Editing Equivalence principle (section)

== Experimental tests ==
=== Tests of the weak equivalence principle ===
Tests of the weak equivalence principle are those that verify the equivalence of gravitational mass and inertial mass. An obvious test is dropping different objects and verifying that they land at the same time. Historically this was the first approach – though probably not by [[Galileo's Leaning Tower of Pisa experiment]]<ref name=Drake>{{cite book|last=Drake|first=Stillman|title=Galileo at Work: His Scientific Biography|date=2003|publisher=Dover publ.|location=Mineola (N.Y.)|isbn=9780486495422|edition=Facsim.}}</ref>{{rp|19–21}} but instead earlier by [[Simon Stevin]],<ref name=Devreese>{{cite book |last1=Devreese |first1=Jozef T. |author-link1=Jozef T. Devreese |last2=Vanden Berghe |first2=Guido |year=2008 |title='Magic Is No Magic': The Wonderful World of Simon Stevin |url=https://books.google.com/books?isbn=1845643917 |page=154 |publisher=WIT Press |isbn=9781845643911 }}</ref> who dropped lead balls of different masses off the [[Nieuwe Kerk (Delft)|Delft churchtower]] and listened for the sound of them hitting a wooden plank.

Newton measured the [[Frequency|period]] of pendulums made with different materials as an alternative test giving the first precision measurements.<ref name=Everitt/> [[Loránd Eötvös]]'s approach in 1908 used a very sensitive [[torsion balance]] to give precision approaching 1 in a billion. Modern experiments have improved this by another factor of a million.

A popular exposition of this measurement was done on the Moon by [[David Scott]] in 1971. He dropped a falcon feather and a hammer at the same time, showing on video<ref>{{cite web | url=https://www.youtube.com/watch?v=MJyUDpm9Kvk |archive-url=https://ghostarchive.org/varchive/youtube/20211221/MJyUDpm9Kvk |archive-date=2021-12-21 |url-status=live |title=Weak Equivalence Principle test on the moon |website=[[YouTube]] |date=18 May 2007 }}{{cbignore}}</ref> that they landed at the same time.

