Editing Curved spacetime (section)

== Experimental test of the sources of spacetime curvature ==
=== Definitions: Active, passive, and inertial mass ===
Bondi distinguishes between different possible types of mass: (1) {{nowrap|1=active mass (<math>m_a</math>)}} is the mass which acts as the '''''source''''' of a gravitational field; (2){{nowrap|1=passive mass (<math>m_p</math>)}} is the mass which '''''reacts to''''' a gravitational field; (3) {{nowrap|1=inertial mass (<math>m_i</math>)}} is the mass which reacts to acceleration.<ref name=Bondi />
* <math>m_p</math> is the same as {{nowrap|1=gravitational mass (<math>m_g</math>)}} in the [[#Basic propositions|discussion of the equivalence principle]].

In Newtonian theory,
* The third law of action and reaction dictates that <math>m_a</math> and <math>m_p</math> must be the same.
* On the other hand, whether <math>m_p</math> and <math>m_i</math> are equal is an empirical result.

In general relativity,
* The equality of <math>m_p</math> and <math>m_i</math> is dictated by the equivalence principle.
* There is no "action and reaction" principle dictating any necessary relationship between <math>m_a</math> and <math>m_p</math>.<ref name="Bondi">{{cite book |last1=Bondi |first1=Hermann |url=http://www.edition-open-sources.org/sources/5/24/index.html |title=The Role of Gravitation in Physics: Report from the 1957 Chapel Hill Conference |date=1957 |publisher=Max Planck Research Library |isbn=978-3-86931-963-6 |editor1-last=DeWitt |editor1-first=Cecile M. |location=Berlin, Germany |pages=159–162 |access-date=1 July 2017 |editor2-last=Rickles |editor2-first=Dean |archive-url=https://web.archive.org/web/20170728121700/http://www.edition-open-sources.org/sources/5/24/index.html |archive-date=28 July 2017 |url-status=live}}</ref>

=== 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>

{{multiple image
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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>

=== Gravitomagnetism ===
[[File:Gravity Probe B Confirms the Existence of Gravitomagnetism.jpg|330px|thumb|Figure 5–11. Gravity Probe B confirmed the existence of gravitomagnetism]]

The existence of gravitomagnetism was proven by [[Gravity Probe B]] {{nowrap|1=(GP-B)}}, a satellite-based mission which launched on 20 April 2004.<ref>
{{cite web
 |url=http://einstein.stanford.edu/content/faqs/faqs.html#launch
 |title=Gravity Probe B: FAQ
 |access-date=2 July 2017
 |archive-date=2 June 2018
 |archive-url=https://web.archive.org/web/20180602231753/http://einstein.stanford.edu/content/faqs/faqs.html#launch
 |url-status=live
}}</ref> The spaceflight phase lasted until <time>2005</time>. The mission aim was to measure spacetime curvature near Earth, with particular emphasis on [[gravitomagnetism]].

Initial results confirmed the relatively large [[geodetic effect]] (which is due to simple spacetime curvature, and is also known as de Sitter precession) to an accuracy of about 1%. The much smaller [[frame-dragging]] effect (which is due to gravitomagnetism, and is also known as [[Lense–Thirring precession]]) was difficult to measure because of unexpected charge effects causing variable drift in the gyroscopes. Nevertheless, by <time datetime="2008-08">August 2008</time>, the frame-dragging effect had been confirmed to within 15% of the expected result,<ref name="Gugliotta2009">
{{cite news
 | last = Gugliotta
 | first = G.
 | title = Perseverance Is Paying Off for a Test of Relativity in Space
 | work = [[New York Times]]
 | date = 16 February 2009
 | url = https://www.nytimes.com/2009/02/17/science/17gravity.html?_r=1
 | access-date = 2 July 2017
 | archive-date = 3 September 2018
 | archive-url = https://web.archive.org/web/20180903151359/https://www.nytimes.com/2009/02/17/science/17gravity.html?_r=1
 | url-status = live
}}</ref> while the geodetic effect was confirmed to better than 0.5%.<ref>{{cite web |last1=Everitt |first1=C. W. F. |last2=Parkinson |first2=B. W. |date=2009 |title=Gravity Probe B Science Results—NASA Final Report |url=http://einstein.stanford.edu/content/final_report/GPB_Final_NASA_Report-020509-web.pdf |url-status=live |archive-url=https://web.archive.org/web/20121023062122/http://einstein.stanford.edu/content/final_report/GPB_Final_NASA_Report-020509-web.pdf |archive-date=23 October 2012 |access-date=2 July 2017}}</ref><ref name=PRL>{{cite journal | author=Everitt | display-authors=etal | date=2011| title=Gravity Probe B: Final Results of a Space Experiment to Test General Relativity| journal=Physical Review Letters| volume = 106 | issue = 22 | page=221101 | doi = 10.1103/PhysRevLett.106.221101 |arxiv =1105.3456 | bibcode=2011PhRvL.106v1101E | pmid=21702590| s2cid=11878715 }}</ref>

