gravitational redshift

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

: $z\; =\; \backslash frac\{\backslash Delta\backslash lambda\}\{\backslash lambda\}\backslash approx\; \backslash frac\{g\backslash Delta\; y\}\{c^2\},$

where $\backslash Delta\; y$ 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 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 m of altitude.

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, such a field is described in general relativity by the Schwarzschild metric, $d\backslash tau^2\; =\; \backslash left(1\; -\; r\_\backslash text\{S\}/R\backslash right)dt^2\; +\; \backslash ldots$, where $d\backslash tau$ is the clock time of an observer at distance R from the center, $dt$ is the time measured by an observer at infinity, $r\_\backslash text\{S\}$ is the Schwarzschild radius $2GM/c^2$, "..." represents terms that vanish if the observer is at rest, $G$ is the Newtonian constant of gravitation, $M$ the mass of the gravitating body, and $c$ the speed of light. The result is that frequencies and wavelengths are shifted according to the ratio

: $1\; +\; z\; =\; \backslash frac\{\backslash lambda\_\backslash infty\}\{\backslash lambda\_\backslash text\{e\}\}\; =\; \backslash left(1\; -\; \backslash frac\{r\_\backslash text\{S\}\}\{R\_\backslash text\{e\}\}\backslash right)^\{-\backslash frac\{1\}\{2\}\}$

where

• $\backslash lambda\_\backslash infty\backslash ,$is the wavelength of the light as measured by the observer at infinity,
• $\backslash lambda\_\backslash text\{e\}\backslash ,$ is the wavelength measured at the source of emission, and
• $R\_\backslash text\{e\}$ is the radius at which the photon is emitted.

This can be related to the redshift parameter conventionally defined as $z\; =\; \backslash lambda\_\backslash infty/\backslash lambda\_\backslash text\{e\}\; -\; 1$.

In the case where neither the emitter nor the observer is at infinity, the transitivity of Doppler shifts allows us to generalize the result to $\backslash lambda\_1/\backslash lambda\_2\; =\; \backslash left[\backslash left(1\; -\; r\_\backslash text\{S\}/R\_1\backslash right)/\backslash left(1\; -\; r\_\backslash text\{S\}/R\_2\backslash right)\backslash right]^\{1/2\}$. The redshift formula for the frequency $\backslash nu\; =\; c/\backslash lambda$ is $\backslash nu\_o/\backslash nu\_\backslash text\{e\}\; =\; \backslash lambda\_\backslash text\{e\}/\backslash lambda\_o$. When $R\_1\; -\; R\_2$ 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 $v\_\backslash text\{e\}$ at $R\_\backslash text\{e\}\; =\; 2GM/v\_\backslash text\{e\}^2$, resulting in the corresponding Lorentz factor:

: $1\; +\; z\; =\; \backslash gamma\_\backslash text\{e\}\; =\; \backslash frac\{1\}\{\backslash sqrt\{1\; -\; (v\_\backslash text\{e\}/c)^2\}\}$.

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 $R\_\backslash text\{e\}$ is larger than $r\_\backslash text\{S\}$. 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.

In the Newtonian limit, i.e. when $R\_\backslash text\{e\}$ is sufficiently large compared to the Schwarzschild radius $r\_\backslash text\{S\}$, the redshift can be approximated as

: $z\; =\; \backslash frac\{\backslash Delta\backslash lambda\}\{\backslash lambda\}\; \backslash approx\; \backslash frac\{1\}\{2\}\backslash frac\{r\_\backslash text\{S\}\}\{R\_\backslash text\{e\}\}\; =\; \backslash frac\{GM\}\{R\_\backslash text\{e\}\; c^2\}\; =\; \backslash frac\{g\; R\_\backslash text\{e\}\}\{c^2\}$

where $g$ is the gravitational acceleration at $R\_\backslash text\{e\}$. For Earth's surface with respect to infinity, z is approximately {{val|7|e=−10}} (the equivalent of a 0.2 m/s radial Doppler shift); for the Moon it is approximately {{val|3|e=−11}} (about 1 cm/s). The value for the surface of the Sun is about {{val|2|e=−6}}, corresponding to 0.64 km/s. (For non-relativistic velocities, the radial 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 $R\_\backslash text\{e\}$, since the gravitational potential is equal to half the square of the escape velocity, thus:

: $z\; \backslash approx\; \backslash frac\{1\}\{2\}\backslash left(\; \backslash frac\{v\_\backslash text\{e\}\}\{c\}\; \backslash right)^2$

where $v\_\backslash text\{e\}$ is the escape velocity at $R\_\backslash text\{e\}$.

It can also be related to the circular orbit velocity $v\_\backslash text\{o\}$ at $R\_\backslash text\{e\}$, which equals $v\_\backslash text\{e\}/\backslash sqrt\{2\}$, thus

: $z\; \backslash approx\; \backslash left(\; \backslash frac\{v\_\backslash text\{o\}\}\{c\}\; \backslash right)^2$.

