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| In [[physics]], the '''gravitational coupling constant''', '''α<sub>G</sub>''', is the [[coupling constant]] characterizing the [[gravitation]]al attraction between two [[elementary particles]] having nonzero [[mass]]. α<sub>G</sub> is a [[fundamental physical constant]] and a [[dimensionless quantity]], so that its numerical value does not vary with the choice of [[units of measurement]].
| | 39 yr old Boarding Run or Cattery Operator Frankie Crochet from Burk's Falls, loves to spend time animals, [http://ganhandodinheironainternet.comoganhardinheiro101.com como ganhar dinheiro na internet] and roller skating. Last month very recently made a journey to Historic Monuments of Dengfeng in “The Centre of Heaven and Earth”. |
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| ==Definition==
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| α<sub>G</sub> can be defined in terms of any pair of [[elementary particle]]s that are stable and well-understood{{clarify|date=January 2013}}. A pair of [[electron]]s, of [[proton]]s, or one electron and one proton all satisfy this criterion. Assuming two electrons, the defining expression and the best current estimate of its value are:
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| :<math>\alpha_G = \frac{G m_e^2}{\hbar c} = \left( \frac{m_e}{m_P} \right)^2 \approx 1.7518 \times 10^{-45} </math>
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| where:
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| * ''G'' is the [[gravitational constant|Newtonian constant of gravitation]];
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| * ''m''<sub>e</sub> is the mass of the [[electron]];
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| * ''c'' is the [[speed of light]] in a vacuum;
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| * ''ħ'' (''"h-bar"'') is the [[reduced Planck constant]];
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| * ''m''<sub>P</sub> is the [[Planck mass]].
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| In [[natural units]], where <math>4\pi G=c=\hbar=\varepsilon_0=1</math>, the expression becomes <math>\alpha_G = \frac{m_e^2}{4\pi}</math>, analogous to the [[fine-structure constant]].
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| == Measurement and uncertainty ==
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| There is no known way of measuring α<sub>G</sub> directly, and [[CODATA]] does not report an estimate of its value. The above estimate is calculated from the [http://physics.nist.gov/cgi-bin/cuu/Category?view=html&All+values.x=65&All+values.y=10 CODATA values] of ''m''<sub>e</sub> and ''m''<sub>P</sub>.
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| While ''m''<sub>e</sub> and ''ħ'' are known to one part in 20,000,000, ''m''<sub>P</sub> is only known to one part in 20,000 (mainly because ''G'' is known to only one part in 10,000). Hence α<sub>G</sub> is known to only four significant digits. By contrast, the [[fine structure constant]] α can be measured directly via the [[quantum Hall effect]] with a precision exceeding one part per billion. 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<sub>G</sub> depends only on that of ''G'', ''ħ'', and ''m''<sub>e</sub>.
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| == Related definitions ==
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| Let μ = ''m''<sub>p</sub>/''m''<sub>e</sub> = 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<sub>G</sub> that have been proposed in the literature differ from the one above merely by a factor of μ or its square;
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| * If α<sub>G</sub> is defined using the mass of one electron, ''m''<sub>e</sub>, and one proton (''m''<sub>p</sub> = μ''m''<sub>e</sub>), then α<sub>G</sub> = μ1.752×10<sup>-45</sup> = 3.217×10<sup>-42</sup>, and α/α<sub>G</sub> ≈ 10<sup>39</sup>. α/α<sub>G</sub> defined in this manner is ''C'' in Eddington (1935: 232), with [[Planck's constant]] replacing the "reduced" Planck constant;
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| * (4.5) in Barrow and Tipler (1986) tacitly defines α/α<sub>G</sub> as ''e''<sup>2</sup>/(''Gm''<sub>p</sub>m<sub>e</sub>) ≈ 10<sup>39</sup>. Even though they do not name the α/α<sub>G</sub> defined in this manner, it nevertheless plays a role in their broad-ranging discussion of [[astrophysics]], [[cosmology]], [[quantum physics]], and the [[anthropic principle]];
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| *''N'' in Rees (2000) is α/α<sub>G</sub> = α/(μ<sup>2</sup>1.752×10<sup>−45</sup>) = α/(5.906×10<sup>−39</sup>) ≈ 10<sup>36</sup>, where the denominator is defined using a pair of protons.
