Draft:Dimensionless gravitational coupling constant

The gravitational coupling constant, α_G, can be defined in terms of the gravitational attraction between two elementary particles:^[1]

$${\displaystyle \alpha _{\mathrm {G} }={\frac {Gm_{\mathrm {p} }^{2}}{\hbar c}}}$$

where:

• G is the gravitational constant;
• m_p is the particle rest mass;
• c is the speed of light in vacuum;
• ħ is the reduced Planck constant.

For a proton the value of the dimensionless constant is approximately 5×10⁻³⁹.^[1][2]: 25

In natural units where c = ħ = 1, the expression becomes^[3]: 1153 $${\displaystyle \alpha _{\mathrm {G} }=Gm_{\mathrm {p} }^{2}\ .}$$

In 1961, a dimensionless gravitational coupling constant was used by Robert H. Dicke.^[1] Dicke's work has been described as perhaps the first explicit discussion of the weak anthropic principle. This principle has various forms. The basic idea is that the constants in physical laws are limited to values which can give rise to physicists. Dicke reasoned that the gravitational coupling must be an extremely small number to ensure that expansion of the universe could continue against the force of gravity caused by all the matter in the universe.^[4]: 7 Dicke's analysis countered an earlier hypothesis by Paul Dirac, suggesting that the gravitational coupling constant depends on cosmic density. In a Big Bang model of the universe, the density varies strongly with time. Dicke noted that sufficient time must elapse for stars to form and this requires a very small gravitational coupling constant.^[3]: 1150

• Coupling constant
• Dimensionless numbers
• Hierarchy problem