{| class="wikitable"
|+ Chronology of weak equivalence principles tests<ref name=CiufoliniWheeler>Ciufolini, Ignazio; Wheeler, John A.; ''Gravitation and Inertia'', Princeton, New Jersey: Princeton University Press, 1995, pp. 117–119</ref>
|-
!Year
!Investigator
!Sensitivity
!Method
|-
||500?
||[[John Philoponus]]<ref>Philoponus, John; "Corollaries on Place and Void", translated by David Furley, Ithaca, New York: Cornell University Press, 1987</ref>
||"small"
||Drop tower
|-
||1585
||[[Simon Stevin]]<ref>Stevin, Simon; ''De Beghinselen der Weeghconst ["Principles of the Art of Weighing"]'', Leyden, 1586; Dijksterhuis, Eduard J.; "The Principal Works of Simon Stevin", Amsterdam, 1955</ref><ref name="Devreese"/>
||{{val|5|e=−2}}
||Drop tower
|-
||1590?
||[[Galileo Galilei]]<ref>Galilei, Galileo; "Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze", Leida: Appresso gli Elsevirii, 1638; "Discourses and Mathematical Demonstrations Concerning Two New Sciences", Leiden: Elsevier Press, 1638</ref><ref name=CiufoliniWheeler/>{{rp|91}}
||{{val|2|e=−3}}
||Pendulum, drop tower
|-
||1686
||[[Isaac Newton]]<ref>Newton, Isaac; "Philosophiae Naturalis Principia Mathematica" [Mathematical Principles of Natural Philosophy and his System of the World], translated by Andrew Motte, revised by Florian Cajori, Berkeley, California: University of California Press, 1934; Newton, Isaac; "The Principia: Mathematical Principles of Natural Philosophy", translated by I. Bernard Cohen and Anne Whitman, with the assistance of Julia Budenz, Berkeley, California: University of California Press, 1999</ref><ref name=CiufoliniWheeler/>{{rp|91}}
||{{val|e=−3}}
||Pendulum
|-
||1832
||[[Friedrich Wilhelm Bessel]]<ref>Bessel, Friedrich W.; "Versuche Uber die Kraft, mit welcher die Erde Körper von verschiedner Beschaffenhelt anzieht", ''Annalen der Physik und Chemie'', Berlin: J. Ch. Poggendorff, 25 401–408 (1832)</ref><ref name=CiufoliniWheeler/>{{rp|91}}
||{{val|2|e=−5}}
||Pendulum
|-
||1908 (1922)
||[[Loránd Eötvös]]<ref>R. v. Eötvös 1890 [https://archive.org/details/mathematischeund818891890magy ''Mathematische und Naturwissenschaftliche Berichte aus Ungarn''], 8, 65; ''Annalen der Physik'' (Leipzig) 68 11 (1922); {{cite journal | doi = 10.1103/PhysRevD.61.022001 | volume=61 | title=Short-range tests of the equivalence principle | year=1999 | journal=Physical Review D  | last1 = Smith | first1 = G. L. | last2 = Hoyle | first2 = C. D. | last3 = Gundlach | first3 = J. H. | last4 = Adelberger | first4 = E. G. | last5 = Heckel | first5 = B. R. | last6 = Swanson | first6 = H. E.| issue=2 | page=022001 | bibcode=1999PhRvD..61b2001S | arxiv = 2405.10982 }}</ref><ref name=CiufoliniWheeler/>{{rp|92}}
||{{val|2|e=−9}}
||Torsion balance
|-
||1910
||Southerns<ref>{{cite journal | last1 = Southerns | first1 = Leonard | year = 1910| title = A Determination of the Ratio of Mass to Weight for a Radioactive Substance | journal = Proceedings of the Royal Society of London | volume = 84 | issue = 571| pages = 325–344 | doi = 10.1098/rspa.1910.0078 | bibcode = 1910RSPSA..84..325S | doi-access = free }}</ref><ref name=CiufoliniWheeler/>{{rp|91}}
||{{val|5|e=−6}}
||Pendulum
|-
||1918
||Zeeman<ref>Zeeman, Pieter (1918) "Some experiments on gravitation: The ratio of mass to weight for crystals and radioactive substances", ''Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen'', Amsterdam 20(4) 542–553</ref><ref name=CiufoliniWheeler/>{{rp|91}}
||{{val|3|e=−8}}
||Torsion balance
|-
||1923
||Potter<ref>{{cite journal | last1 = Potter | first1 = Harold H. | year = 1923 | title = Some Experiments on the Proportionality of Mass and Weight | journal = Proceedings of the Royal Society of London | volume = 104 | issue = 728| pages = 588–610 | doi = 10.1098/rspa.1923.0130 | bibcode = 1923RSPSA.104..588P | doi-access = free }}</ref><ref name=CiufoliniWheeler/>{{rp|91}}
||{{val|3|e=−6}}
||Pendulum
|-
||1935
||Renner<ref>{{cite journal | last1 = Renner | first1 = János | year = 1935 | title = Kísérleti vizsgálatok a tömegvonzás és tehetetlenség arányosságáról | journal = Mathematikai és Természettudományi Értesítő | volume = 53 | page = 569 }}</ref><ref name=CiufoliniWheeler/>{{rp|92}}
||{{val|2|e=−9}}
||Torsion balance
|-
||1964
||Roll, Krotkov, [[Robert H. Dicke|Dicke]]<ref name="RollKrotkovDicke">Roll, Peter G.; Krotkov, Robert; Dicke, Robert H.; ''The equivalence of inertial and passive gravitational mass'', Annals of Physics, Volume 26, Issue 3, 20 February 1964, pp. 442–517</ref>
||{{val|3|e=−11}}
||Torsion balance
|-
||1972
||[[Vladimir Braginsky|Braginsky]], Panov<ref>{{cite journal | last1 = Braginski | first1 = Vladimir Borisovich | last2 = Panov | first2 = Vladimir Ivanovich | year = 1971 | title = Журнал Экспериментальной и Теоретической Физики | journal = (Zhurnal Éksperimental'noĭ I Teoreticheskoĭ Fiziki, Journal of Experimental and Theoretical Physics) | volume = 61 | page = 873 }}</ref><ref name=CiufoliniWheeler/>{{rp|92}}
||{{val|e=−12}}
||Torsion balance
|-
||1976
||Shapiro, et al.<ref>{{cite journal|last1=Shapiro |first1=Irwin I. |last2=Counselman |first2=III |last3=Charles |first3=C. |last4=King |first4=Robert W. |year=1976 |title=Verification of the principle of equivalence for massive bodies |url=http://prl.aps.org/pdf/PRL/v36/i11/p555_1 |archive-url=https://archive.today/20140122182435/http://prl.aps.org/pdf/PRL/v36/i11/p555_1 |url-status=dead |archive-date=2014-01-22 |journal=Physical Review Letters |volume=36 |issue= 11|pages=555–558 |doi=10.1103/physrevlett.36.555 |bibcode=1976PhRvL..36..555S }}</ref><ref name=CiufoliniWheeler/>{{rp|92}}
||{{val|e=−12}}
||Lunar laser ranging
|-
||1979
||Keiser, Faller<ref>{{cite journal | last1 = Keiser | first1 = George M. | last2 = Faller | first2 = James E. | year = 1979 | title=New approach to the Eötvös experiment | journal = Bulletin of the American Physical Society | volume = 24 | page = 579 }}</ref><ref name=CiufoliniWheeler/>{{rp|93}}
||{{val|4|e=−11}}
||Fluid support
|-
||1987
||Niebauer, et al.<ref>{{cite journal | last1 = Niebauer | first1 = Timothy M. | last2 = McHugh | first2 = Martin P. | last3 = Faller | first3 = James E. | year = 1987 | title = Galilean test for the fifth force | url = https://zenodo.org/record/1233860| journal = Physical Review Letters | volume = 59 | issue = 6| pages = 609–612 | doi=10.1103/physrevlett.59.609|bibcode = 1987PhRvL..59..609N | pmid=10035824| type = Submitted manuscript }}</ref><ref name=CiufoliniWheeler/>{{rp|95}}
||{{val|e=−10}}
||Drop tower
|-
||1989
||Stubbs, et al.<ref>{{cite journal | last1 = Stubbs | first1 = Christopher W. | last2 = Adelberger | first2 = Eric G. | last3 = Heckel | first3 = Blayne R. | last4 = Rogers | first4 = Warren F. | last5 = Swanson | first5 = H. Erik | last6 = Watanabe | first6 = R. | last7 = Gundlach | first7 = Jens H. | last8 = Raab | first8 = Frederick J. | year = 1989 | title = Limits on Composition-Dependent Interactions Using a Laboratory Source: Is There a "Fifth Force" Coupled to Isospin? | journal = Physical Review Letters | volume = 62 | issue = 6| pages = 609–612 | doi=10.1103/physrevlett.62.609|bibcode = 1989PhRvL..62..609S | pmid=10040283}}</ref><ref name=CiufoliniWheeler/>{{rp|93}}
||{{val|e=−11}}
||Torsion balance
|-
||1990
||Adelberger, Eric G.; et al.<ref>{{cite journal | last1 = Adelberger | first1 = Eric G. | last2 = Stubbs | first2 = Christopher W. | last3 = Heckel | first3 = Blayne R. | last4 = Su | first4 = Y. | last5 = Swanson | first5 = H. Erik | last6 = Smith | first6 = G. L. | last7 = Gundlach | first7 = Jens H. | last8 = Rogers | first8 = Warren F. | year = 1990 | title = Testing the equivalence principle in the field of the Earth: Particle physics at masses below 1 μeV? | journal = Physical Review D | volume = 42 | issue = 10| pages = 3267–3292 | doi=10.1103/physrevd.42.3267| pmid = 10012726 |bibcode = 1990PhRvD..42.3267A }}</ref><ref name=CiufoliniWheeler/>{{rp|95}}
||{{val|e=−12}}
||Torsion balance
|-
||1999
||Baessler, et al.<ref>{{cite journal | last1 = Baeßler | first1 = Stefan | display-authors = etal | year = 2001 | title = Remarks by Heinrich Hertz (1857–94) on the equivalence principle| journal = Classical and Quantum Gravity | volume = 18 | issue = 13| page = 2393 | doi = 10.1088/0264-9381/18/13/301 | bibcode = 2001CQGra..18.2393B | s2cid = 250758089 }}</ref><ref>{{cite journal | last1 = Baeßler | first1 = Stefan | last2 = Heckel | first2 = Blayne R. | last3 = Adelberger | first3 = Eric G. | last4 = Gundlach | first4 = Jens H. | last5 = Schmidt | first5 = Ulrich | last6 = Swanson | first6 = H. Erik | year = 1999 | title = Improved Test of the Equivalence Principle for Gravitational Self-Energy | journal = Physical Review Letters | volume = 83 | issue = 18| page = 3585 | doi = 10.1103/physrevlett.83.3585 | bibcode = 1999PhRvL..83.3585B }}</ref>
||{{val|5|e=−14}}
||Torsion balance
|-
||2008
||Schlamminger, et al.<ref>{{cite journal |last1=Schlamminger |first1=Stephan |last2=Choi |first2=Ki-Young |last3=Wagner |first3=Todd A. |last4=Gundlach |first4=Jens H. |last5=Adelberger |first5=Eric G. |title=Test of the Equivalence Principle Using a Rotating Torsion Balance |journal=Physical Review Letters |volume=100 |issue=4 |year=2008 |doi=10.1103/PhysRevLett.100.041101 |bibcode=2008PhRvL.100d1101S |arxiv=0712.0607 |pmid=18352252 |page=041101|s2cid=18653407 }}</ref>
||{{val|e=−13}}
||Torsion balance
|-
||2017
||[[MICROSCOPE]]<ref>
{{cite journal
 |title=MICROSCOPE Mission: First Results of a Space Test of the Equivalence Principle
 |last1= Touboul|first1= Pierre
 |last2= Métris|first2= Gilles
 |last3= Rodrigues|first3= Manuel
 |last4= André|first4= Yves
 |last5= Baghi|first5= Quentin
 |last6= Bergé|first6= Joël
 |last7= Boulanger|first7= Damien
 |last8= Bremer|first8= Stefanie
 |last9= Carle|first9= Patrice
 |last10= Chhun|first10= Ratana
 |last11= Christophe|first11= Bruno
 |last12= Cipolla|first12= Valerio
 |last13= Damour|first13= Thibault
 |last14= Danto|first14= Pascale
 |last15= Dittus|first15= Hansjoerg
 |last16= Fayet|first16= Pierre
 |last17= Foulon|first17= Bernard
 |last18= Gageant|first18= Claude
 |last19= Guidotti|first19= Pierre-Yves
 |last20= Hagedorn|first20= Daniel
 |last21= Hardy|first21= Emilie
 |last22= Huynh|first22= Phuong-Anh
 |last23= Inchauspe|first23= Henri
 |last24= Kayser|first24= Patrick
 |last25= Lala|first25= Stéphanie
 |last26= Lämmerzahl|first26= Claus
 |last27= Lebat|first27= Vincent
 |last28= Leseur|first28= Pierre
 |last29= Liorzou|first29= Françoise
 |last30= List|first30= Meike
 |display-authors= 29
 |journal= Physical Review Letters
 |volume= 119
 |issue= 23
 |year= 2017
 |pages= 231101
 |arxiv= 1712.01176
 |doi= 10.1103/PhysRevLett.119.231101
 |pmid= 29286705|s2cid= 6211162
 |bibcode=2017PhRvL.119w1101T
}}</ref><ref>
{{cite journal |last1=Touboul |first1=Pierre |last2=Métris |first2=Gilles |last3=Rodrigues |first3=Manuel |last4=Bergé |first4=Joel |last5=Robert |first5=Alain |last6=Baghi |first6=Quentin |last7=André |first7=Yves |last8=Bedouet |first8=Judicaël |last9=Boulanger |first9=Damien |last10=Bremer |first10=Stefanie |last11=Carle |first11=Patrice |year=2022 |title=MICROSCOPE Mission: Final Results of the Test of the Equivalence Principle. |journal=Physical Review Letters |volume= 129 |issue=12 |pages=121102 |doi=10.1103/PhysRevLett.129.121102 |pmid=36179190 |arxiv=2209.15487 |bibcode=2022PhRvL.129l1102T |s2cid=252468544 |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.129.121102 }}</ref>
||10<sup>−15</sup>
||Earth orbit
|}

