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In Einstein's general theory of relativity, the gravitational redshift is the phenomenon that as a photon travels up through a gravitational well it will lose energy. This loss of energy causes a decrease in the photon’s frequency, which equivalently causes the wavelength of the photon to increase, or ‘redshift’ as it moves through a medium. As a result if two clocks are operating at different gravitational potentials, the clock at the higher gravitational potential will be expected to ‘tick’ faster, that is it will have a higher measured frequency than the clock at the lower gravitational potential.

The gravitational redshift is a simple consequence of Einstein's equivalence principle (that gravity and acceleration are equivalent) and was found by Einstein eight years before the full theory of relativity.

Observing the gravitational redshift in the solar system is one of the classical tests of general relativity. Gravitational redshifts are an important effect in satellite-based navigation systems such as GPS. If the effects of general relativity were not taken into account, such systems would not work at all. Measuring the gravitational redshift to high precision with atomic clocks can serve as a test of Lorentz symmetry and guide searches for dark matter.

Prediction by the equivalence principle and general relativity

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

$${\frac {\Delta \lambda }{\lambda }}\approx {\frac {g\Delta y}{c^{2}}},}$$

where $$\Delta y$$ 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.

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, $$d\tau ^{2}=\left(1-r_{\text{S}}/R\right)dt^{2}+\ldots }$$ , where $$d\tau$$ is the clock time of an observer at distance R from the center, $$dt$$ is the time measured by an observer at infinity, $$r_{\text{S}}}$$ is the Schwarzschild radius $$2GM/c^{2}}$$, "..." represents terms that vanish if the observer is at rest, G is Newton's gravitational constant, 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

$${\frac {\lambda _{\infty }}{\lambda _{\text{e}}}}=\left(1-{\frac {r_{\text{S}}}{R_{\text{e}}}}\right)^{-{\frac {1}{2}}},}$$

where

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

This can be related to the redshift parameter conventionally defined as $$z=\lambda _{\infty }/\lambda _{\text{e}}-1}$$. 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 $$\lambda _{1}/\lambda _{2}=\left[\left(1-r_{\text{S}}/R_{1}\right)/\left(1-r_{\text{S}}/R_{2}\right)\right]^{1/2}}$$. The redshift formula for the frequency $$\nu = c/\lambda$$ is $$\nu _{o}/\nu _{\text{e}}=\lambda _{\text{e}}/\lambda _{o}}$$. When $$R_{1}-R_{2}}$$ is small, these results are consistent with the equation given above based on the equivalence principle.

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 $$R_{\text{e}}}$$ is larger than $$r_{\text{S}}}$$. 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 $$R_{\text{e}}}$$ is sufficiently large compared to the Schwarzschild radius $$r_{\text{S}}}$$, the redshift can be approximated as

$$z\approx {\frac {1}{2}}{\frac {r_{\text{S}}}{R_{\text{e}}}}={\frac {GM}{c^{2}R_{\text{e}}}}}$$

Experimental verification
Initial observations of gravitational redshift of white dwarf stars

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. However, measurements by Adams have been criticized as being too low 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±19 km/s, with more accurate measurements by the Hubble Space Telescope, showing 80.4±4.8 km/s.
Terrestrial tests

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. 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.

A very accurate gravitational redshift experiment was performed in 1976, where a hydrogen maser clock on a rocket was launched to a height of 10,000 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.
Later astronomical measurements

James W. Brault, a graduate student of Robert Dicke at Princeton University, measured the gravitational redshift of the sun using optical methods in 1962.

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.
Ground based optical clock measurement

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

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

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

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) 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—if clocks at different points had different rates.

This was precisely Einstein's conclusion in 1911. 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). Nowadays, this can be easily shown in accelerated coordinates. The metric tensor in units where the speed of light is one is:

$$ds^{2}=-r^{2}dt^{2}+dr^{2}\,$$

and for an observer at a constant value of r, the rate at which a clock ticks, R(r), is the square root of the time coefficient, R(r)=r. The acceleration at position r is equal to the curvature of the hyperbola at fixed r, and like the curvature of the nested circles in polar coordinates, it is equal to 1/r.

So at a fixed value of g, the fractional rate of change of the clock-rate, the percentage change in the ticking at the top of an accelerating box vs at the bottom, is:

$${R(r+dr)-R(r) \over R}={dr \over r}=gdr\,}$$

The rate is faster at larger values of R, away from the apparent direction of acceleration. The rate is zero at r=0, which is the location of the acceleration horizon.

Using the equivalence principle, Einstein concluded that the same thing holds in any gravitational field, that the rate of clocks R at different heights was altered according to the gravitational field g. When g is slowly varying, it gives the fractional rate of change of the ticking rate. If the ticking rate is everywhere almost this same, the fractional rate of change is the same as the absolute rate of change, so that:

$${dR \over dx}=g=-{dV \over dx}\,}$$

Since the rate of clocks and the gravitational potential have the same derivative, they are the same up to a constant. The constant is chosen to make the clock rate at infinity equal to 1. Since the gravitational potential is zero at infinity:

$$R(x)=1-{V(x) \over c^{2}}\,}$$

where the speed of light has been restored to make the gravitational potential dimensionless.

