Gravitational coupling constant


In physics, a gravitational coupling constant is a constant characterizing the gravitational attraction between a given pair of elementary particles. The electron mass is typically used, and the associated constant typically denoted. It is a dimensionless quantity, with the result that its numerical value does not vary with the choice of units of measurement, only with the choice of particle.

Definition

is typically defined in terms of the gravitational attraction between two electrons. More precisely,
where:
In Planck units, where, the expression becomes the square of the electron mass
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 electrostatic repulsion between two particles with equal charge, the magnitude of which is equal to the square of the elementary charge, this gravitational coupling constant measures the gravitational attraction between two electrons. This is one manner of expressing that "Gravity is a far weaker force than the electromagnetic interaction" since is 42 orders of magnitude smaller than.

Measurement and uncertainty

There is no known way of measuring directly, and CODATA does not report an estimate of its value. The above estimate is calculated from the of and.
While is known to one part in and has an exact value by definition, is only known to one part in . Hence αG is known to only four significant digits. By contrast, the fine structure constant can be measured via the anomalous magnetic dipole moment of electron with a precision of a few parts per 1010. Also, the meter and second are now defined in a way such that has an exact value by definition. Hence the precision of depends only on that of, and.

Related definitions

Let 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 that have been proposed in the literature differ from the one above merely by a factor of or its square;
There is an arbitrariness in the choice of which particle's mass to use. In this article is defined in terms of a pair of electrons unless stated otherwise. And while the relationship between and gravitation is somewhat analogous to that of the fine-structure constant and electromagnetism, the important difference is that the standard definition of describes a ratio in terms of electron mass alone, whereas the fine-structure constant relates to the elementary charge, which is a quantum that is independent of the choice of particle.
The electron is a stable particle possessing one elementary charge and one electron mass. Hence the ratio measures the relative strengths of the electrostatic and gravitational forces between two electrons. Expressed in natural units, the constants become and, resulting in a meaningful ratio. Thus the ratio of the electron charge to the electron mass determines the relative strengths of electromagnetic and gravitational interaction between two electrons.
is 43 orders of magnitude greater than calculated for two electrons. The electrostatic force between two charged elementary particles is vastly greater than the corresponding gravitational force between them. 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.
has a simple physical interpretation: it is the square of the electron mass, measured in units of Planck mass. By virtue of this, is connected to the Higgs mechanism, which determines the rest masses of the elementary particles. can only be measured with relatively low precision, and is seldom mentioned in the physics literature.
Because
where is the Planck time, is related to, the Compton angular frequency of the electron.