Experiments are still being performed at the [[University of Washington]] which have placed limits on the differential acceleration of objects towards the Earth, the Sun and towards [[dark matter]] in the [[Galactic Center]].<ref>{{Cite web|url=http://www.npl.washington.edu/eotwash/|title=The Eöt-Wash Group &#124; Laboratory Tests of Gravitational and sub-Gravitational Physics|website=www.npl.washington.edu}}</ref> Future satellite experiments<ref name="Dittus">{{cite conference|last1=Dittus |first1=Hansjörg |first2=Claus |last2=Lāmmerzahl |title=Experimental Tests of the Equivalence Principle and Newton's Law in Space |page=95 |volume=758 |doi=10.1063/1.1900510 |conference=Gravitation and Cosmology: 2nd Mexican Meeting on Mathematical and Experimental Physics, AIP Conference Proceedings |url=http://www.zarm.uni-bremen.de/2forschung/gravi/publications/papers/2005DittusLaemmerzahl.pdf |bibcode=2005AIPC..758...95D |url-status=dead |archive-url=https://web.archive.org/web/20081217040845/http://www.zarm.uni-bremen.de/2forschung/gravi/publications/papers/2005DittusLaemmerzahl.pdf |archive-date=17 December 2008|year=2005 }}</ref> – [[STEP (satellite)|Satellite Test of the Equivalence Principle]]<ref>{{cite web | url=http://einstein.stanford.edu/STEP/ | title=S T e P}}</ref> and Galileo Galilei – will test the weak equivalence principle in space, to much higher accuracy.<ref>{{cite web | url=http://eotvos.dm.unipi.it/nobili/ | title="GALILEO GALILEI" GG Small Mission Project}}</ref>

