and coupling strength
To further understand the relationship for the fundamental gravitational and electrostatic forces, we will study the Einstein B coefficient oscillator strength. The oscillator strength f expresses the probability for creating a stimulated change in momentum or stimulated force. Einstein B  coefficient expressed in terms of oscillator strength ƒ is;


 * $$B= \frac{k_e\;e^2}{m_e\;h\;\upsilon_{}}\;f$$

Using Einstein B coefficient, force can be expressed as;


 * $$Force\;_B=\frac{c^2}{B}=\frac{h}{c}\;\upsilon^2=\frac{m_e\;c^2\;h\;\upsilon_{}}{k_e\;e^2}\;\frac{1}{f} $$

Solving for oscillator strength as a function of frequency we get;


 * $$f =\frac{m_e\;c^3}{k_e\;e^2}\frac{1}{\upsilon}=\frac{1.1\times 10^{23}}{\upsilon}\frac{1}{s}$$

Solving oscillator strength for specific frequencies,

Compton Frequency
 * $$\upsilon=\upsilon_{B_e}\;\;{then}\;\;\;f^{-1}_{B_e}=\frac{1}{\sqrt{\alpha}}\;\;\;f^{-2}_{B_e}=\frac{1}$$

Electron B Frequency


 * $$\upsilon=\bar{\upsilon}_C\;\;{then}\;\;\;f_C^{-1}={\alpha}\;\;\;f_C^{-2}={\alpha}^2$$

Planck Frequency
 * $$\upsilon=\upsilon_P\;\;{then}\;\;\;f_P^{-1}=\frac{\alpha}{\sqrt{\alpha_G}}\;\;\;f_P^{-2}=\frac{\alpha^2}$$

We can extend the gravitational to the electrostatic force relationship;


 * $$\frac{G\;{m^2_e}}{k_e\;q_e^2}=\frac{B_\kappa}{B_e}=\frac{\upsilon_{Be}^2}{\upsilon_{P}^2}=\frac{F_{G_e}}{F_{B_e}}=\frac{F_{B_e}}{F_P}=\frac{f^2_P}{f^2_{B_e}}=\frac{\alpha_G}{\alpha}=\alpha_e=2.4\times 10^{-43}$$

This relationship allows us to define the following coupling constants in terms of Einstein's B coefficient frequency squared;
 * $$\alpha_G=\frac{\bar{\upsilon}^2_C}{\upsilon^2_{P}}\;\;\;\alpha=\frac{\bar{\upsilon}^2_C}{\upsilon^2_{B_e}}\;\;\;\alpha_e=\frac{{\upsilon}^2_{B_e}}{\upsilon^2_{P}}$$

Conclusion
Using Einstein's Gravitational Coefficient and dimensional balancing we have demonstrated that the frequency squared dimension reduction used in Newton's law of universal gravitation plays a significant role in the relative strength of the of the gravitational force to the electrostatic force.

These concepts have also been extended to extend our understanding of the coupling constants and their dependence on the frequency squared relationships.

Given the probabilistic nature of Einstein's Coefficients, the hope is that the generalization of Einstein's Gravitational Coefficient can be used to converge the classic and quantum understanding of gravity.

Additional the generalization of Einstein's Gravitational Coefficient should lend itself to string theory models