View source for Gravitational redshift

{{Short description|Shift of wavelength of a photon to longer wavelength}}
{{About-distinguish|redshift caused by gravitation|Gravitational wave#Redshifting{{!}}Redshifting of gravitational waves}}
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[[File:Gravitational red-shifting2.png|thumb|200px|The gravitational [[redshift]] of a light wave as it moves upwards against a gravitational field (produced by the yellow star below). The effect is greatly exaggerated in this diagram.]]

In [[physics]] and [[general relativity]], '''gravitational redshift''' (known as '''Einstein shift''' in older literature)<ref>{{cite web|title=Einstein shift definition and meaning {{!}} Collins English Dictionary|url=https://www.collinsdictionary.com/dictionary/english/einstein-shift|access-date=2021-01-21|website=www.collinsdictionary.com|language=en}}</ref><ref name=":4">{{Cite journal|last=Eddington|first=A. S.|date=1926|title=Einstein Shift and Doppler Shift|journal=Nature|language=en|volume=117|issue=2933|pages=86|doi=10.1038/117086a0|bibcode=1926Natur.117...86E|s2cid=4092843|issn=1476-4687|doi-access=free}}</ref> is the phenomenon that [[Electromagnetic radiation|electromagnetic waves]] or [[photon]]s travelling out of a [[gravity well|gravitational well]] lose [[energy]]. This loss of energy corresponds to a decrease in the wave [[frequency]] and increase in the [[wavelength]], known more generally as a ''[[redshift]]''. The opposite effect, in which photons gain energy when travelling into a gravitational well, is known as a '''gravitational blueshift''' (a type of ''[[blueshift]]''). The effect was first described by [[Albert Einstein|Einstein]] in 1907,<ref>{{Cite journal |last=Einstein |first=Albert |date=1907 |title=Relativitätsprinzip und die aus demselben gezogenen Folgerungen |trans-title=On the Relativity Principle and the Conclusions Drawn from It |url=http://www.soso.ch/wissen/hist/SRT/E-1907.pdf |journal=Jahrbuch der Radioaktivität |issue=4 |pages=411–462}}</ref><ref>{{Cite journal|last=Valente|first=Mário Bacelar|date=2018-12-06|title=Einstein's redshift derivations: its history from 1907 to 1921|url=https://revistas.pucsp.br/index.php/circumhc/article/view/36437|journal=Circumscribere: International Journal for the History of Science|language=en|volume=22|pages=1–16|doi=10.23925/1980-7651.2018v22;1-16|s2cid=239568887 |issn=1980-7651|doi-access=free}}</ref> eight years before his publication of [[General relativity|the full theory of relativity]].

Gravitational redshift can be interpreted as a consequence of the [[equivalence principle]] (that gravitational effects are locally equivalent to inertial effects and the redshift is caused by the [[Relativistic Doppler effect|Doppler effect]])<ref name=":1">{{Cite web|last=Florides|first=Petros S.|date=|title=Einstein's Equivalence Principle and the Gravitational Red Shift|url=https://www.maths.tcd.ie/report_series/tcdmath/tcdm1111.pdf|access-date=|website=School of Mathematics, Trinity College, Ireland}}</ref> or as a consequence of the [[mass–energy equivalence]] and conservation of energy ('falling' photons gain energy),<ref>{{Cite journal |last=Chang |first=Donald C. |date=2018 |title=A quantum mechanical interpretation of gravitational redshift of electromagnetic wave |url=https://linkinghub.elsevier.com/retrieve/pii/S0030402618312774 |journal=Optik |language=en |volume=174 |pages=636–641 |doi=10.1016/j.ijleo.2018.08.127|bibcode= |s2cid=126341445 }}</ref><ref>{{cite arXiv|last1=Evans|first1=R. F.|last2=Dunning-Davies|first2=J.|date=2004|title=The Gravitational Red-Shift|eprint=gr-qc/0403082}}</ref> though there are numerous subtleties that complicate a rigorous derivation.<ref name=":1" /><ref name=":2">{{Cite conference|last=Scott|first=Robert B|date=2015|title=Teaching the gravitational redshift: lessons from the history and philosophy of physics|journal=Journal of Physics: Conference Series |conference=Spanish Relativity Meeting (ERE 2014)|language=en|volume=600|issue=1|page=012055|doi=10.1088/1742-6596/600/1/012055|bibcode=2015JPhCS.600a2055S|doi-access=free}}</ref> A gravitational redshift can also equivalently be interpreted as [[gravitational time dilation]] at the source of the radiation:<ref name=":2" /><ref name=":4" /> if two [[electronic oscillator|oscillator]]s (attached to [[transmitter]]s producing electromagnetic radiation) are operating at different [[gravitational potential]]s, the oscillator at the higher gravitational potential (farther from the attracting body) will tick faster; that is, when observed from the same location, it will have a higher measured frequency than the oscillator at the lower gravitational potential (closer to the attracting body).