With the first successful production of antimatter, in particular anti-hydrogen, a new approach to test the weak equivalence principle has been proposed. Experiments to compare the gravitational behavior of matter and antimatter are currently being developed.<ref>{{cite journal|title=Testing the Weak Equivalence Principle with an antimatter beam at CERN|journal=Journal of Physics: Conference Series|date=2015|volume=631|issue=1|page=012047|url=http://stacks.iop.org/1742-6596/631/i=1/a=012047|bibcode = 2015JPhCS.631a2047K |doi = 10.1088/1742-6596/631/1/012047 |last1=Kimura|first1=M.|last2=Aghion|first2=S.|last3=Amsler|first3=C. |last4=Ariga|first4=A.|last5=Ariga|first5=T.|last6=Belov|first6=A.|last7=Bonomi|first7=G. |last8=Bräunig|first8=P. |last9=Bremer|first9=J.|last10=Brusa|first10=R. S.|last11=Cabaret|first11=L.|last12=Caccia |first12=M. |last13=Caravita|first13=R. |last14=Castelli|first14=F.|last15=Cerchiari|first15=G. |last16=Chlouba|first16=K. |last17=Cialdi|first17=S. |last18=Comparat|first18=D. |last19=Consolati|first19=G.|last20=Demetrio|first20=A. |last21=Derking|first21=H. |last22=Di Noto|first22=L. |last23=Doser |first23=M.|last24=Dudarev|first24=A. |last25=Ereditato|first25=A. |last26=Ferragut|first26=R.|last27=Fontana|first27=A. |last28=Gerber|first28=S. |last29=Giammarchi|first29=M. |last30=Gligorova|first30=A.|display-authors=29|doi-access=free|hdl=2434/457743|hdl-access=free}}</ref>

