Editing Curved spacetime (section)

=== Pressure as a gravitational source ===
[[File:Cavendish and Kreuzer Torsion Balance Diagrams.svg|thumb|330px|Figure 5–9. (A) Cavendish experiment, (B) Kreuzer experiment]]
The classic experiment to measure the strength of a gravitational source (i.e. its active mass) was first conducted in 1797 by [[Cavendish experiment|Henry Cavendish]] (Fig.&nbsp;5-9a). Two small but dense balls are suspended on a fine wire, making a [[torsion balance]]. Bringing two large test masses close to the balls introduces a detectable torque. Given the dimensions of the apparatus and the measurable spring constant of the torsion wire, the gravitational constant ''G'' can be determined.

To study pressure effects by compressing the test masses is hopeless, because attainable laboratory pressures are insignificant in comparison with the {{nowrap|1=mass-energy}} of a metal ball.

However, the repulsive electromagnetic pressures resulting from protons being tightly squeezed inside atomic nuclei are typically on the order of 10<sup>28</sup>&nbsp;atm ≈ 10<sup>33</sup>&nbsp;Pa ≈ 10<sup>33</sup>&nbsp;kg·s<sup>−2</sup>m<sup>−1</sup>. This amounts to about 1% of the nuclear mass density of approximately 10<sup>18</sup>kg/m<sup>3</sup> (after factoring in c<sup>2</sup> ≈ 9×10<sup>16</sup>m<sup>2</sup>s<sup>−2</sup>).<ref>{{cite book |last1=Crowell |first1=Benjamin |title=General Relativity |date=2000 |publisher=Light and Matter |location=Fullerton, CA |pages=241–258 |url=http://www.lightandmatter.com/ |access-date=30 June 2017 |archive-date=18 June 2017 |archive-url=https://web.archive.org/web/20170618200413/http://www.lightandmatter.com/ |url-status=live }}</ref>

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 | footer            = Figure 5-10. Lunar laser ranging experiment. (left) This [[retroreflector]] was left on the Moon by astronauts on the [[Apollo 11]] mission. (right) Astronomers all over the world have bounced laser light off the retroreflectors left by Apollo astronauts and Russian lunar rovers to measure precisely the Earth-Moon distance.
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If pressure does not act as a gravitational source, then the ratio <math>m_a/m_p</math> should be lower for nuclei with higher atomic number ''Z'', in which the electrostatic pressures are higher. {{nowrap|1=L. B. Kreuzer}} (1968) did a Cavendish experiment using a Teflon mass suspended in a mixture of the liquids trichloroethylene and dibromoethane having the same buoyant density as the Teflon (Fig.&nbsp;5-9b). Fluorine has atomic number {{math|1=''Z'' = 9}}, while bromine has {{math|1=''Z'' = 35}}. Kreuzer found that repositioning the Teflon mass caused no differential deflection of the torsion bar, hence establishing active mass and passive mass to be equivalent to a precision of 5×10<sup>−5</sup>.<ref>{{cite journal |last1=Kreuzer |first1=L. B. |title=Experimental measurement of the equivalence of active and passive gravitational mass |journal=Physical Review |date=1968 |volume=169 |issue=5 |pages=1007–1011|bibcode = 1968PhRv..169.1007K |doi = 10.1103/PhysRev.169.1007 }}</ref>

Although Kreuzer originally considered this experiment merely to be a test of the ratio of active mass to passive mass, Clifford Will (1976) reinterpreted the experiment as a fundamental test of the coupling of sources to gravitational fields.<ref>{{cite journal|last1=Will|first1=C. M.|title=Active mass in relativistic gravity-Theoretical interpretation of the Kreuzer experiment|journal=The Astrophysical Journal|date=1976|volume=204|pages=224–234|url=http://articles.adsabs.harvard.edu/full/seri/ApJ../0204/1976ApJ...204..224W.html|bibcode=1976ApJ...204..224W|doi=10.1086/154164|access-date=2 July 2017|archive-date=28 September 2018|archive-url=https://web.archive.org/web/20180928123722/http://articles.adsabs.harvard.edu/full/seri/ApJ../0204/1976ApJ...204..224W.html|url-status=live|doi-access=free}}</ref>

In 1986, Bartlett and Van Buren noted that [[lunar laser ranging]] had detected a 2&nbsp;km offset between the moon's center of figure and its center of mass. This indicates an asymmetry in the distribution of Fe (abundant in the Moon's core) and Al (abundant in its crust and mantle). If pressure did not contribute equally to spacetime curvature as does mass–energy, the moon would not be in the orbit predicted by classical mechanics. They used their measurements to tighten the limits on any discrepancies between active and passive mass to about 10<sup>−12</sup>.<ref>{{cite journal |last1=Bartlett |first1=D. F. |last2=Van Buren |first2=Dave |date=1986 |title=Equivalence of active and passive gravitational mass using the moon |journal=Physical Review Letters |volume=57 |issue=1 |pages=21–24 |bibcode=1986PhRvL..57...21B |doi=10.1103/PhysRevLett.57.21 |pmid=10033347}}</ref> With decades of additional lunar laser ranging data, Singh et al. (2023) reported improvement on these limits by a factor of about 100.<ref name="Singh_2023">{{cite journal |last1=Singh |first1=Vishwa Vijay |last2=Müller |first2=Jürgen |last3=Biskupek |first3=Liliane |last4=Hackmann |first4=Eva |last5=Lämmerzahl |first5=Claus |title=Equivalence of Active and Passive Gravitational Mass Tested with Lunar Laser Ranging |journal=Physical Review Letters |date=2023 |volume=131 |issue=2 |page=021401 |doi=10.1103/PhysRevLett.131.021401 |pmid=37505941 |access-date=7 March 2024 |url=https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.131.021401|arxiv=2212.09407 |bibcode=2023PhRvL.131b1401S }}</ref>