To first approximation, gravitational redshift is proportional to the difference in [[gravitational potential]] divided by the [[speed of light]] squared, <math>z = \Delta U / c^2</math>, thus resulting in a very small effect. Light escaping from the surface of the Sun was predicted by Einstein in 1911 to be redshifted by roughly 2 [[Parts per million|ppm]] or 2 × 10<sup>−6</sup>.<ref name=":0">{{Cite news|last=Gräfe|first=Franziska|date=23 October 2020|title=New study verifies prediction from Einstein's General Theory of Relativity — English|language=en|work=Leibniz Institute for Astrophysics Potsdam|url=https://www.aip.de/en/news/science/new-study-verifies-prediction-from-einsteins-general-theory-of-relativity#:~:text=In%201911,%20Einstein%20predicted%20a,spectrum%20reflected%20by%20the%20moon|access-date=2021-01-14}}</ref> Navigational signals from [[Global Positioning System|GPS satellites]] orbiting at {{val|20,000|u=km}} altitude are perceived blueshifted by approximately 0.5 [[part per billion|ppb]] or 5 × 10<sup>−10</sup>,<ref>{{Cite web|last=Ashby|first=Neil|date=July 20–21, 2006|title=Relativity in the Global Positioning System|url=https://www.aapt.org/doorway/tgrutalks/ashby/AshbyTalk3of6.htm#gravitational%20redshift|access-date=2021-01-14|website=American Association of Physics Teachers}}</ref> corresponding to a (negligible) increase of less than 1&nbsp;Hz in the frequency of a 1.5&nbsp;GHz GPS radio signal (however, the accompanying [[gravitational time dilation]] affecting the atomic clock in the satellite ''is'' crucially important for accurate navigation<ref>{{Cite journal |last=Ashby |first=Neil |date=2003 |title=Relativity in the Global Positioning System |journal=Living Reviews in Relativity |volume=6 |issue=1 |pages=1 |doi=10.12942/lrr-2003-1 |doi-access=free |issn=1433-8351 |pmc=5253894 |pmid=28163638|bibcode=2003LRR.....6....1A }}</ref>). On the surface of the Earth the gravitational potential is proportional to height, <math>\Delta U = g \Delta h</math>, and the corresponding redshift is roughly 10<sup>−16</sup> (0.1 [[parts per quadrillion]]) per meter of change in [[elevation]] and/or [[altitude]].

In [[astronomy]], the magnitude of a gravitational redshift is often expressed as the velocity that would create an equivalent shift through the [[relativistic Doppler effect]]. In such units, the 2 ppm sunlight redshift corresponds to a 633&nbsp;m/s receding velocity, roughly of the same magnitude as convective motions in the Sun, thus complicating the measurement.<ref name=":0" /> The GPS satellite gravitational blueshift velocity equivalent is less than 0.2&nbsp;m/s, which is negligible compared to the actual Doppler shift resulting from its orbital velocity. In astronomical objects with strong gravitational fields the redshift can be much greater; for example, light from the surface of a [[white dwarf]] is gravitationally redshifted on average by around (50&nbsp;km/s)/''c'' (around 170&nbsp;ppm).<ref>{{cite web|last1=Trimble|first1=Virginia|last2=Barstow|first2=Martin|date=November 2020|title=Gravitational redshift and White Dwarf stars|url=https://www.einstein-online.info/en/spotlight/redshift_white_dwarfs/|access-date=2021-01-16|website=[[Einstein Online]]|publisher=[[Max Planck Institute for Gravitational Physics]]|language=en}}</ref>