Proposals that may lead to a [[quantum gravity|quantum theory of gravity]] such as [[string theory]] and [[loop quantum gravity]] predict violations of the weak equivalence principle because they contain many light [[scalar field]]s with long [[Compton wavelength]]s, which should generate [[fifth force]]s and variation of the fundamental constants. Heuristic arguments suggest that the magnitude of these equivalence principle violations could be in the 10<sup>−13</sup> to 10<sup>−18</sup> range.<ref name="Overduin2009" />

Currently envisioned tests of the weak equivalence principle are approaching a degree of sensitivity such that ''non-discovery'' of a violation would be just as profound a result as discovery of a violation. Non-discovery of equivalence principle violation in this range would suggest that gravity is so fundamentally different from other forces as to require a major reevaluation of current attempts to unify gravity with the other forces of nature. A positive detection, on the other hand, would provide a major guidepost towards unification.<ref name="Overduin2009">{{Cite journal | last1 = Overduin | first1 = James | last2 = Everitt | first2 = Francis | last3 = Mester | first3 = John | last4 = Worden | first4 = Paul | title = The Science Case for STEP | doi = 10.1016/j.asr.2009.02.012 | journal = Advances in Space Research | volume = 43 | issue = 10 | pages = 1532–1537 | year = 2009 |arxiv = 0902.2247 |bibcode = 2009AdSpR..43.1532O | s2cid = 8019480 }}</ref>

=== Tests of the Einstein equivalence principle ===
In addition to the tests of the weak equivalence principle, the Einstein equivalence principle requires testing the local Lorentz invariance and local positional invariance conditions.

