Editing Gravitational redshift (section)

== History ==

The gravitational weakening of light from high-gravity stars was predicted by [[John Michell]] in 1783 and [[Pierre-Simon Laplace]] in 1796, using [[Isaac Newton]]'s concept of light corpuscles (see: [[Emission theory (relativity)|emission theory]]) and who predicted that some stars would have a gravity so strong that light would not be able to escape. The effect of gravity on light was then explored by [[Johann Georg von Soldner]] (1801), who calculated the amount of deflection of a light ray by the Sun, arriving at the Newtonian answer which is half the value predicted by [[general relativity]]. All of this early work assumed that light could slow down and fall, which is inconsistent with the modern understanding of light waves.

Einstein's 1917 paper on general relativity proposed three tests: the timing of the perihelion of Mercury, the bending of light around the Sun, and the shift in frequency of light emerging from a different gravitational potential, now called the gravitational redshift. Of these, the redshift proved difficult for physicist to understand and to measure convincingly.<ref>{{Cite journal |last=Earman |first=John |last2=Glymour |first2=Clark |date=September 1980 |title=The gravitational red shift as a test of general relativity: History and analysis |url=https://linkinghub.elsevier.com/retrieve/pii/0039368180900254 |journal=Studies in History and Philosophy of Science Part A |language=en |volume=11 |issue=3 |pages=175–214 |doi=10.1016/0039-3681(80)90025-4|url-access=subscription }}</ref> A confusing mix of complex and subtle issues plague even famous textbook descriptions of the phenomenon.<ref>{{Cite journal |last=Scott |first=Robert B |date=2015-04-28 |title=Teaching the gravitational redshift: lessons from the history and philosophy of physics |url=https://iopscience.iop.org/article/10.1088/1742-6596/600/1/012055 |journal=Journal of Physics: Conference Series |volume=600 |pages=012055 |doi=10.1088/1742-6596/600/1/012055 |issn=1742-6588|doi-access=free }}</ref>

Once it became accepted that light was an electromagnetic wave, it was clear that the frequency of light should not change from place to place, since waves from a source with a fixed frequency keep the same frequency everywhere. One way around this conclusion would be if time itself were altered{{snd}} if clocks at different points had different rates. This was precisely Einstein's conclusion in 1911.<ref name=E1911>{{cite journal |author=Einstein, A. |title=On the Influence of Gravitation on the Propagation of Light |journal=Annalen der Physik |year=1911 |volume=35 |issue=10 |pages=898–908 |doi=10.1002/andp.19113401005 |bibcode=1911AnP...340..898E |url=https://zenodo.org/record/1424213 }}</ref> He considered an accelerating box, and noted that according to the [[special theory of relativity]], the clock rate at the "bottom" of the box (the side away from the direction of acceleration) was slower than the clock rate at the "top" (the side toward the direction of acceleration). Indeed, in a frame moving (in <math>x</math> direction) with velocity <math>v</math> relative to the rest frame, the clocks at a nearby position <math>dx</math> [[Lorentz transformation#Physical formulation of Lorentz boosts|are ahead]] by <math>(dx/c)(v/c)</math> (to the first order); so an acceleration <math>g</math> (that changes speed by <math>g/dt</math> per time <math>dt</math>) makes clocks at the position <math>dx</math> to be ahead by <math>(dx/c)(g/c)dt</math>, that is, tick at a rate 
: <math>R=1+(g/c^2)dx</math>

The equivalence principle implies that this change in clock rate is the same whether the acceleration <math>g</math> is that of an accelerated frame without gravitational effects, or caused by a gravitational field in a stationary frame. Since acceleration due to gravitational potential <math>V</math> is <math>-dV/dx</math>, we get
: <math>{dR \over dx} = g/c^2 = - {dV/c^2 \over dx}</math>
so – in weak fields – the change <math>\Delta R</math> in the clock rate is equal to <math>-\Delta V/c^2</math>.

The changing rates of clocks allowed Einstein to conclude that light waves change frequency as they move, and the frequency/energy relationship for photons allowed him to see that this was best interpreted as the effect of the gravitational field on the [[mass–energy equivalence|mass–energy]] of the photon. To calculate the changes in frequency in a nearly static gravitational field, only the time component of the metric tensor is important, and the lowest order approximation is accurate enough for ordinary stars and planets, which are much bigger than their [[Schwarzschild radius]].