Observing the gravitational redshift in the [[Solar System]] is one of the [[classical tests of general relativity]].<ref>{{Cite web|last=Alley|first=Carrol Overton|date=|title=GPS Setup Showed General Relativistic Effects on Light Operate at Emission and Reception, Not In-Flight as Required by Big Bang's Friedman-Lemaitre Spacetime Expansion Paradigm|url=http://www.theorionfoundation.com/Proof/gps.pdf|access-date=|website=The Orion Foundation}}</ref> Measuring the gravitational redshift to high precision with [[atomic clock]]s can serve as a test of [[Lorentz covariance|Lorentz symmetry]] and guide searches for [[dark matter]].

== Prediction by the equivalence principle and general relativity ==
=== Uniform gravitational field or acceleration ===
Einstein's theory of general relativity incorporates the [[equivalence principle]], which can be stated in various different ways. One such statement is that gravitational effects are locally undetectable for a free-falling observer. Therefore, in a laboratory experiment at the surface of the Earth, all gravitational effects should be equivalent to the effects that would have been observed if the laboratory had been accelerating through outer space at ''g''. One consequence is a gravitational [[Doppler effect]]. If a light pulse is emitted at the floor of the laboratory, then a free-falling observer says that by the time it reaches the ceiling, the ceiling has accelerated away from it, and therefore when observed by a detector fixed to the ceiling, it will be observed to have been Doppler shifted toward the red end of the spectrum. This shift, which the free-falling observer considers to be a kinematical Doppler shift, is thought of by the laboratory observer as a gravitational redshift. Such an effect was verified in the 1959 [[Pound–Rebka experiment]]. In a case such as this, where the gravitational field is uniform, the change in wavelength is given by
: <math>z = \frac{\Delta\lambda}{\lambda}\approx \frac{g\Delta y}{c^2},</math>
where <math>\Delta y</math> is the change in height. Since this prediction arises directly from the equivalence principle, it does not require any of the mathematical apparatus of general relativity, and its verification does not specifically support general relativity over any other theory that incorporates the equivalence principle.

On Earth's surface (or in a spaceship accelerating at 1&nbsp;''g''), the gravitational redshift is approximately {{val|1.1|e=−16}}, the equivalent of a {{val|3.3|e=−8|u=m/s}} Doppler shift for every 1&nbsp;m of altitude.

=== Spherically symmetric gravitational field ===
When the field is not uniform, the simplest and most useful case to consider is that of a spherically symmetric field. By [[Birkhoff's theorem (relativity)|Birkhoff's theorem]], such a field is described in general relativity by the [[Schwarzschild metric]], <math>d\tau^2 = \left(1 - r_\text{S}/R\right)dt^2 + \ldots</math>, where <math>d\tau</math> is the clock time of an observer at distance ''R'' from the center, <math>dt</math> is the time measured by an observer at infinity, <math>r_\text{S}</math> is the Schwarzschild radius <math>2GM/c^2</math>, "..." represents terms that vanish if the observer is at rest, <math>G</math> is the [[Newtonian constant of gravitation]], <math>M</math> the [[mass]] of the gravitating body, and <math>c</math> the [[speed of light]]. The result is that frequencies and wavelengths are shifted according to the ratio
: <math>1 + z = \frac{\lambda_\infty}{\lambda_\text{e}} = \left(1 - \frac{r_\text{S}}{R_\text{e}}\right)^{-\frac{1}{2}}</math>
where
* <math>\lambda_\infty\,</math>is the wavelength of the light as measured by the observer at infinity,
* <math>\lambda_\text{e}\,</math> is the wavelength measured at the source of emission, and
* <math>R_\text{e}</math> is the radius at which the photon is emitted.

This can be related to the [[Redshift|redshift parameter]] conventionally defined as <math>z = \lambda_\infty/\lambda_\text{e} - 1</math>.