Testing local Lorentz invariance amounts to testing special relativity, a theory with vast number of existing tests.<ref name=Will2014/>{{rp|12}} Nevertheless, attempts to look for quantum gravity require even more precise tests. The modern tests include looking for directional variations in the [[speed of light]] (called "clock anisotropy tests") and new forms of the [[Michelson–Morley experiment]]. The anisotropy measures less than one part in 10<sup>−20</sup>.<ref name=Will2014/>{{rp|14}}

Testing local positional invariance divides in to tests in space and in time.<ref name=Will2014/>{{rp|17}} Space-based tests use measurements of the [[gravitational redshift]], the classic is the [[Pound–Rebka experiment]] in the 1960s. The most precise measurement was done in 1976 by flying a hydrogen maser and comparing it to one on the ground. The [[Global Positioning System]] requires compensation for this redshift to give accurate position values.

Time-based tests search for variation of [[dimensionless]] [[fundamental physical constants|constants and mass ratios]].<ref name="Uzan">{{Cite journal |last=Uzan |first=Jean-Philippe |date=2003-04-07 |title=The fundamental constants and their variation: observational and theoretical status |url=https://link.aps.org/doi/10.1103/RevModPhys.75.403 |journal=Reviews of Modern Physics |language=en |volume=75 |issue=2 |pages=403–455 |doi=10.1103/RevModPhys.75.403 |arxiv=hep-ph/0205340 |bibcode=2003RvMP...75..403U |s2cid=118684485 |issn=0034-6861}}</ref> For example, Webb et al.<ref>{{cite journal |first1=John K. |last1=Webb |first2=Michael T. |last2=Murphy |first3=Victor V. |last3=Flambaum |first4=Vladimir A. |last4=Dzuba |first5=John D. |last5=Barrow |first6=Chris W. |last6=Churchill |first7=Jason X. |last7=Prochaska |first8=Arthur M. |last8=Wolfe |doi=10.1103/PhysRevLett.87.091301 |journal=Physical Review Letters |title=Further Evidence for Cosmological Evolution of the Fine Structure Constant |volume=87 |issue=9 |year=2001 |arxiv=astro-ph/0012539 |pmid=11531558 |bibcode=2001PhRvL..87i1301W |pages=091301|s2cid=40461557 }}</ref> reported detection of variation (at the 10<sup>−5</sup> level) of the fine-structure constant from measurements of distant [[quasar]]s. Other researchers dispute these findings.<ref>{{Cite journal |last1=Rocha |first1=G |last2=Trotta |first2=R |last3=Martins |first3=C.J.A.P |last4=Melchiorri |first4=A |last5=Avelino |first5=P.P |last6=Viana |first6=P.T.P |date=Nov 2003 |title=New constraints on varying α |url=https://linkinghub.elsevier.com/retrieve/pii/S1387647303001532 |journal=New Astronomy Reviews |language=en |volume=47 |issue=8–10 |pages=863–869 |doi=10.1016/j.newar.2003.07.018|arxiv=astro-ph/0309205 |bibcode=2003NewAR..47..863R |s2cid=9280269 }}</ref>

The present best limits on the variation of the fundamental constants have mainly been set by studying the naturally occurring [[Oklo]] [[natural nuclear fission reactor]], where nuclear reactions similar to ones we observe today have been shown to have occurred underground approximately two billion years ago. These reactions are extremely sensitive to the values of the fundamental constants.

{| class="wikitable"
|+ Tests of changes in fundamental constants<ref name=Will2014/>{{rp|19}}
|-
!Constant
!Year
!Method
!Limit on fractional change per year
|-
||[[weak interaction]] constant
||1976
||Oklo
||10<sup>−11</sup>
|-
||[[fine-structure constant]]
||1976
||Oklo
||10<sup>−16</sup>
|-
||[[electron]]–[[proton]] mass ratio
||2002
||quasars
||10<sup>−15</sup>
|}

=== Tests of the strong equivalence principle ===
The strong equivalence principle can be tested by 1) finding orbital variations in massive bodies (Sun-Earth-Moon), 2) variations in the gravitational constant (''G'') depending on nearby sources of gravity or on motion, or 3) searching for a variation of Newton's gravitational constant over the life of the universe<ref name=Will2014/>{{rp|47}}

