Gravitational coupling constant

α_G is typically defined in terms of the gravitational attraction between two electrons. More precisely,

${\displaystyle \alpha _{\mathrm {G} }={\frac {Gm_{\mathrm {e} }^{2}}{\hbar c}}=\left({\frac {m_{\mathrm {e} }}{m_{\mathrm {P} }}}\right)^{2}\approx 1.7518\times 10^{-45}}$

where:

• G is the gravitational constant;
• m_e is the electron rest mass;
• c is the speed of light in vacuum;
• ħ is the reduced Planck constant;
• m_P is the Planck mass.

In Planck units, where G = c = ħ = 1, the expression becomes the square of the electron mass

${\displaystyle \alpha _{\mathrm {G} }=m_{\mathrm {e} }^{2}\ .}$

This shows that the gravitational coupling constant can be thought of as the analogue of the fine-structure constant (also expressed in Planck units, extended to include the normalization 4πε₀ = 1):

${\displaystyle \alpha =e^{2}\approx 7.29735\times 10^{-3}\ .}$

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 α_G is 42 orders of magnitude smaller than α.

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

While m_e is known to one part in 3000000000 and ħ has an exact value by definition, m_P is only known to one part in 100000 (mainly because G is known to only one part in 50000). Hence α_G 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 a few parts per 10¹⁰.[1] Also, the meter and second are now defined in a way such that c has an exact value by definition. Hence the precision of α_G depends only on that of G, and m_e.

Let μ = m_p/m_e = 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 α_G that have been proposed in the literature differ from the one above merely by a factor of μ or its square;

• If α_G is defined using the mass of one electron, m_e, and one proton (m_p = μm_e), then α_G = 1.752×10⁻⁴⁵μ = 3.217×10⁻⁴², and α/α_G ≈ 10³⁹. α/α_G 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/α_G as e²/Gm_pm_e ≈ 10³⁹. Even though they do not name the α/α_G 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/α_G = α/1.752×10⁻⁴⁵μ² = 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 α is a function of the elementary charge, α_G is normally a function of the electron rest mass). In this article α_G is defined in terms of a pair of electrons unless stated otherwise. And while the relationship between α_G and gravitation is somewhat analogous to that of the fine-structure constant and electromagnetism, the important difference is that the standard definition of α_G 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 α/α_G measures the relative strengths of the electrostatic and gravitational forces between two electrons. Expressed in natural units (so that 4πG = c = ħ = ε₀ = 1), the constants become α = e²/4π and α_G = m_e²/4π, resulting in a meaningful ratio α/α_G = (e/m_e)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.

α is 43 orders of magnitude greater than α_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. 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.

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

Because

${\displaystyle \alpha _{\text{G}}={\frac {Gm_{\text{e}}^{2}}{\hbar c}}=\left(t_{\text{P}}\omega _{\text{C}}\right)^{2}}$

where t_P is the Planck time, α_G is related to ω_C, the Compton angular frequency of the electron.

• Coupling constant
• Dimensionless numbers
• Hierarchy problem

• Barrow, John D.; Tipler, Frank J. (1986). The Anthropic Cosmological Principle 1st edition 1986 (revised 1988). Oxford University Press. ISBN 978-0-19-282147-8. LCCN 87028148.
• Barrow, John D. (2002). The Constants of Nature. Pantheon Books. ISBN 0-375-42221-8.
• Eddington, Arthur (1935). New Pathways in Science. Cambridge Univ. Press.
• Rees, Martin (2000). Just Six Numbers: The Deep Forces That Shape the Universe. ISBN 0-465-03673-2.