Dirac's Constants and Fundamental Physical Ratios

Fundamental Forces:
$$F_{Electric} = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{r^2}$$ $$F_{Gravitation} = G\frac{m_p m_e}{r^2}$$ Operating a ratio between the two expressions, we have an indicative ratio of the intensity of the electric force relative to the gravitational force. In this context, the order of magnitude observed by Dirac appears for the first time. We have: $$\frac{F_{Coulomb}}{F_{Gravitation}} := \frac{e^2}{4\pi\varepsilon_0 G m_p m_e} \approx 10^{40}$$ Ratio between Electron Radius and Observable Universe
Another heuristic ratio that may have a certain interest is the ratio between the radius of the visible Universe and the classical radius of the electron. In previous chapters we introduced Hubble's constant $H_0$ and we saw how this is dimensionally an inverse of time. In particular $H_0^{-1}$ therefore represents a time called Hubble time or expansion time of the Universe and is considered indicative of the age of the Universe. This, according to recent observations of Hubble's constant, is 13.7 billion years. $$\frac{1}{H_0} = 13.7 \times 10^9 \text{ years}$$ Multiplying this value by the speed of light $c$, we obtain what is normally indicated as the cosmological horizon or radius of the observable universe which we will denote as $R_U$. $$R_U \approx c H_0^{-1}$$ Conversely, the classical radius of the electron indicates the dimensions that an electron, within the limits of classical physics, should have if its mass were simply composed of electrostatic energy. $$R_e \approx \frac{1}{4\pi\varepsilon_0}\frac{e^2}{m_e c^2}$$ Applying the ratio between the two quantities, we again have an order of magnitude close to $10^{40}$. Indeed: $$\frac{R_U}{R_e} \approx 10^{40}$$ Ratio between Compton Length and Schwarzschild Radius of the Proton
Finally, the last ratio calculated by Dirac is that between the Compton Length of the proton and its Schwarzschild radius. The Compton length of a particle indicates the wavelength associated with it by De Broglie in his hypothesis on the wave behavior of matter. From a theoretical point of view, the Compton length of a particle indicates the dimension for which the quantum behavior of the particle can no longer be neglected to describe its functioning. This length therefore indicates in a certain way the limit in the scale of dimensions from which the typical description of quantum mechanics of the particle begins to be valid. In practice, the Compton length of a particle is nothing other than the wavelength that a photon with energy equal to that of the particle would have. $$E = h\nu = \frac{hc}{\lambda} = mc^2$$ In particular, therefore, for the proton we have: $$\lambda_p = \frac{h}{m_p c}$$ The Schwarzschild radius is conversely a characteristic length of each mass coming from the General Relativity environment. If a certain mass $m$ is compacted in an area smaller than its Schwarzschild radius, it, according to General Relativity, is destined to collapse and generate a singularity in the space-time fabric. From a theoretical point of view, the Schwarzschild radius represents a limit beyond which the laws of General Relativity cease to be valid. From a practical point of view this limit is very simple to calculate being: $$r_s = \frac{2Gm_p}{c^2}$$ If we therefore calculate the ratio between the two expressions, we obtain something that morally approaches a ratio between validity scales between Quantum Mechanics and General Relativity and in which the order of magnitude of Dirac appears again: $$\frac{\lambda_p}{r_s} = \frac{hc}{2Gm_p^2} \approx 10^{40}$$