=== Astronomical observations ===
{{see also|Tests of general relativity}}

A number of experimenters initially claimed to have identified the effect using astronomical measurements, and the effect was considered to have been finally identified in the spectral lines of the star [[Sirius B]] by [[Walter Sydney Adams|W.S. Adams]] in 1925.<ref name= "Hetherington1980">Hetherington, N. S., [http://adsabs.harvard.edu/full/1980QJRAS..21..246H "Sirius B and the gravitational redshift - an historical review"], ''Quarterly Journal Royal Astronomical Society'', vol. 21, Sept. 1980, pp. 246–252. Accessed 6 April 2017.</ref> However, measurements by Adams have been criticized as being too low<ref name= "Hetherington1980"/><ref name= "Holberg2010">Holberg, J. B., [http://articles.adsabs.harvard.edu//full/2010JHA....41...41H/0000041.000.html "Sirius B and the Measurement of the Gravitational Redshift"], ''Journal for the History of Astronomy'', vol. 41, 1, 2010, pp. 41–64. Accessed 6 April 2017.</ref> and these observations are now considered to be measurements of spectra that are unusable because of scattered light from the primary, Sirius A.<ref name= "Holberg2010" /> The first accurate measurement of the gravitational redshift of a white dwarf was done by Popper in 1954, measuring a 21&nbsp;km/s gravitational redshift of [[40 Eridani]] B.<ref name= "Holberg2010" /> The redshift of [[Sirius|Sirius B]] was finally measured by Greenstein ''et al.'' in 1971, obtaining the value for the gravitational redshift of 89±16&nbsp;km/s, with more accurate measurements by the Hubble Space Telescope, showing 80.4±4.8&nbsp;km/s.<ref>[https://ui.adsabs.harvard.edu/abs/1971ApJ...169..563G/abstract Effective Temperature, Radius, and Gravitational Redshift of Sirius B], J. L. Greenstein, J.B. Oke, H. L. Shipman, ''Astrophysical Journal'' '''169''' (Nov. 1, 1971), pp. 563&ndash;566.</ref>{{Citation needed|date=January 2021}}

[[James W. Brault]], a graduate student of [[Robert Dicke]] at [[Princeton University]], measured the gravitational redshift of the sun using optical methods in 1962.<ref>{{cite thesis |type=PhD |last=Brault |first=James W. |date=1962 |title=The Gravitational Redshift in the Solar Spectrum |url=https://www.proquest.com/docview/302083560|via=ProQuest |id={{ProQuest|302083560}} }}</ref> In 2020, a team of scientists published the most accurate measurement of the solar gravitational redshift so far, made by analyzing [[Iron|Fe]] spectral lines in sunlight reflected by the Moon; their measurement of a mean global 638 ± 6&nbsp;m/s lineshift is in agreement with the theoretical value of 633.1&nbsp;m/s.<ref>{{Cite journal|last1=Hernández|first1=J. I. González|last2=Rebolo|first2=R.|last3=Pasquini|first3=L.|last4=Curto|first4=G. Lo|last5=Molaro|first5=P.|last6=Caffau|first6=E.|last7=Ludwig|first7=H.-G.|last8=Steffen|first8=M.|last9=Esposito|first9=M.|last10=Mascareño|first10=A. Suárez|last11=Toledo-Padrón|first11=B.|date=2020-11-01|title=The solar gravitational redshift from HARPS-LFC Moon spectra - A test of the general theory of relativity|url=https://www.aanda.org/articles/aa/abs/2020/11/aa38937-20/aa38937-20.html|journal=Astronomy & Astrophysics|language=en|volume=643|pages=A146|doi=10.1051/0004-6361/202038937|arxiv=2009.10558|bibcode=2020A&A...643A.146G |s2cid=221836649|issn=0004-6361}}</ref><ref name=":3">{{Cite journal|last=Smith|first=Keith T.|date=2020-12-18|title=Editors' Choice|quote=Gravitational redshift of the Sun|journal=Science|language=en|volume=370|issue=6523|pages=1429–1430|doi=10.1126/science.2020.370.6523.twil|bibcode=2020Sci...370Q1429S|issn=0036-8075|doi-access=free}}</ref> Measuring the solar redshift is complicated by the Doppler shift caused by the motion of the Sun's surface, which is of similar magnitude as the gravitational effect.<ref name=":3" />

