Gravitational coupling constant

In physics, a gravitational coupling constant can refer to any of a large number of possible constants characterizing the gravitational attraction between a pair of particles, which need not even be elementary particles. A pair of electrons, a pair of protons, or a pair of one each is often used. As a consequence, it cannot be considered to be a fundamental physical constant. 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 particles. It is not a coupling constant in the sense used in quantum field theory.

Definition

$α G$ for two selected particles is typically defined^{[citation needed]} in terms of the gravitational attraction between them. More precisely,

α_{G} = \frac{G m_{x} m_{y}}{ℏ c} = \frac{m_{x} m_{y}}{m_{P}^{2}},

where:

$m x$ and $m y$ are the rest masses of two particles;
$G$ is the Newtonian constant of gravitation;
$c$ is the speed of light;
$ħ$ is the reduced Planck constant;
$m P$ is the Planck mass.

Measurement and uncertainty

CODATA does not report an estimate of any of the possible values of $α G$ . An estimate calculated for a pair of electrons from the CODATA values of $m e$ ^[1], $G$ ^[2], $c$ ^[3] and $ħ$ ^[4]. 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, giving $α G = 1.7518 \times 10 -45$ , with the predominant source of uncertainty being that of $G$ , which is thus 2.2×10⁻⁵.

Variations

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 $α G = 3.217 \times 10 -42$ , and $α / α G \approx 10 39$ . $α / α G$ defined in this manner is $C$ in Eddington (1935: 232), with the Planck constant replacing the "reduced" Planck constant;
(4.5) in Barrow and Tipler (1986) tacitly defines $α / α G$ as $k e e 2 / Gm p m e \approx 10 39$ . 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 $α / α G = α / 5.906 \times 10 -39 \approx 10 36$ , where the denominator is defined using a pair of protons.

Given that there are innumerable candidate particles with diverse masses and that no given particle mass can be considered either fundamental or quantized in the same way that the elementary charge is, $α G$ cannot be considered to be fundamental in any sense.

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 a function of the masses of any chosen pair of particles. The relationship between $α G$ and gravitation may be seen as somewhat analogous in form to that of the fine-structure constant and electromagnetism, but an important difference is that the $α G$ is defined in terms of particle masses of some choice of particles, 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 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.

$α$ is 43 orders of magnitude greater than $α G$ calculated for two electrons (or 37 orders, for two protons: a difference of six orders of magnitude). The electrostatic force between two electrons (or protons) is vastly greater than the corresponding gravitational force between them. The gravitational attraction among subatomic particles can typically be ignored. Gravitation dominates for macroscopic objects because they are electrostatically neutral to a very high degree.

$α G$ for electrons may be interpreted as the square of ratio of the electron mass to the Planck mass. $α G$ is seldom mentioned in the physics literature.

References

↑ "2018 CODATA Value: electron mass in u". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
↑ "2018 CODATA Value: Newtonian constant of gravitation". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
↑ "2018 CODATA Value: speed of light in vacuum". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
↑ "2018 CODATA Value: reduced Planck constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-28.

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

This article "Gravitational coupling constant" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Gravitational coupling constant. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

[physconst-me-1] "2018 CODATA Value: electron mass in u". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.

[physconst-G-2] "2018 CODATA Value: Newtonian constant of gravitation". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.

[physconst-c-3] "2018 CODATA Value: speed of light in vacuum". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.

[physconst-hbar-4] "2018 CODATA Value: reduced Planck constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-28.

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Gravitational coupling constant

Contents

Definition

Measurement and uncertainty

Variations

Discussion

See also

References

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