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In Newton's law, it is the proportionality constant connecting the [[gravitational force]] between two bodies with the product of their [[mass]]es and the [[inverse-square law|inverse square]] of their [[distance]]. In the [[Einstein field equations]], it quantifies the relation between the geometry of spacetime and the energy–momentum tensor (also referred to as the [[stress–energy tensor]]).
The measured value of the constant is known with some certainty to four significant digits. In [[SI units]], its value is approximately <!--{{math|''G''}} = -->{{physconst|G|round=4|unit=no|after= N⋅m<sup>2</sup>/kg<sup>2</sup>.
The modern notation of Newton's law involving {{math|''G''}} was introduced in the 1890s by [[C. V. Boys]]. The first implicit measurement with an accuracy within about 1% is attributed to [[Henry Cavendish]] in a [[Cavendish experiment|1798 experiment]].{{efn|Cavendish determined the value of ''G'' indirectly, by reporting a value for the [[Earth's mass]], or the average density of Earth, as {{val|5.448|u=g.cm-3}}.|name=|group=}}
== Definition ==
According to [[Newton's law of universal gravitation]], the [[Norm (mathematics)#Euclidean norm|magnitude]] of the attractive [[force]] ({{math|''F''}}) between two bodies each with a spherically symmetric [[density]] distribution is directly proportional to the product of their [[mass]]es, {{math|''m''<sub>1</sub>}} and {{math|''m''<sub>2</sub>}}, and [[Inverse-square law|inversely proportional to the square of the distance]], {{math|''r''}}, directed along the line connecting their [[centre of mass|centres of mass]]:
<math display="block">F=G\frac{m_1m_2}{r^2}.</math>
The [[Proportionality (mathematics)|constant of proportionality]], {{math|''G''}}, in this non-relativistic formulation is the gravitational constant. Colloquially, the gravitational constant is also called "Big G", distinct from "small g" ({{math|''g''}}), which is the [[Gravity of Earth|local gravitational field of Earth]] (
<math display="block">g = G\frac{M_\oplus}{r_\oplus^2}.</math>
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<math display="block">G_{\mu \nu} + \Lambda g_{\mu \nu} = \kappa T_{\mu \nu} \,,</math>
where {{math|''G''{{sub|''μν''}}}} is the [[Einstein tensor]] (not the gravitational constant despite the use of {{mvar|G}}), {{math|Λ}} is the [[cosmological constant]], {{mvar|g{{sub|μν}}}} is the [[metric tensor (general relativity)|metric tensor]], {{mvar|T{{sub|μν}}}} is the [[stress–energy tensor]], and {{math|''κ''}} is the [[Einstein gravitational constant]], a constant originally introduced by [[Albert Einstein|Einstein]] that is directly related to the Newtonian constant of gravitation:<ref name="ein" /><ref>{{cite book |title= Introduction to General Relativity |url= https://archive.org/details/introductiontoge00adle |url-access= limited |first1=Ronald |last1=Adler |first2=Maurice |last2=Bazin |first3=Menahem |last3=Schiffer |publisher= McGraw-Hill |location= New York |year= 1975 |edition= 2nd |isbn= 978-0-07-000423-8 |page= [https://archive.org/details/introductiontoge00adle/page/n360 345]}}</ref>{{efn|Depending on the choice of definition of the Einstein tensor and of the stress–energy tensor it can alternatively be defined as {{math|1=''κ'' = {{sfrac|8π''G''|''c''<sup>2</sup>}} ≈ {{val|1.866|e=-26|u=m⋅kg<sup>−1</sup>}}}}}}
<math display="block">\kappa = \frac{8\pi G}{c^4} \approx 2.