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

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

The formula for the gravitational red shift in the Newtonian limit can also be derived using the properties of a photon: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

In a gravitational field $\backslash vec\{g\}$ a particle of mass $m$ and velocity $\backslash vec\{v\}$ changes it's energy $E$ according to:

: $\backslash frac\{\backslash mathrm\; dE\}\{\backslash mathrm\; dt\}\; =\; m\; \backslash vec\{g\}\backslash cdot\; \backslash vec\{v\}\; =\; \backslash vec\{g\}\backslash cdot\backslash vec\{p\}$.

For a massless photon described by its energy $E\; =\; h\; \backslash nu\; =\; \backslash hbar\; \backslash omega$ and momentum $\backslash vec\{p\}\; =\; \backslash hbar\backslash vec\{k\}$ this equation becomes after dividing by the Planck constant $\backslash hbar$:

: $\backslash frac\{\backslash mathrm\; d\; \backslash omega\}\{\backslash mathrm\; dt\}\; =\; \backslash vec\{g\}\backslash cdot\; \backslash vec\{k\}$

Inserting the gravitational field of a spherical body of mass $M$ within the distance $\backslash vec\{r\}$

: $\backslash vec\{g\}\; =\; -G\; M\; \backslash frac\{\backslash vec\{r\}\}\{r^3\}$

and the wave vector of a photon leaving the gravitational field in radial direction

: $\backslash vec\{k\}\; =\; \backslash frac\{\backslash omega\}\{c\}\; \backslash frac\{\backslash vec\{r\}\}\{r\}$

the energy equation becomes

: $\backslash frac\{\backslash mathrm\; d\; \backslash omega\}\{\backslash mathrm\; dt\}\; =\; -\backslash frac\{G\; M\}\{c\}\; \backslash frac\{\backslash omega\}\{r^2\}.$

Using $\backslash mathrm\; dr\; =\; c\; \backslash ,\backslash mathrm\; dt$ an ordinary differential equation which is only dependent on the radial distance $r$ is obtained:

: $\backslash frac\{\backslash mathrm\; d\; \backslash omega\}\{\backslash mathrm\; dr\}\; =\; -\backslash frac\{G\; M\}\{c^2\}\; \backslash frac\{\backslash omega\}\{r^2\}$

For a photon starting at the surface of a spherical body with a Radius $R\_e$ with a frequency $\backslash omega\_0\; =\; 2\; \backslash pi\; \backslash nu\_0$ the analytical solution is:

: $\backslash frac\{\backslash mathrm\; d\; \backslash omega\}\{\backslash mathrm\; dr\}\; =\; -\backslash frac\{G\; M\}\{c^2\}\; \backslash frac\{\backslash omega\}\{r^2\}\; \backslash quad\; \backslash Rightarrow\; \backslash quad\; \backslash omega(r)\; =\; \backslash omega\_0\; \backslash exp\; \backslash left\; (\; -\backslash frac\{G\; M\}\{c^2\}\; \backslash left(\; \backslash frac\{1\}\{R\_e\}\; -\; \backslash frac\{1\}\{r\}\; \backslash right)\; \backslash right)$

In a large distance from the body $r\; \backslash rightarrow\; \backslash infty$ an observer measures the frequency :

: $\backslash omega\_\backslash text\{obs\}\; =\; \backslash omega\_0\; \backslash exp\; \backslash left\; (\; -\backslash frac\{G\; M\}\{c^2\}\; \backslash left(\; \backslash frac\{1\}\{R\_e\}\; \backslash right)\; \backslash right)\; \backslash simeq\; \backslash omega\_0\; \backslash left(\; 1\; -\; \backslash frac\{G\; M\}\{R\_e\; c^2\}\; +\; \backslash frac\{1\}\{2\}\; \backslash frac\{G^2\; M^2\}\{R\_e^2\; c^4\}\; -\; \backslash ldots\; \backslash right).$

Therefore, the red shift is:

: $z\; =\; \backslash frac\{\backslash omega\_0\; -\; \backslash omega\_\backslash text\{obs\}\}\{\backslash omega\_\backslash text\{obs\}\}$

= \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)}

In the linear approximation

: $z\; =\; \backslash frac\{\; \backslash frac\{G\; M\}\{R\_e\; c^2\}\; -\; \backslash frac\{1\}\{2\}\; \backslash frac\{G^2\; M^2\}\{R\_e^2\; c^4\}\; +\; \backslash dots\}\{\; 1\; -\; \backslash frac\{G\; M\}\{R\_e\; c^2\}\; +\; \backslash frac\{1\}\{2\}\; \backslash frac\{G^2\; M^2\}\{R\_e^2\; c^4\}\; -\; \backslash ldots\; \}\; \backslash simeq\; \backslash frac\{\; \backslash frac\{G\; M\}\{R\_e\; c^2\}\; \}\{\; 1\; -\; \backslash frac\{G\; M\}\{R\_e\; c^2\}\; +\; \backslash frac\{1\}\{2\}\; \backslash frac\{G^2\; M^2\}\{R\_e^2\; c^4\}\; -\; \backslash dots\}\; \backslash simeq\; \backslash frac\{G\; M\}\{c^2\; R\_e\}$