In 2011, the group of Radek Wojtak of the Niels Bohr Institute at the University of Copenhagen collected data from 8000 galaxy clusters and found that the light coming from the cluster centers tended to be red-shifted compared to the cluster edges, confirming the energy loss due to gravity.<ref>{{cite web|last=Bhattacharjee|first=Yudhijit|year=2011|title=Galaxy Clusters Validate Einstein's Theory|url=https://www.science.org/content/article/galaxy-clusters-validate-einsteins-theory|access-date=2013-07-23|publisher=News.sciencemag.org}}</ref>

In 2018, the star [[S2 (star)|S2]] made its closest approach to [[Sagittarius A*|Sgr A*]], the 4-million solar mass [[supermassive black hole]] at the centre of the [[Milky Way]], reaching 7650&nbsp;km/s or about 2.5% of [[the speed of light]] while passing the black hole at a distance of just 120 [[Astronomical unit|AU]], or 1400 [[Schwarzschild radius|Schwarzschild radii]]. Independent analyses by the GRAVITY collaboration<ref>{{Cite journal|last1=Abuter|first1=R.|last2=Amorim|first2=A.|last3=Anugu|first3=N.|last4=Bauböck|first4=M.|last5=Benisty|first5=M.|last6=Berger|first6=J. P.|last7=Blind|first7=N.|last8=Bonnet|first8=H.|last9=Brandner|first9=W.|last10=Buron|first10=A.|last11=Collin|first11=C.|date=2018-07-01|title=Detection of the gravitational redshift in the orbit of the star S2 near the Galactic centre massive black hole|url=https://www.aanda.org/articles/aa/abs/2018/07/aa33718-18/aa33718-18.html|journal=Astronomy & Astrophysics|language=en|volume=615|pages=L15|doi=10.1051/0004-6361/201833718|arxiv=1807.09409|bibcode=2018A&A...615L..15G|s2cid=118891445|issn=0004-6361}}</ref><ref>{{Cite journal|last=Witze|first=Alexandra|date=2018-07-26|title=Milky Way's black hole provides long-sought test of Einstein's general relativity|journal=Nature|language=en|volume=560|issue=7716|pages=17|doi=10.1038/d41586-018-05825-3|pmid=30065325|bibcode=2018Natur.560...17W|s2cid=51888156|doi-access=free}}</ref><ref>{{Cite web|title=Tests of General Relativity|url=https://www.mpe.mpg.de/7260308/Tests-of-General-Relativity|access-date=2021-01-17|website=www.mpe.mpg.de|language=en}}</ref><ref>{{Cite web|last=|title=First Successful Test of Einstein's General Relativity Near Supermassive Black Hole - Culmination of 26 years of ESO observations of the heart of the Milky Way|url=https://www.eso.org/public/news/eso1825/|access-date=2021-01-17|website=www.eso.org|language=en}}</ref> (led by [[Reinhard Genzel]]) and the KECK/UCLA Galactic Center Group<ref>{{Cite journal|last1=Do|first1=Tuan|last2=Hees|first2=Aurelien|last3=Ghez|first3=Andrea|last4=Martinez|first4=Gregory D.|last5=Chu|first5=Devin S.|last6=Jia|first6=Siyao|last7=Sakai|first7=Shoko|last8=Lu|first8=Jessica R.|last9=Gautam|first9=Abhimat K.|last10=O’Neil|first10=Kelly Kosmo|last11=Becklin|first11=Eric E.|date=2019-08-16|title=Relativistic redshift of the star S0-2 orbiting the Galactic center supermassive black hole|url=https://www.science.org/doi/10.1126/science.aav8137|journal=Science|volume=365|issue=6454|pages=664–668|language=en|doi=10.1126/science.aav8137|issn=0036-8075|pmid=31346138|arxiv=1907.10731|bibcode=2019Sci...365..664D|s2cid=198901506}}</ref><ref>{{Cite web|last=Siegel|first=Ethan|date=2019-08-01|title=General Relativity Rules: Einstein Victorious In Unprecedented Gravitational Redshift Test|url=https://medium.com/starts-with-a-bang/general-relativity-rules-einstein-victorious-in-unprecedented-gravitational-redshift-test-7ab4076bcd61|access-date=2021-01-17|website=Medium|language=en}}</ref> (led by [[Andrea M. Ghez|Andrea Ghez]]) revealed a combined [[Transverse Doppler effect|transverse Doppler]] and gravitational redshift up to 200&nbsp;km/s/c, in agreement with general relativity predictions.