Orbital variations due to gravitational self-energy should cause a "polarization" of solar system orbits called the [[Nordtvedt effect]]. This effect has been sensitively tested by [[Lunar Laser Ranging experiments]].<ref>{{cite web | url=http://funphysics.jpl.nasa.gov/technical/grp/lunar-laser.html | title=Fundamental Physics of Space – Technical Details | access-date=7 May 2005 | archive-url=https://web.archive.org/web/20161128185551/http://funphysics.jpl.nasa.gov/technical/grp/lunar-laser.html | archive-date=28 November 2016 | url-status=dead }}</ref><ref>{{cite journal |last1=Viswanathan |first1=V |last2=Fienga |first2=A |last3=Minazzoli |first3=O |last4=Bernus |first4=L |last5=Laskar |first5=J |last6=Gastineau |first6=M |title=The new lunar ephemeris INPOP17a and its application to fundamental physics |journal=Monthly Notices of the Royal Astronomical Society |date=May 2018 |volume=476 |issue=2 |pages=1877–1888 |doi=10.1093/mnras/sty096|doi-access=free |arxiv=1710.09167 |bibcode=2018MNRAS.476.1877V |s2cid=119454879 }}</ref> Up to the limit of one part in 10<sup>13</sup> there is no Nordtvedt effect.

A tight bound on the effect of nearby gravitational fields on the strong equivalence principle comes from modeling the orbits of binary stars and comparing the results to [[pulsar]] timing data.<ref name=Will2014/>{{rp|49}} In 2014, astronomers discovered a stellar triple system containing a millisecond pulsar [[PSR J0337+1715]] and two [[white dwarf]]s orbiting it. The system provided them a chance to test the strong equivalence principle in a strong gravitational field with high accuracy.<ref>{{cite journal |url=http://www.nature.com/nature/journal/vaop/ncurrent/full/nature12917.html#ref7 |title=A millisecond pulsar in a stellar triple system |first1=Scott M. |last1=Ransom |display-authors=etal |journal=Nature |year=2014|doi=10.1038/nature12917 |arxiv = 1401.0535 |bibcode = 2014Natur.505..520R |volume=505 |issue=7484 |pages=520–524 |pmid=24390352|s2cid=4468698 }}</ref><ref>{{cite journal|title=Universality of free fall from the orbital motion of a pulsar in a stellar triple system|author=Anne M. Archibald|author-link=Anne Archibald|display-authors=etal |journal=Nature|volume=559 |issue=7712|pages=73–76 |date=4 July 2018|doi=10.1038/s41586-018-0265-1|pmid=29973733 |arxiv=1807.02059 | bibcode=2018Natur.559...73A|s2cid=49578025}}</ref><ref>{{cite news|title=Even Phenomenally Dense Neutron Stars Fall like a Feather – Einstein Gets It Right Again|url=https://public.nrao.edu/news/neutron-stars-fall/ |publisher=NRAO|date=4 July 2018 |work=Charles Blue, Paul Vosteen}}</ref> If there is any departure from the strong equivalence principle, it is no more than two [[parts per million]].<ref>{{Cite journal|last1=Voisin|first1=G. |last2=Cognard|first2=I. |last3=Freire|first3=P. C. C.|last4=Wex|first4=N. |last5=Guillemot|first5=L. |last6=Desvignes|first6=G. |last7=Kramer|first7=M. |last8=Theureau|first8=G. |date=2020-06-01|title=An improved test of the strong equivalence principle with the pulsar in a triple star system |url=https://www.aanda.org/articles/aa/abs/2020/06/aa38104-20/aa38104-20.html |journal=Astronomy & Astrophysics|language=en |volume=638 |pages=A24 |arxiv=2005.01388|doi=10.1051/0004-6361/202038104|bibcode=2020A&A...638A..24V|s2cid=218486794 |issn=0004-6361}}</ref>

Most alternative theories of gravity predict a change in the gravity constant over time.  Studies of [[Big Bang nucleosynthesis]], analysis of pulsars, and the lunar laser ranging data have shown that ''G'' cannot have varied by more than 10% since the creation of the universe. The best data comes from studies of the [[ephemeris]] of Mars, based on three successive NASA missions, [[Mars Global Surveyor]], [[Mars Odyssey]], and [[Mars Reconnaissance Orbiter]].<ref name=Will2014/>{{rp|50}}