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

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) 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.{{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}} A confusing mix of complex and subtle issues plague even famous textbook descriptions of the phenomenon.{{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 }}

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.{{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 }} 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 $x$ direction) with velocity $v$ relative to the rest frame, the clocks at a nearby position $dx$ are ahead by $(dx/c)(v/c)$ (to the first order); so an acceleration $g$ (that changes speed by $g/dt$ per time $dt$) makes clocks at the position $dx$ to be ahead by $(dx/c)(g/c)dt$, that is, tick at a rate

: $R=1+(g/c^2)dx$

The equivalence principle implies that this change in clock rate is the same whether the acceleration $g$ 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 $V$ is $-dV/dx$, we get

: $\{dR\; \backslash over\; dx\}\; =\; g/c^2\; =\; -\; \{dV/c^2\; \backslash over\; dx\}$

so – in weak fields – the change $\backslash Delta\; R$ in the clock rate is equal to $-\backslash Delta\; V/c^2$.

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 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|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 W.S. Adams in 1925.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. However, measurements by Adams have been criticized as being too lowHolberg, 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. and these observations are now considered to be measurements of spectra that are unusable because of scattered light from the primary, Sirius A. The first accurate measurement of the gravitational redshift of a white dwarf was done by Popper in 1954, measuring a 21 km/s gravitational redshift of 40 Eridani B. The redshift of Sirius B was finally measured by Greenstein et al. in 1971, obtaining the value for the gravitational redshift of 89±16 km/s, with more accurate measurements by the Hubble Space Telescope, showing 80.4±4.8 km/s.[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. 563–566.{{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.{{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}} }} In 2020, a team of scientists published the most accurate measurement of the solar gravitational redshift so far, made by analyzing Fe spectral lines in sunlight reflected by the Moon; their measurement of a mean global 638 ± 6 m/s lineshift is in agreement with the theoretical value of 633.1 m/s.{{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}}{{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}} 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.

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.{{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}}

In 2018, the star S2 made its closest approach to Sgr A*, the 4-million solar mass supermassive black hole at the centre of the Milky Way, reaching 7650 km/s or about 2.5% of the speed of light while passing the black hole at a distance of just 120 AU, or 1400 Schwarzschild radii. Independent analyses by the GRAVITY collaboration{{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}}{{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}}{{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}}{{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}} (led by Reinhard Genzel) and the KECK/UCLA Galactic Center Group{{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}}{{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}} (led by Andrea Ghez) revealed a combined transverse Doppler and gravitational redshift up to 200 km/s/c, in agreement with general relativity predictions.

In 2021, Mediavilla (IAC, Spain) & Jiménez-Vicente (UGR, Spain) were able to use measurements of the gravitational redshift in quasars 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%.{{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 }}

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.

{{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

}}

{{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.{{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}}

The effect is now considered to have been definitively verified by the experiments of Pound, Rebka and Snider between 1959 and 1965. The Pound–Rebka experiment of 1959 measured the gravitational redshift in spectral lines using a terrestrial ⁵⁷Fe gamma source over a vertical height of 22.5 metres.{{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 }} 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.{{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}}

A very accurate gravitational redshift experiment was performed in 1976,{{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}} 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.{{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}}

In 2010, an experiment placed two aluminum-ion quantum clocks close to each other, but with the second elevated 33 cm compared to the first, making the gravitational red shift effect visible in everyday lab scales.{{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 }}

{{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

}}

In 2020, a group at the University of Tokyo measured the gravitational redshift of two strontium-87 optical lattice clocks.{{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 }} The measurement took place at Tokyo Skytree where the clocks were separated by approximately 450 m and connected by telecom fibers. The gravitational redshift can be expressed as

: $z\; =\; \backslash frac\{\backslash Delta\backslash nu\}\{\backslash nu\_\{1\}\}\; =\; (1+\backslash alpha)\backslash frac\{\backslash Delta\; U\}\{c^2\}$,

where $\backslash Delta\backslash nu=\backslash nu\_\{2\}-\backslash nu\_\{1\}$ is the gravitational redshift, $\backslash nu\_\{1\}$ is the optical clock transition frequency, $\backslash Delta\; U=\; U\_\{2\}-\; U\_\{1\}$ is the difference in gravitational potential, and $\backslash alpha$ denotes the violation from general relativity. By Ramsey spectroscopy of the strontium-87 optical clock transition (429 THz, 698 nm) the group determined the gravitational redshift between the two optical clocks to be 21.18 Hz, corresponding to a z-value of approximately 5 × 10⁻¹⁴. Their measured value of $\backslash alpha$, $(1.4\; \backslash pm\; 9.1)\backslash times\; 10^\{-5\}$, is an agreement with recent measurements made with hydrogen masers in elliptical orbits.{{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 }}{{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 }}

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 ⁸⁷Sr clock transition between the top and the bottom of a millimeter-tall ultracold cloud of 100,000 strontium atoms in an optical lattice.{{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}}{{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}}

{{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)

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Category:Albert Einstein

Category:Effects of gravity