Gravitational Coupling Constant

Much like there is a coupling constant for the electromagnetic force (also called the fine structure constant), there is one for the gravitational force, called, believe it or not, the gravitational coupling constant. It is used to define the gravitational attraction between two elementary particles having some mass.


The way it is defined is as the gravity between two electrons and is a unitless quantity.

Where:

• G is the gravitation constant  (6.67x10^-11 m³/s²kg)
• m_e is the mass of an electron (9.109x10^-31 kg)
• ℏ is the Planck constant over 2π (called the reduce Planck constant, 1.05457x10^-34 J*s)
• c is the speed of light (3x108 m/s)

Compare this to the fine coupling constant which is approximately 1/137 and we can see that the electromagnetic force is 10⁴³ times stronger than the gravitational force for electrons. Depending on what elementary particles are used (proton-electron or proton-proton), the ratio between them can vary, but in all cases, the electromagnetic coupling constant is magnitudes greater than that for gravity. In Martin Rees' case, he compares the fine structure constant to the gravitational coupling constant for two protons, and the ratio is 10³⁶. Any variance of this ratio can lead to the universe not being the way we see it.


To calculate the gravitational coupling constant for two protons, replace m_e with m_p in the top equation. To calculate it for the attraction between a proton and an electron, replace one m_e with m_p.