# Gravitational coupling constant

In physics, the gravitational coupling constant, αG, is the coupling constant characterizing the gravitational attraction between two elementary particles having nonzero mass. αG is a fundamental physical constant and a dimensionless quantity, so that its numerical value does not vary with the choice of units of measurement.

## Definition

αG is defined in terms of the gravitational attraction between pair of electrons. Precisely,

$\alpha_G = \frac{G m_e^2}{\hbar c} = \left( \frac{m_e}{m_P} \right)^2 \approx 1.7518 \times 10^{-45}$

where:

In natural units, where $4\pi G=c=\hbar=\varepsilon_0=1$, the expression becomes $\alpha_G = \frac{m_e^2}{4\pi}$. This shows that the gravitational coupling constant can be thought of as the analogue of the fine-structure constant; while the fine-structure constant measures the electromagnetic repulsion between two electrons, the gravitational coupling constant measures the gravitational attraction.

## Measurement and uncertainty

There is no known way of measuring αG directly, and CODATA does not report an estimate of its value. The above estimate is calculated from the CODATA values of me and mP.

While me and ħ are known to one part in 20,000,000, mP is only known to one part in 20,000 (mainly because G is known to only one part in 10,000). Hence αG is known to only four significant digits. By contrast, the fine structure constant α can be measured directly via the quantum Hall effect with a precision exceeding one part per billion. Also, the meter and second are now defined in a way such that c has an exact value by definition. Hence the precision of αG depends only on that of G, ħ, and me.

## Related definitions

Let μ = mp/me = 1836.15267247(80) be the dimensionless proton-to-electron mass ratio, the ratio of the rest mass of the proton to that of the electron. Other definitions of αG that have been proposed in the literature differ from the one above merely by a factor of μ or its square;

• If αG is defined using the mass of one electron, me, and one proton (mp = μme), then αG = μ1.752×10−45 = 3.217×10−42, and α/αG ≈ 1039. α/αG defined in this manner is C in Eddington (1935: 232), with Planck's constant replacing the "reduced" Planck constant;
• (4.5) in Barrow and Tipler (1986) tacitly defines α/αG as e2/(Gmpme) ≈ 1039. Even though they do not name the α/αG defined in this manner, it nevertheless plays a role in their broad-ranging discussion of astrophysics, cosmology, quantum physics, and the anthropic principle;
• N in Rees (2000) is α/αG = α/(μ21.752×10−45) = α/(5.906×10−39) ≈ 1036, where the denominator is defined using a pair of protons.

## Discussion

There is an arbitrariness in the choice of which particle's mass to use (whereas $\alpha$ is a function of the elementary charge, $\alpha_G$ is normally a function of the electron rest mass). In this article $\alpha_G$ is defined in terms of a pair of electrons unless stated otherwise. For such a system, $\alpha_G$ is to gravitation as the fine-structure constant is to electromagnetism[dubious ].

The electron is a stable particle possessing one elementary charge and one electron mass. Hence the ratio $\frac{\alpha}{\alpha_G}$ measures the relative strengths of the electrostatic and gravitational forces between two electrons. Expressed in natural units (so that $4\pi G = c = \hbar = \varepsilon_0 = 1$), the coupling constants become $\alpha=\frac{e^2}{4\pi}$ and $\alpha_G=\frac{m_e^2}{4\pi}$, resulting in a meaningful ratio $\frac{\alpha}{\alpha_G}=\left(\frac{e}{m_e}\right)^2$. Thus the ratio of the electron charge to the electron mass (in natural units) determines the relative strengths of electromagnetic and gravitational interaction between two electrons.

$\alpha$ is 43 orders of magnitude greater than $\alpha_G$ calculated for two electrons (or 37 orders, for two protons). The electrostatic force between two charged elementary particles is vastly greater than the corresponding gravitational force between them. This is so because a charged elementary particle has in the order of one Planck charge, but a mass many orders of magnitude smaller than the Planck mass. The gravitational attraction among elementary particles, charged or not, can hence be ignored. Gravitation dominates for macroscopic objects because they are electrostatically neutral to a very high degree.

$\alpha_G$ has a surprisingly simple physical interpretation: it is the square of the electron mass, measured in units of Planck mass. By virtue of this, $\alpha_G$ is connected to the Higgs mechanism, which determines the rest masses of the elementary particles. $\alpha_G$ can only be measured with relatively low precision, and is seldom mentioned in the physics literature.

Because $\alpha_G=\frac{G m_e^2}{\hbar c}=\left( t_P \omega_C \right)^2$, where $t_P$ is the Planck time, $\alpha_G$ is related to $\omega_C$, the Compton angular frequency of the electron.