A relativity/quantum bridge

v.2 n.1

When special relativity is combined with Newtonian gravity at the elementary particle scale

(1- v^2/c^2)^-1 = (n/2) ?c / G(m1+m2)m2 ....... (1)

γ^2 = (n/2)(α?G)^-1 ............ (v→c) .................. (1a) *

where the left side has a relativistic character, and the right side a quantum. The tangential velocity, v, of mass m2 relative to mass m1 is v→c (no less). For instance, within the proton let n = 1, m1 ≈ m2 ≈ 10^-29 kg ≈ light quark mass; then v →c. When the mass term in Equation (1) is that of two electrons or two protons, α?G indicates the gravitational coupling constant. **

Alpha sub-G and alpha. The gravitational coupling constant, one of the two bridge pillars here, is a dimensionless constant of nature, along with the fine structure constant, α, for instance. Comparing α?G and α,

α?G = G (m?e)^2 / ?c ...................................... (2)

α = k e^2 / ?c ≈ [(m?e)/(m?q)]^3/2 ..... (3)

where m?e is electron mass and k is the coulomb force constant. Among other interpretations, alpha may also be interpreted as a ratio of electron mass to light quark mass; with Equation (3), three universal constants are related to the fundamental masses. ***

In closing. The gravitational coupling constant and the fine structure constant are derived from combinations of special relativity and Newtonian gravity, and singularities are avoided in both theories, since n in Equations (1) and (1a) cannot have values less than 1, so that v→c and v ≠ c.

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* γ = (1- v^2/c^2)^-1/2

** DOI:?10.13140/RG.2.2.20258.35524

*** (8) Yet another interpretation, and (brief) derivation, of the fine structure constant | LinkedIn