Physics: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 α_G. 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

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

[math]\displaystyle{ \alpha_\mathrm{G} = \frac{G m_\mathrm{e}^2}{\hbar c} = \left( \frac{m_\mathrm{e}}{m_\mathrm{P}} \right)^2 \approx 1.7518 \times 10^{-45} }[/math]

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

[math]\displaystyle{ \alpha_\mathrm{G} = m_\mathrm{e}^2 \ . }[/math]

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):

[math]\displaystyle{ \alpha = e^2 \approx 7.29735 \times 10^{-3} \ . }[/math]

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.

Measurement and uncertainty

CODATA does not report an estimate of the value of α_G. The above estimate is calculated from the CODATA values of G, m_e, c and ħ. Their respective relative uncertainties are u_r(G) = 2.2×10⁻⁵, u_r(m_e) = 3.0×10⁻¹⁰, u_r(c) = 0 and u_r(ħ) = 0, with the prediminant source of uncertainty being that of G, which is thus 2.2×10⁻⁵.

Related definitions

Other definitions of α_G that have been proposed in the literature differ from the one above;

• If α_G is defined using the mass of one electron, m_e, and one proton, m_p, then α_G = 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/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 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 α_G is defined in terms of electron mass, 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 ratio of the electron mass to the Planck mass. α_G is seldom mentioned in the physics literature.

See also

• Coupling constant
• Dimensionless numbers
• Hierarchy problem

References

• Barrow, John D.; Tipler, Frank J. (1988). The Anthropic Cosmological Principle. Oxford University Press. ISBN 978-0-19-282147-8. 
• Barrow, John D. (2002). The Constants of Nature. Pantheon Books. ISBN 0-375-42221-8. https://archive.org/details/constantsofnatur0000barr. 
• Eddington, Arthur (1935). New Pathways in Science. Cambridge Univ. Press. https://archive.org/details/in.ernet.dli.2015.203149. 
• Rees, Martin (2000). Just Six Numbers: The Deep Forces That Shape the Universe. ISBN 0-465-03673-2. https://archive.org/details/justsixnumbersde00rees_0.