The coefficient of the $$dt^{2}$$ in the metric tensor is the square of the clock rate, which for small values of the potential is given by keeping only the linear term:

R 2 = 1 − 2 V {\displaystyle R^{2}=1-2V\,} R^{2}=1-2V\, \)

and the full metric tensor is:

$$ds^{2}=-\left(1-{2V(r) \over c^{2}}\right)c^{2}dt^{2}+dx^{2}+dy^{2}+dz^{2}}$$

where again the C's have been restored. This expression is correct in the full theory of general relativity, to lowest order in the gravitational field, and ignoring the variation of the space-space and space-time components of the metric tensor, which only affect fast moving objects.

Using this approximation, Einstein reproduced the incorrect Newtonian value for the deflection of light in 1909. But since a light beam is a fast moving object, the space-space components contribute too. After constructing the full theory of general relativity in 1916, Einstein solved for the space-space components in a post-Newtonian approximation 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 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.

Tests of general relativity
Equivalence principle
Gravitational time dilation
Redshift
Gravitational wave#Redshifting (Redshifting of Gravitational waves due to speed or cosmic expansion)

Notes

Hetherington, N. S., "Sirius B and the gravitational redshift - an historical review", Quarterly Journal Royal Astronomical Society, vol. 21, Sept. 1980, p. 246-252. Accessed 6 April 2017.
Holberg, J. B., "Sirius B and the Measurement of the Gravitational Redshift", Journal for the History of Astronomy, Vol. 41, 1, 2010, p. 41-64. Accessed 6 April 2017.
Pound, R.; Rebka, G. (1960). "Apparent Weight of Photons". Physical Review Letters. 4 (7): 337–341. Bibcode:1960PhRvL...4..337P. doi:10.1103/PhysRevLett.4.337.
Pound, R. V.; Snider J. L. (November 2, 1964). "Effect of Gravity on Nuclear Resonance". Physical Review Letters. 13 (18): 539–540. Bibcode:1964PhRvL..13..539P. doi:10.1103/PhysRevLett.13.539.
Vessot, R. F. C.; M. W. Levine; E. M. Mattison; E. L. Blomberg; T. E. Hoffman; G. U. Nystrom; B. F. Farrel; R. Decher; et al. (December 29, 1980). "Test of Relativistic Gravitation with a Space-Borne Hydrogen Maser". Physical Review Letters. 45 (26): 2081–2084. Bibcode:1980PhRvL..45.2081V. doi:10.1103/PhysRevLett.45.2081.
Ashby, Neil (2003). "Relativity in the Global Positioning System". Living Reviews in Relativity. 6 (1): 1. Bibcode:2003LRR.....6....1A. doi:10.12942/lrr-2003-1. PMC 5253894. PMID 28163638.
Bhattacharjee, Yudhijit (2011). "Galaxy Clusters Validate Einstein's Theory". News.sciencemag.org. Retrieved 2013-07-23.
"Test of general relativity by a pair of transportable optical lattice clocks".
"Gravitational Redshift Test Using Eccentric Galileo Satellites".

"Test of the Gravitational Redshift with Galileo Satellites in an Eccentric Orbit".

Primary sources

Michell, John (1784). "On the means of discovering the distance, magnitude etc. of the fixed stars". Philosophical Transactions of the Royal Society. 74: 35–57. Bibcode:1784RSPT...74...35M. doi:10.1098/rstl.1784.0008.

Laplace, Pierre-Simon (1796). The system of the world (English translation 1809). 2. London: Richard Phillips. pp. 366–368.

Soldner, Johann Georg von (1804). "On the deflection of a light ray from its rectilinear motion, by the attraction of a celestial body at which it nearly passes by" . Berliner Astronomisches Jahrbuch: 161–172.

Albert Einstein, "Relativity: the Special and General Theory." (@Project Gutenberg).
Pound, R.V.; Rebka, G.A.; Jr (1959). "Gravitational Red-Shift in Nuclear Resonance". Physical Review Letters 3 (9): 439–441. Bibcode:1959PhRvL...3..439P. doi:10.1103/physrevlett.3.439.
Pound, R.V.; Snider, J.L. (1965). "Effect of gravity on gamma radiation". Phys. Rev. B. 140 (3B): 788–803. Bibcode:1965PhRv..140..788P. doi:10.1103/physrev.140.b788.
Pound, R.V. (2000). "Weighing Photons" (2000)". Classical and Quantum Gravity. 17 (12): 2303–2311. Bibcode:2000CQGra..17.2303P. doi:10.1088/0264-9381/17/12/301.

References
Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973), Gravitation, San Francisco: W. H. Freeman, ISBN 978-0-7167-0344-0.

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