In 2021, Mediavilla ([[Instituto de Astrofísica de Canarias|IAC]], Spain) & Jiménez-Vicente ([[University of Granada|UGR]], Spain) were able to use measurements of the gravitational redshift in [[quasar]]s up to cosmological redshift of {{nowrap|''z'' ≈ 3}} to confirm the predictions of [[Einstein's equivalence principle]] and the lack of cosmological evolution within 13%.<ref>{{Cite journal|last1=Mediavilla|first1=E.|last2=Jiménez-Vicente|first2=J.|year=2021|title=Testing Einstein's Equivalence Principle and Its Cosmological Evolution from Quasar Gravitational Redshifts|journal=The Astrophysical Journal|volume=914|issue=2|pages=112|arxiv=2106.11699|doi=10.3847/1538-4357/abfb70|bibcode=2021ApJ...914..112M |s2cid=235593322 |doi-access=free }}</ref>

In 2024, Padilla et al. have estimated the gravitational redshifts of supermassive black holes (SMBH) in eight thousand quasars and one hundred Seyfert type 1 galaxies from the full width at half maximum (FWHM) of their emission lines, finding {{nowrap|log ''z'' ≈ −4}}, compatible with SMBHs of ~ 1 billion solar masses and broadline regions of ~ 1 parsec radius. This same gravitational redshift was directly measured by these authors in the SAMI sample of [[LINER]] galaxies, using the redshift differences between lines emitted in central and outer regions.<ref name="padillaetal2024">
{{cite journal
 |author1=N. D. Padilla |author2=S. Carneiro|author3=J. Chaves-Montero |author4= C. J. Donzelli|author5=C. Pigozzo|author6=P. Colazo|author7=J. S. Alcaniz| date=2024
 |title= Active galactic nuclei and gravitational redshifts
 |journal= Astronomy and Astrophysics
 |volume= 683 |pages=120–126
 |doi=10.1051/0004-6361/202348146 |bibcode=2024A&A...683A.120P
 |arxiv=2304.13036
}}</ref>

=== Terrestrial tests ===
{{For|experiments measuring the slowing of clocks|Gravitational time dilation#Experimental confirmation}}
Between 1925 and 1955, very few attempts were made to measure the gravitational redshift.<ref name=Hentschel-1996>{{Cite journal |last=Hentschel |first=Klaus |date=May 1996 |title=Measurements of gravitational redshift between 1959 and 1971 |url=https://www.tandfonline.com/doi/full/10.1080/00033799600200211 |journal=Annals of Science |language=en |volume=53 |issue=3 |pages=269–295 |doi=10.1080/00033799600200211 |issn=0003-3790|url-access=subscription }}</ref>
The effect is now considered to have been definitively verified by the experiments of [[Robert Pound|Pound]], Rebka and Snider between 1959 and 1965. The [[Pound–Rebka experiment]] of 1959 measured the gravitational redshift in spectral lines using a terrestrial [[Iron-57|<sup>57</sup>Fe]] [[gamma rays|gamma]] source over a vertical height of 22.5 metres.<ref name="Pound–Rebka">{{cite journal | doi = 10.1103/PhysRevLett.4.337 | title = Apparent Weight of Photons | date = 1960 | last1 = Pound | first1 = R. | last2 = Rebka | first2 = G. | journal = Physical Review Letters | volume = 4 | issue = 7 | pages = 337–341 | bibcode=1960PhRvL...4..337P| doi-access = free }}</ref> This paper was the first determination of the gravitational redshift which used measurements of the change in wavelength of gamma-ray photons generated with the [[Mössbauer effect]], which generates radiation with a very narrow line width. The accuracy of the gamma-ray measurements was typically 1%.