Subsequent measurements of frame dragging by laser-ranging observations of the [[LARES (satellite)|LARES]], {{nowrap|1=[[LAGEOS]]-1}} and {{nowrap|1=LAGEOS-2}} satellites has improved on the {{nowrap|1=GP-B}} measurement, with results (as of 2016) demonstrating the effect to within 5% of its theoretical value,<ref>{{cite journal |last1=Ciufolini |first1=Ignazio |last2=Paolozzi |first2=Antonio Rolf Koenig |last3=Pavlis |first3=Erricos C. |last4=Koenig |first4=Rolf |date=2016 |title=A test of general relativity using the LARES and LAGEOS satellites and a GRACE Earth gravity model |journal=European Physical Journal C |volume=76 |issue=3 |page=120 |arxiv=1603.09674 |bibcode=2016EPJC...76..120C |doi=10.1140/epjc/s10052-016-3961-8 |pmc=4946852 |pmid=27471430}}</ref> although there has been some disagreement on the accuracy of this result.<ref>{{cite journal|last=Iorio|first=L.|title=A comment on "A test of general relativity using the LARES and LAGEOS satellites and a GRACE Earth gravity model. Measurement of Earth's dragging of inertial frames," by I. Ciufolini et al|journal=The European Physical Journal C|date=February 2017|volume=77|issue=2|pages=73|doi=10.1140/epjc/s10052-017-4607-1|bibcode = 2017EPJC...77...73I |arxiv = 1701.06474 |s2cid=118945777}}</ref>

Another effort, the Gyroscopes in General Relativity (GINGER) experiment, seeks to use three 6&nbsp;m [[ring lasers]] mounted at right angles to each other 1400&nbsp;m below the Earth's surface to measure this effect.<ref>{{cite web |last1=Cartlidge |first1=Edwin |title=Underground ring lasers will put general relativity to the test |url=http://physicsworld.com/cws/article/news/2016/jan/20/underground-ring-lasers-will-put-general-relativity-to-the-test |website=physicsworld.com |date=20 January 2016 |publisher=Institute of Physics |access-date=2 July 2017 |archive-date=12 July 2017 |archive-url=https://web.archive.org/web/20170712144159/http://physicsworld.com/cws/article/news/2016/jan/20/underground-ring-lasers-will-put-general-relativity-to-the-test |url-status=live }}</ref><ref>{{cite web |title=Einstein right using the most sensitive Earth rotation sensors ever made |url=https://phys.org/news/2017-05-einstein-sensitive-earth-rotation-sensors.html |website=Phys.org |publisher=Science X network |access-date=2 July 2017 |archive-date=10 May 2017 |archive-url=https://web.archive.org/web/20170510173326/https://phys.org/news/2017-05-einstein-sensitive-earth-rotation-sensors.html |url-status=live }}</ref> The first ten years of experience with a prototype ring laser gyroscope array, GINGERINO, established that the full scale experiment should be able to measure gravitomagnetism due to the Earth's rotation to within a 0.1% level or even better.<ref>{{cite book |last1=Altucci |first1=C. |last2=Bajardi |first2=F. |last3=Basti |first3=A. |last4=Beverini |first4=N. |last5=Capozziello |first5=S.  |title=The Ginger project – preliminary results. Proceedings of the MG16 Meeting on General Relativity Online; 5–10 July 2021 |date=2021 |isbn=978-981-12-6977-6 |pages=3956–3962 |doi=10.1142/9789811269776_0329 |url=https://doi.org/10.1142/9789811269776_0329 |access-date=7 September 2024}}</ref>