Gravitational constant: Difference between revisions

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 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=}}

<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''²}} ≈ {{val|1.866|e=-26|u=m⋅kg⁻¹}}}}}}

 

== 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&nbsp;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 10³⁹ times greater than the force of gravity—roughly the same ratio as the [[Solar mass|mass of the Sun]] to a microgram.|name=|group=}}

 

 

 

=== Natural units ===

<math display="block"> GM=\frac{3\pi V}{P^2} ,</math>

where {{math|''V''}} is the volume inside the radius of the orbit. It follows that

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.

 

 

== References ==

 

{{reflist|30em}}