An improved experiment was done by Pound and Snider in 1965, with an accuracy better than the 1% level.<ref>{{cite journal|last=Pound| first=R. V.|author2=Snider J. L. | date= November 2, 1964| title=Effect of Gravity on Nuclear Resonance| journal=[[Physical Review Letters]]| volume = 13 | issue = 18 | pages=539–540 | doi = 10.1103/PhysRevLett.13.539 | bibcode=1964PhRvL..13..539P| doi-access=free}}</ref>

A very accurate gravitational redshift experiment was performed in 1976,<ref>{{cite journal | display-authors=8 | last=Vessot | first= R. F. C.| date= December 29, 1980| title=Test of Relativistic Gravitation with a Space-Borne Hydrogen Maser | journal=Physical Review Letters | volume = 45 | issue = 26 | pages=2081–2084 | doi = 10.1103/PhysRevLett.45.2081 | author2 = M. W. Levine | author3 = E. M. Mattison | author4 = E. L. Blomberg| author5 = T. E. Hoffman | author6 = G. U. Nystrom| author7 = B. F. Farrel | author8 = R. Decher| author9 = P. B. Eby | author10 = C. R. Baugher| author11 = J. W. Watts | author12 = D. L. Teuber | author13 = F. D. Wills | name-list-style = amp | bibcode=1980PhRvL..45.2081V}}</ref> where a [[hydrogen]] [[maser]] clock on a rocket was launched to a height of {{val|10,000|u=km}}, and its rate compared with an identical clock on the ground. It tested the gravitational redshift to 0.007%.

Later tests can be done with the [[Global Positioning System]] (GPS), which must account for the gravitational redshift in its timing system, and physicists have analyzed timing data from the GPS to confirm other tests. When the first satellite was launched, it showed the predicted shift of 38 microseconds per day. This rate of the discrepancy is sufficient to substantially impair the function of GPS within hours if not accounted for. An excellent account of the role played by general relativity in the design of GPS can be found in Ashby 2003.<ref>{{cite journal|title=Relativity in the Global Positioning System|journal=Living Reviews in Relativity|volume=6|issue=1|pages=1|doi=10.12942/lrr-2003-1|pmid=28163638|pmc=5253894|year = 2003|last1 = Ashby|first1 = Neil|doi-access=free |bibcode=2003LRR.....6....1A}}</ref>

In 2010, an experiment placed two aluminum-ion quantum clocks close to each other, but with the second elevated 33&nbsp;cm compared to the first, making the gravitational red shift effect visible in everyday lab scales.<ref>{{cite journal|last1=Chou|first1=C.W.|last2=Hume|first2=D.B.|last3=Rosenband|first3=T.|last4=Wineland|first4=D.J.|title=Optical Clocks and Relativity|journal=Science|year=2010|volume=329|issue=5999|pages=1630–1633|doi=10.1126/science.1192720|pmid=20929843 |bibcode=2010Sci...329.1630C |s2cid=125987464 |url=https://zenodo.org/record/1230910 }}</ref><ref>
{{cite press release
 |url=https://arstechnica.com/science/2010/09/einsteins-relativity-measured-in-newtons-domain/
 |title=Einstein's time dilation apparent when obeying the speed limit
 |publisher=[[Ars Technica]] 
 |date=24 September 2010
 |access-date=2015-04-10
}}</ref>

