Gravitational coupling constant

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

αG can be defined in terms of any pair of charged elementary particles that are permanent and well-understood. A pair of electrons, of protons, or one electron and one proton all satisfy this criterion. Assuming two electrons, the defining expression and the currently known value are:

\alpha_G = \frac{G m_e^2}{\hbar c} = \left( \frac{m_e}{m_P} \right)^2 \approx 1.752 \times 10^{-45}

where:

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[edit] Measurement and uncertainty

The Quantum Hall Effect makes it possible to measure the value of the fine structure constant α directly, to better than 1 part per billion. On the other hand, there is no known way of measuring αG directly. The meter and second are now defined so that c has an exact value by definition. Therefore, the value of αG depends on measurements of G, \hbar and me. While me and \hbar are known to better than one part in 5,000,000, G is known to only about one part in 7000. Hence αG is normally expressed to only four significant digits.

[edit] Related definitions

Let β = mp/me = 1836.15267261 be the dimensionless ratio of the rest mass of the proton to that of the electron. αG can also be defined using the mass of one proton and one electron, in which case αG = β1.752×10-45 = 3.217×10-42, and α/αG ≈ 1039. α/αG defined in this manner is (C) in Eddington (1935: 232) (except that Planck's constant appears in place of Dirac's), (4.5) in Barrow and Tipler (1986), and ν in Rees (1999).

Barrow and Tipler (1986), following Eddington, replace me with the mass of the proton mp = βme, in which case αG = β21.752×10-45 ≈ 10-39. They invoke the resulting αG freely, without naming it. Note that the three definitions of αG proposed here differ merely by a factor of β or its square.

The physics literature seldom mentions αG. This may be due to the arbitrariness of the choice among particles to use (whereas α is a function of the elementary charge e, about which there is no debate), and the relatively low precision with which αG can be measured.

[edit] Discussion

αG is to gravitation what the fine-structure constant is to electromagnetism and quantum electrodynamics.

Because αG is the square of the electron's mass (in units of Planck mass), αG plays a role in the Higgs mechanism by which the masses of the elementary particles are determined.

Because \alpha_G = G m_e^2 / \hbar c = t_P^2 \omega_C^2, where tP is the Planck time, αG is related to ωC, the Compton angular frequency of the electron.

The proton and the electron are stable, have nonzero mass, and each carry one unit of elementary charge e. Hence the ratio α/αG measures the relative strengths of the gravitational and electrostatic attraction between these elementary particles. Assuming Planck units (so that G=c=\hbar=4\pi\epsilon_0=1), defining αG in terms of a pair of electrons, and recalling that α = e2, then αG = me2 and α/αG = (e/me)2. Thus the ratio of the electron's mass to its charge, when both are measured in Planck units, grounds the relative strengths of gravitation and electromagnetism.

Empirically, α is 43 orders of magnitude greater than αG; the electrostatic force between charged subatomic particles is vastly stronger than the corresponding gravitational attraction. This is so because the charge of a charged subatomic particle is approximately one Planck unit of charge, but its mass is many orders of magnitude smaller than the Planck unit of mass. The gravitational attraction among subatomic particles, charged or not, can hence be ignored. That gravitation is relevant for macroscopic objects proves that they are electrostatically neutral to a very high degree.

αG plays a central role in Barrow and Tipler's (1986) broad-ranging discussion of astrophysics, cosmology, quantum physics, teleology, and the anthropic principle.

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