- Gravitational coupling constant
The

**gravitational coupling constant**,**α**, is a_{G}fundamental physical constant , thecoupling constant characterizing thegravitation al attraction between twoelementary particles withcharge and nonzeromass . α_{G}is also adimensionless quantity , so that its numerical value does not vary with the choice ofunits of measurement .α

_{G}can be defined in terms of any pair of chargedelementary particle s that are permanent and well-understood. A pair ofelectron s, ofproton s, or one electron and one proton all satisfy this criterion. Assuming two electrons, the defining expression and the currently known value are::$alpha\_G\; =\; frac\{G\; m\_e^2\}\{hbar\; c\}\; =\; left(\; frac\{m\_e\}\{m\_P\}\; ight)^2\; approx\; 1.752\; imes\; 10^\{-45\}$

where:

* "G" = Newtonian constant of gravitation;

* "m"_{e}= mass of theelectron ;

* "c" =speed of light in a vacuum;

* $hbar$ =Dirac's constant or the "reduced"Planck's constant ;

* "m"_{P}=Planck mass .**Measurement and uncertainty**The

Quantum Hall Effect makes it possible to measure the value of thefine structure constant α directly, to better than 1 part per billion. On the other hand, there is no known way of measuring α_{G}directly. Themeter andsecond are now defined so that "c" has an exact value by definition. Therefore, the value of α_{G}depends on measurements of "G", $hbar$ and "m"_{e}. While "m"_{e}and $hbar$ are known to better than one part in 5,000,000, "G" is known to only about one part in 7000. Hence α_{G}is normally expressed to only four significant digits.**Related definitions**Let β = "m"

_{p}/"m"_{e}= 1836.15267261 be the dimensionless ratio of therest mass of the proton to that of the electron. α_{G}can also be defined using the mass of one proton and one electron, in which case α_{G}= β1.752×10^{-45}= 3.217×10^{-42}, and α/α_{G}≈ 10^{39}. α/α_{G}defined in this manner is (C) in Eddington (1935: 232) (except thatPlanck'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 "m"

_{e}with the mass of theproton "m"_{p}= β"m"_{e}, in which case α_{G}= β^{2}1.752×10^{-45}≈ 10^{-39}. They invoke the resulting α_{G}freely, without naming it. Note that the three definitions of α_{G}proposed here differ merely by a factor of β or its square.The physics literature seldom mentions α

_{G}. This may be due to the arbitrariness of the choice among particles to use (whereas α is a function of theelementary charge "e", about which there is no debate), and the relatively low precision with which α_{G}can be measured.**Discussion**α

_{G}is togravitation what thefine-structure constant is toelectromagnetism andquantum electrodynamics .Because α

_{G}is the square of the electron's mass (in units ofPlanck mass ), α_{G}plays a role in theHiggs mechanism by which the masses of theelementary particle s are determined.Because $alpha\_G\; =\; G\; m\_e^2\; /\; hbar\; c\; =\; t\_P^2\; omega\_C^2$, where $t\_P$ is the

Planck time , α_{G}is related to $omega\_C$, theCompton angular frequency of the electron.The

proton and theelectron are stable, have nonzero mass, and each carry one unit ofelementary charge "e". Hence the ratio α/α_{G}measures the relative strengths of thegravitation al andelectrostatic attraction between these elementary particles. AssumingPlanck units (so that $G=c=hbar=4piepsilon\_0=1$), defining α_{G}in terms of a pair of electrons, and recalling that α = "e"^{2}, then α_{G}= "m"_{e}^{2}and α/α_{G}= ("e"/"m"_{e})^{2}. Thus the ratio of the electron's mass to its charge, when both are measured inPlanck units , grounds the relative strengths ofgravitation andelectromagnetism .Empirically, α is 43

orders of magnitude greater than α_{G}; theelectrostatic force between chargedsubatomic particle s is vastly stronger than the corresponding gravitational attraction. This is so because the charge of a chargedsubatomic particle is approximately one Planck unit of charge, but its mass is manyorders 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.α

_{G}plays a central role in Barrow and Tipler's (1986) broad-ranging discussion ofastrophysics ,cosmology ,quantum physics ,teleology , and theanthropic principle .**ee also***

CODATA

*fine structure constant

*gravitational constant

*dimensionless numbers **References***

John D. Barrow andFrank J. Tipler , 1986. "The Anthropic Cosmological Principle". Oxford University Press.

*John D. Barrow , 2002. "The Constants of Nature". Pantheon Books.

*Arthur Eddington , 1935. "New Pathways in Science". Cambridge Univ. Press.**External links*** "Hyperphysics": [

*http://hyperphysics.phy-astr.gsu.edu/HBASE/forces/couple.html#c5 Gravitational coupling constant.*]

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