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 αG. 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.
- G is the gravitational constant;
- me is the electron rest mass;
- c is the speed of light in vacuum;
- ħ ("h-bar") is the reduced Planck constant;
- mP is the Planck mass.
In natural units, where , the expression becomes . 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 particles with equal charge, the magnitude of which is equal to the elementary charge, this gravitational coupling constant measures the gravitational attraction between two electrons.
Measurement and uncertainty
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 via the anomalous magnetic dipole moment of electron with a precision of few parts per 1010. 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.
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.
There is an arbitrariness in the choice of which particle's mass to use (whereas is a function of the elementary charge, is normally a function of the electron rest mass). 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 of both the proton and electron.
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 (so that ), the coupling constants become and , resulting in a meaningful ratio . 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.
is 43 orders of magnitude greater than 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.
has a surprisingly 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.
- Coupling constant
- Dimensionless numbers
- Fine structure constant
- Gravitational constant
- Hierarchy problem
- "CODATA Recommended Values of the Fundamental Physical Constants: 2010" (PDF). National Institute of Standards and Technology, USA.
- Barrow, John D.; Tipler, Frank J. (1988). The Anthropic Cosmological Principle. Oxford University Press. ISBN 978-0-19-282147-8. LCCN 87028148.
- John D. Barrow, 2002. The Constants of Nature. Pantheon Books.
- Arthur Eddington, 1935. New Pathways in Science. Cambridge Univ. Press.
- Martin Rees, 2000. Just Six Numbers: The Deep Forces That Shape the Universe. ISBN 0-465-03673-2
- Hyperphysics: Gravitational coupling constant.