== Value and uncertainty ==
The gravitational constant is a physical constant that is difficult to measure with high accuracy.<ref name=gillies>{{cite journal|first=George T. |last=Gillies |title=The Newtonian gravitational constant: recent measurements and related studies |journal=Reports on Progress in Physics |date=1997 |volume=60 |issue=2 |pages=151–225 |doi=10.1088/0034-4885/60/2/001|bibcode = 1997RPPh...60..151G |s2cid=250810284 }}. A lengthy, detailed review. See Figure 1 and Table 2 in particular.</ref> This is because the gravitational force is an extremely weak force as compared to other [[fundamental forces]] at the laboratory scale.{{efn|For example, the gravitational force between an [[electron]] and a [[proton]] 1 m apart is approximately {{val|e=−67|ul=N}}, whereas the [[electromagnetic force]] between the same two particles is approximately {{val|e=−28|u=N}}. The electromagnetic force in this example is in the order of 10<sup>39</sup> times greater than the force of gravity—roughly the same ratio as the [[Solar mass|mass of the Sun]] to a microgram.|name=|group=}}
In [[International System of Units|SI]] units, the [[CODATA]]-recommended value of the gravitational constant is:{{physconst|G|ref=only}}
: <math>G</math> = {{physconst|G|ref=no}}
=== Natural units ===
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In [[orbital mechanics]], the period {{math|''P''}} of an object in circular orbit around a spherical object obeys
<math display="block"> GM=\frac{3\pi V}{P^2} ,</math>
where {{math|''V''}} is the volume inside the radius of the orbit, and {{math|''M''}} is the total mass of the two objects. It follows that
: <math> P^2=\frac{3\pi}{G}\frac{V}{M}\approx 10.896 \, \mathrm{ h^2 {\cdot} g {\cdot} cm^{-3} \,}\frac{V}{M}.</math>
This way of expressing {{math|''G''}} shows the relationship between the average density of a planet and the period of a satellite orbiting just above its surface.
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|-
!scope="row"|2014
| 6.67408(31) || 46 ppm || <ref>{{cite journal | last1=Mohr | first1=Peter J. | last2=Newell | first2=David B. | last3=Taylor | first3=Barry N. | title=CODATA Recommended Values of the Fundamental Physical Constants: 2014 | journal=Journal of Physical and Chemical Reference Data | volume=45 | issue=4 | year=2016 | pages=1527–1605 | issn=0047-2689 | doi=10.1063/1.4954402 | bibcode=2016JPCRD..45d3102M | arxiv=1203.5425 }}</ref>
|-
!scope="row"|2018
| 6.67430(15) || 22 ppm || <ref>Eite Tiesinga, Peter J. Mohr, David B. Newell, and Barry N. Taylor (2019), "[http://physics.nist.gov/constants The 2018 CODATA Recommended Values of the Fundamental Physical Constants]" (Web Version 8.0). Database developed by J. Baker, M. Douma, and [[Svetlana Kotochigova|S. Kotochigova]]. National Institute of Standards and Technology, Gaithersburg, MD 20899.</ref>
|-
!scope="row"|2022
| 6.67430(15) || 22 ppm || <ref>{{citation |author1=Mohr, P. |author2=Tiesinga, E. |author3=Newell, D. |author4=Taylor, B. |date=2024-05-08 |title=Codata Internationally Recommended 2022 Values of the Fundamental Physical Constants |url=https://www.nist.gov/publications/codata-internationally-reconmmended-2022-values-fundamental-physical-constants |access-date=2024-05-15 }}</ref>
|-
|}
In the January 2007 issue of ''[[Science (journal)|Science]]'', Fixler et al. described a measurement of the gravitational constant by a new technique, [[atom interferometry]], reporting a value of {{
{{cite journal |last1=Rosi |first1=G.
|last2=Sorrentino |first2=F. |last3=Cacciapuoti |first3=L. |last4=Prevedelli |first4=M.
|last5=Tino |first5=G. M. |journal=Nature |volume=510 |issue=7506 |date=26 June 2014 |pages=518–521
|url=http://www2.fisica.unlp.edu.ar/materias/FisGral2semestre2/Rosi.pdf |url-status=live
|doi=10.1038/nature13433 |pmid=24965653 |arxiv=1412.7954
|s2cid=4469248
▲ |url=http://www2.fisica.unlp.edu.ar/materias/FisGral2semestre2/Rosi.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www2.fisica.unlp.edu.ar/materias/FisGral2semestre2/Rosi.pdf |archive-date=2022-10-09 |url-status=live|bibcode=2014Natur.510..518R }}</ref><ref>
|bibcode=2014Natur.510..518R
}}</ref><ref>
{{cite journal
|last1=Schlamminger |first1=Stephan
|title=Fundamental constants: A cool way to measure big G
|journal=Nature |volume=510 |issue=7506 |pages=478–480
|date=18 June 2014 |url=https://www.nature.com/articles/nature13507.pdf
|doi=10.1038/nature13507 |doi-access=free |bibcode=2014Natur.510..478S |pmid=24965646▼
|url-status=live |doi=10.1038/nature13507 |doi-access=free
}}</ref> Although much closer to the accepted value (suggesting that the Fixler ''et al.'' measurement was erroneous), this result was 325 ppm below the recommended 2014 CODATA value, with non-overlapping [[standard uncertainty]] intervals.
<!-- 6.67191(99) vs. 6.67408(31) [2014], a difference of 0.00217(104). Also *barely* not overlapping with the
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== References ==
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