In 2020, a group at the [[University of Tokyo]] measured the gravitational redshift of two strontium-87 [[optical lattice]] clocks.<ref>{{cite journal |author=Takamoto, M. |author2=Ushijima, I. |author3=Ohmae, N. |display-authors=etal |date=6 April 2020|title=Test of general relativity by a pair of transportable optical lattice clocks|doi=10.1038/s41566-020-0619-8|journal=Nat. Photonics|volume=14|issue=7 |pages=411–415|bibcode=2020NaPho..14..411T |s2cid=216309660 }}</ref> The measurement took place at [[Tokyo Skytree]] where the clocks were separated by approximately 450&nbsp;m and connected by telecom fibers. The gravitational redshift can be expressed as
: <math> z = \frac{\Delta\nu}{\nu_{1}} = (1+\alpha)\frac{\Delta U}{c^2} </math>,
where <math>\Delta\nu=\nu_{2}-\nu_{1}</math> is the gravitational redshift, <math>\nu_{1}</math> is the optical clock transition frequency, <math>\Delta U= U_{2}- U_{1}</math> is the difference in gravitational potential, and <math>\alpha</math> denotes the violation from general relativity. By [[Ramsey interferometry|Ramsey spectroscopy]] of the strontium-87 optical clock transition (429&nbsp;THz, 698&nbsp;nm) the group determined the gravitational redshift between the two optical clocks to be 21.18&nbsp;Hz, corresponding to a ''z''-value of approximately 5 × 10<sup>−14</sup>. Their measured value of <math>\alpha</math>, <math>(1.4 \pm 9.1)\times 10^{-5} </math>, is an agreement with recent measurements made with hydrogen masers in elliptical orbits.<ref>{{cite journal |author=Sven Herrmann |author2=Felix Finke |author3=Martin Lülf |author4=Olga Kichakova |author5=Dirk Puetzfeld |author6=Daniela Knickmann |author7=Meike List |author8=Benny Rievers |author9=Gabriele Giorgi |author10=Christoph Günther |author11=Hansjörg Dittus |author12=Roberto Prieto-Cerdeira |author13=Florian Dilssner |author14=Francisco Gonzalez |author15=Erik Schönemann |author16=Javier Ventura-Traveset |author17=Claus Lämmerzahl|title=Test of the Gravitational Redshift with Galileo Satellites in an Eccentric Orbit |journal=Physical Review Letters |volume=121 |issue=23 |date=December 2018 |page=231102 |doi=10.1103/PhysRevLett.121.231102|pmid=30576165 |arxiv=1812.09161 |bibcode=2018PhRvL.121w1102H |s2cid=58537350 }}</ref><ref>{{cite journal |author=P. Delva |author2=N. Puchades |author3=E. Schönemann |author4=F. Dilssner |author5=C. Courde |author6=S. Bertone |author7=F. Gonzalez |author8=A. Hees |author9=Ch. Le Poncin-Lafitte |author10=F. Meynadier |author11=R. Prieto-Cerdeira |author12=B. Sohet |author13=J. Ventura-Traveset |author14=P. Wolf|title=Gravitational Redshift Test Using Eccentric Galileo Satellites |journal=Physical Review Letters |volume=121 |issue=23 |date=December 2018 |page=231101 |doi=10.1103/PhysRevLett.121.231101|pmid=30576203 |arxiv=1812.03711 |bibcode=2018PhRvL.121w1101D |s2cid=58666075 }}</ref>

In October 2021, a group at [[JILA]] led by physicist [[Jun Ye]] reported a measurement of gravitational redshift in the submillimeter scale. The measurement is done on the <sup>87</sup>Sr clock transition between the top and the bottom of a millimeter-tall ultracold cloud of 100,000 [[strontium]] atoms in an [[optical lattice]].<ref>{{cite journal |last1=Bothwell |first1=Tobias |last2=Kennedy |first2=Colin J. |last3=Aeppli |first3=Alexander |last4=Kedar |first4=Dhruv |last5=Robinson |first5=John M. |last6=Oelker |first6=Eric |last7=Staron |first7=Alexander |last8=Ye |first8=Jun |year=2022 |title=Resolving the gravitational redshift across a millimetre-scale atomic sample |url=https://jila.colorado.edu/sites/default/files/2022-02/Redshift%201%20mm_Nature%202022.pdf |journal=Nature |volume=602 |issue=7897 |pages=420–424 |arxiv=2109.12238 |bibcode=2022Natur.602..420B |doi=10.1038/s41586-021-04349-7 |pmid=35173346 |s2cid=237940816}}</ref><ref>{{Cite web|last=McCormick|first=Katie|date=2021-10-25|title=An Ultra-Precise Clock Shows How to Link the Quantum World With Gravity|url=https://www.quantamagazine.org/an-atomic-clock-promises-link-between-quantum-world-and-gravity-20211025/|access-date=2021-10-29|website=Quanta Magazine|language=en}}</ref>