In the case where neither the emitter nor the observer is at infinity, the [[transitive relation|transitivity]] of Doppler shifts allows us to generalize the result to <math>\lambda_1/\lambda_2 = \left[\left(1 - r_\text{S}/R_1\right)/\left(1 - r_\text{S}/R_2\right)\right]^{1/2}</math>. The redshift formula for the frequency <math>\nu = c/\lambda</math> is <math>\nu_o/\nu_\text{e} = \lambda_\text{e}/\lambda_o</math>. When <math>R_1 - R_2</math> is small, these results are consistent with the equation given above based on the equivalence principle.

The redshift ratio may also be expressed in terms of a (Newtonian) escape velocity <math>v_\text{e}</math> at <math>R_\text{e} = 2GM/v_\text{e}^2</math>, resulting in the corresponding [[Lorentz factor]]:
: <math>1 + z = \gamma_\text{e} = \frac{1}{\sqrt{1 - (v_\text{e}/c)^2}}</math>.

For an object compact enough to have an [[event horizon]], the redshift is not defined for photons emitted inside the Schwarzschild radius, both because signals cannot escape from inside the horizon and because an object such as the emitter cannot be stationary inside the horizon, as was assumed above. Therefore, this formula only applies when <math>R_\text{e}</math> is larger than <math>r_\text{S}</math>. When the photon is emitted at a distance equal to the Schwarzschild radius, the redshift will be ''infinitely'' large, and it will not escape to ''any'' finite distance from the Schwarzschild sphere. When the photon is emitted at an infinitely large distance, there is no redshift.

==== Newtonian limit ====
In the Newtonian limit, i.e. when <math>R_\text{e}</math> is sufficiently large compared to the Schwarzschild radius <math>r_\text{S}</math>, the redshift can be approximated as
: <math>z = \frac{\Delta\lambda}{\lambda} \approx \frac{1}{2}\frac{r_\text{S}}{R_\text{e}} = \frac{GM}{R_\text{e} c^2} = \frac{g R_\text{e}}{c^2}</math>
where <math>g</math> is the [[gravitational acceleration]] at <math>R_\text{e}</math>. For Earth's surface with respect to infinity, ''z'' is approximately {{val|7|e=−10}} (the equivalent of a 0.2&nbsp;m/s radial Doppler shift); for the Moon it is approximately {{val|3|e=−11}} (about 1&nbsp;cm/s). The value for the surface of the Sun is about {{val|2|e=−6}}, corresponding to 0.64&nbsp;km/s. (For non-relativistic velocities, the radial [[Relativistic Doppler effect|Doppler equivalent velocity]] can be approximated by multiplying ''z'' with the speed of light.)

The z-value can be expressed succinctly in terms of the [[escape velocity]] at <math>R_\text{e}</math>, since the [[gravitational potential]] is equal to half the square of the [[escape velocity]], thus:
: <math>z \approx \frac{1}{2}\left( \frac{v_\text{e}}{c} \right)^2</math>
where <math>v_\text{e}</math> is the escape velocity at <math>R_\text{e}</math>.

It can also be related to the circular orbit velocity <math>v_\text{o}</math> at <math>R_\text{e}</math>, which equals <math>v_\text{e}/\sqrt{2}</math>, thus
: <math>z \approx \left( \frac{v_\text{o}}{c} \right)^2</math>.

For example, the gravitational blueshift of distant starlight due to the Sun's gravity, which the Earth is orbiting at about 30&nbsp;km/s, would be approximately 1 × 10<sup>−8</sup> or the equivalent of a 3&nbsp;m/s radial Doppler shift.

For an object in a (circular) orbit, the gravitational redshift is of comparable magnitude as the [[transverse Doppler effect]], <math>z \approx \tfrac{1}{2} \beta^2</math> where {{nowrap|1=''β'' = ''v''/''c''}}, while both are much smaller than the [[Relativistic Doppler effect|radial Doppler effect]], for which <math>z \approx \beta</math>.