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| == Discussion ==
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| There is an arbitrariness in the choice of which particle's mass to use (whereas <math>\alpha</math> is a function of the [[elementary charge]], <math>\alpha_G</math> is normally a function of the [[electron rest mass]]). In this article <math>\alpha_G</math> is defined in terms of a pair of [[electron]]s unless stated otherwise. For such a system, <math>\alpha_G</math> is to [[gravitation]] as the [[fine-structure constant]] is to [[electromagnetism]]{{dubious|date=January 2013}}.
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| The [[electron]] is a stable particle possessing one [[elementary charge]] and one [[electron mass]]. Hence the ratio <math>\frac{\alpha}{\alpha_G}</math> measures the relative strengths of the [[electrostatic]] and [[gravitation]]al forces between two electrons. Expressed in [[natural units]] (so that <math>4\pi G = c = \hbar = \varepsilon_0 = 1</math>), the coupling constants become <math>\alpha=\frac{e^2}{4\pi}</math> and <math>\alpha_G=\frac{m_e^2}{4\pi}</math>, resulting in a meaningful ratio <math>\frac{\alpha}{\alpha_G}=\left(\frac{e}{m_e}\right)^2</math>. 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.
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| <math>\alpha</math> is 43 [[orders of magnitude]] greater than <math>\alpha_G</math> calculated for two electrons (or 37 orders, for two protons). The [[electrostatic]] force between two charged [[elementary particle]]s is vastly greater than the corresponding [[gravitation]]al force between them. This is so because a charged elementary particle has in the order of one [[Planck charge]], but a mass many orders of magnitude smaller than the [[Planck mass]]. 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.
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| <math>\alpha_G</math> has a surprisingly simple physical interpretation: it is the square of the [[electron mass]], measured in units of [[Planck mass]]. By virtue of this, <math>\alpha_G</math> is connected to the [[Higgs mechanism]], which determines the rest masses of the [[elementary particle]]s. <math>\alpha_G</math> can only be measured with relatively low precision, and is seldom mentioned in the physics literature.
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| Because <math>\alpha_G=\frac{G m_e^2}{\hbar c}=\left( t_P \omega_C \right)^2 </math>, where <math>t_P</math> is the [[Planck time]], <math>\alpha_G</math> is related to <math>\omega_C</math>, the [[Compton angular frequency]] of the [[electron]].
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| ==See also==
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| *[[Coupling constant]]
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| *[[Dimensionless numbers]]
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| *[[Fine structure constant]]
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| *[[Gravitational constant]]
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| ==References==
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| *{{BarrowTipler1986}}
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| *[[John D. Barrow]], 2002. ''The Constants of Nature''. Pantheon Books.
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| * [[Arthur Eddington]], 1935. ''New Pathways in Science''. Cambridge Univ. Press.
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| *[[Martin Rees]], 2000. ''Just Six Numbers: The Deep Forces That Shape the Universe''. ISBN 0-465-03673-2
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| ==External links==
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| * ''Hyperphysics'': [http://hyperphysics.phy-astr.gsu.edu/HBASE/forces/couple.html#c5 Gravitational coupling constant.]
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| [[Category:Fundamental constants]]
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| [[Category:Gravitation]]
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| [[Category:Dimensionless numbers]]
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| [[Category:Anthropic principle]]
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39 yr old Boarding Run or Cattery Operator Frankie Crochet from Burk's Falls, loves to spend time animals, como ganhar dinheiro na internet and roller skating. Last month very recently made a journey to Historic Monuments of Dengfeng in “The Centre of Heaven and Earth”.