==== Prediction of the Newtonian limit using the properties of photons ====
The formula for the gravitational red shift in the Newtonian limit can also be derived using the properties of a photon:<ref>A. Malcherek: ''Elektromagnetismus und Gravitation'', Vereinheitlichung und Erweiterung der klassischen Physik. 2. Edition, Springer-Vieweg, Wiesbaden, 2023, ISBN 978-3-658-42701-6. [[doi:10.1007/978-3-658-42702-3]]</ref>

In a gravitational field <math>\vec{g}</math> a particle of mass <math>m</math> and velocity  <math>\vec{v}</math> changes it's  energy <math>E</math> according to:
: <math>\frac{\mathrm dE}{\mathrm dt} = m \vec{g}\cdot \vec{v} = \vec{g}\cdot\vec{p}</math>.

For a massless photon described by its energy <math>E = h \nu = \hbar \omega</math> and momentum   <math>\vec{p} = \hbar\vec{k}</math> this equation becomes after dividing by the Planck constant <math>\hbar</math>:
: <math>\frac{\mathrm d \omega}{\mathrm dt} = \vec{g}\cdot \vec{k}</math>

Inserting the gravitational field of a spherical body of mass <math>M</math> within the distance  <math>\vec{r}</math>
: <math>\vec{g} = -G M \frac{\vec{r}}{r^3}</math>
and the wave vector of a photon leaving the gravitational field in radial direction
: <math>\vec{k} = \frac{\omega}{c} \frac{\vec{r}}{r}</math>
the energy equation becomes
: <math>\frac{\mathrm d \omega}{\mathrm dt} = -\frac{G M}{c} \frac{\omega}{r^2}.</math>
Using <math>\mathrm dr = c \,\mathrm dt</math> an ordinary differential equation which is only dependent on the radial distance <math>r</math> is obtained:
: <math>\frac{\mathrm d \omega}{\mathrm dr} = -\frac{G M}{c^2} \frac{\omega}{r^2} </math>

For a photon starting at the surface of a spherical body with a Radius <math>R_e</math> with a frequency <math>\omega_0 = 2 \pi \nu_0</math> the analytical solution is:
: <math>\frac{\mathrm d \omega}{\mathrm dr} = -\frac{G M}{c^2} \frac{\omega}{r^2} \quad \Rightarrow \quad \omega(r) = \omega_0 \exp \left ( -\frac{G M}{c^2} \left( \frac{1}{R_e} - \frac{1}{r} \right) \right) </math>

In a large distance from the body <math>r \rightarrow \infty</math> an observer  measures the frequency :
: <math>\omega_\text{obs} = \omega_0 \exp \left ( -\frac{G M}{c^2} \left( \frac{1}{R_e} \right) \right) \simeq \omega_0 \left( 1 - \frac{G M}{R_e c^2} + \frac{1}{2} \frac{G^2 M^2}{R_e^2 c^4} - \ldots \right). </math>

Therefore, the red shift is:
: <math> z = \frac{\omega_0 - \omega_\text{obs}}{\omega_\text{obs}}
= \frac{1 - \exp \left( -\frac{G M}{R_e c^2} \right)}{\exp \left( -\frac{G M}{R_e c^2} \right)} = \frac{1 - \exp \left( -\frac{r_S}{2 R_e} \right)}{\exp \left( -\frac{r_S}{2 R_e} \right)} </math>

In the linear approximation 
: <math>z = \frac{ \frac{G M}{R_e c^2} - \frac{1}{2} \frac{G^2 M^2}{R_e^2 c^4} + \dots}{ 1 - \frac{G M}{R_e c^2} + \frac{1}{2} \frac{G^2 M^2}{R_e^2 c^4} - \ldots } \simeq \frac{ \frac{G M}{R_e c^2} }{ 1 - \frac{G M}{R_e c^2} + \frac{1}{2} \frac{G^2 M^2}{R_e^2 c^4} - \dots} \simeq \frac{G M}{c^2 R_e} </math>

the Newtonian limit for the gravitational red shift of General Relativity is obtained.

== Experimental verification ==
{{see also|Tests of general relativity}}

=== Astronomical observations ===
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}}
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>

== Early historical development of the theory ==
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.

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>.

Since the light would be slowed down by gravitational time dilation (as seen by outside observer), the regions with lower gravitational potential would act like a medium with higher refractive index causing light to [[refraction|deflect]]. This reasoning allowed Einstein in 1911 to reproduce the incorrect Newtonian value for the deflection of light.<ref name=E1911/> At the time he only considered the time-dilating manifestation of gravity, which is the dominating contribution at non-relativistic speeds; however relativistic objects travel through space a comparable amount as they do though time, so purely spatial curvature becomes just as important. After constructing the full theory of general relativity, Einstein solved in 1915<ref>{{cite web|url=https://einsteinpapers.press.princeton.edu/vol6-trans/124|title=Explanation of the Perihelion Motion of Mercury from the General Theory of Relativity}}</ref> the full [[post-Newtonian approximation]] for the Sun's gravity and calculated the correct amount of light deflection – double the Newtonian value. Einstein's prediction was confirmed by many experiments, starting with [[Arthur Eddington]]'s 1919 solar eclipse expedition.

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]].

== See also ==
{{Portal|Astronomy|Physics}}
* [[Tests of general relativity]]
* [[Equivalence principle]]
* [[Gravitational time dilation]]
* [[Redshift]]
* {{slink|Gravitational wave#Redshifting}} (redshifting of gravitational waves due to speed or cosmic expansion)

== Citations ==
{{reflist}}

== References ==
=== Primary sources ===
* {{cite journal
 |last=Michell, John
 |date=1784
 |title=On the means of discovering the distance, magnitude etc. of the fixed stars
 |journal=Philosophical Transactions of the Royal Society
 |pages = 35–57
 |doi=10.1098/rstl.1784.0008
 |volume=74
 |doi-access=free
 |bibcode=1784RSPT...74...35M
}}
* {{cite book
 |last=Laplace, Pierre-Simon
 |date=1796
 |title=The system of the world
 |volume=2
 |edition=1809 English translation
 |pages = 366–368
 |location=London
 |publisher=Richard Phillips
 |url=https://books.google.com/books?id=f7Kv2iFUNJoC
}}
* {{cite journal
 |last=von Soldner, Johann Georg
 |date=1804
 |title=On the deflection of a light ray from its rectilinear motion, by the attraction of a celestial body at which it nearly passes by
 |journal=Berliner Astronomisches Jahrbuch
 |pages =161–172|title-link=s:On the Deflection of a Light Ray from its Rectilinear Motion
}}
* Albert Einstein, "Relativity: the Special and General Theory". [[gutenberg:5001|(@Project Gutenberg)]].
* {{cite journal | last1 = Pound | first1 = R.V. | last2 = Rebka | first2 = G.A. Jr. | year = 1959 | title = Gravitational Red-Shift in Nuclear Resonance | journal = Phys. Rev. Lett. | volume = 3 | issue = 9| pages = 439–441 | doi=10.1103/physrevlett.3.439|bibcode = 1959PhRvL...3..439P | doi-access = free }}
* {{cite journal | last1 = Pound | first1 = R.V. | last2 = Snider | first2 = J.L. | year = 1965 | title = Effect of gravity on gamma radiation | doi = 10.1103/physrev.140.b788 | journal = Phys. Rev. B | volume = 140 | issue = 3B| pages = 788–803 |bibcode = 1965PhRv..140..788P }}
* {{cite journal | last1 = Pound | first1 = R.V. | year = 2000| title = Weighing Photons" (2000) | journal = Classical and Quantum Gravity | volume = 17 | issue = 12| pages = 2303–2311 | doi=10.1088/0264-9381/17/12/301|bibcode = 2000CQGra..17.2303P | s2cid = 250886562 }}

=== Other sources ===
* {{cite book | last1 = Misner | first1 = Charles W. | last2 = Thorne | first2 = Kip S. | last3 = Wheeler | first3 = John Archibald | title = Gravitation | publisher = [[W. H. Freeman]] | location = San Francisco |date=1973-09-15 | isbn = 978-0-7167-0344-0 }}

{{Authority control}}

[[Category:Albert Einstein]]
[[Category:Effects of gravity]]