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Gravitational redshift

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Physical effect in general relativity

This article is about redshift caused by gravitation and is not to be confused withRedshifting of gravitational waves.

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The gravitationalredshift 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.

Inphysics andgeneral relativity,gravitational redshift (known asEinstein shift in older literature)^[1]^[2] is the phenomenon thatelectromagnetic waves orphotons travelling out of agravitational well loseenergy. This loss of energy corresponds to a decrease in the wavefrequency and increase in thewavelength, known more generally as aredshift. The opposite effect, in which photons gain energy when travelling into a gravitational well, is known as agravitational blueshift (a type ofblueshift). The effect was first described byEinstein in 1907,^[3]^[4] eight years before his publication ofthe full theory of relativity. Observing the gravitational redshift in theSolar System is one of theclassical tests of general relativity.^[5]

Gravitational redshift can be interpreted as a consequence of theequivalence principle (that gravitational effects are locally equivalent to inertial effects and the redshift is caused by theDoppler effect)^[6] or as a consequence of themass–energy equivalence and conservation of energy ('falling' photons gain energy),^[7]^[8] though there are numerous subtleties that complicate a rigorous derivation.^[6]^[9] A gravitational redshift can also equivalently be interpreted asgravitational time dilation at the source of the radiation:^[9]^[2] if twooscillators (attached totransmitters producing electromagnetic radiation) are operating at differentgravitational potentials, 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).

Magnitudes

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To first approximation, gravitational redshift is proportional to the difference ingravitational potential divided by thespeed of light squared, $z=\Delta U/c^{2}$ , 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 2ppm or 2 × 10⁻⁶.^[10] Navigational signals fromGPS satellites orbiting at20000 km altitude are perceived blueshifted by approximately 0.5ppb or 5 × 10⁻¹⁰,^[11] corresponding to a (negligible) increase of less than 1 Hz in the frequency of a 1.5 GHz GPS radio signal (however, the accompanyinggravitational time dilation affecting the atomic clock in the satelliteis crucially important for accurate navigation^[12]). On the surface of the Earth the gravitational potential is proportional to height, $\Delta U=g\Delta h$ , and the corresponding redshift is roughly 10⁻¹⁶ (0.1parts per quadrillion) per meter of change inelevation and/oraltitude.

Inastronomy, the magnitude of a gravitational redshift is often expressed as the velocity that would create an equivalent shift through therelativistic Doppler effect. In such units, the 2 ppm sunlight redshift corresponds to a 633 m/s receding velocity, roughly of the same magnitude as convective motions in the Sun, thus complicating the measurement.^[10] The GPS satellite gravitational blueshift velocity equivalent is less than 0.2 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 awhite dwarf is gravitationally redshifted on average by around (50 km/s)/c (around 170 ppm).^[13]

Prediction by the equivalence principle and general relativity

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Uniform gravitational field or acceleration

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Einstein's theory of general relativity incorporates theequivalence 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 atg. One consequence is a gravitationalDoppler 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 1959Pound–Rebka experiment. In a case such as this, where the gravitational field is uniform, the change in wavelength is given by

z={\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.

On Earth's surface (or in a spaceship accelerating at 1 g), the gravitational redshift is approximately1.1×10⁻¹⁶, the equivalent of a3.3×10⁻⁸ m/s Doppler shift for every 1 m of altitude.

Spherically symmetric gravitational field

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When the field is not uniform, the simplest and most useful case to consider is that of a spherically symmetric field. ByBirkhoff's theorem, such a field is described in general relativity by theSchwarzschild 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 distanceR from the center, $d t {\displaystyle dt}$ 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, $G {\displaystyle G}$ is theNewtonian constant of gravitation, $M {\displaystyle M}$ themass of the gravitating body, and $c {\displaystyle c}$ thespeed of light. The result is that frequencies and wavelengths are shifted according to the ratio

1+z={\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
$R_{\text{e}}$ is the radius at which the photon is emitted.

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

In the case where neither the emitter nor the observer is at infinity, thetransitivity 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.

The redshift ratio may also be expressed in terms of a (Newtonian) escape velocity $v_{\text{e}}$ at $R_{\text{e}}=2GM/v_{\text{e}}^{2}$ , resulting in the correspondingLorentz factor:

1+z=\gamma _{\text{e}}={\frac {1}{\sqrt {1-(v_{\text{e}}/c)^{2}}}}

For an object compact enough to have anevent 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 beinfinitely large, and it will not escape toany finite distance from the Schwarzschild sphere. When the photon is emitted at an infinitely large distance, there is no redshift.

Newtonian limit

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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={\frac {\Delta \lambda }{\lambda }}\approx {\frac {1}{2}}{\frac {r_{\text{S}}}{R_{\text{e}}}}={\frac {GM}{R_{\text{e}}c^{2}}}={\frac {gR_{\text{e}}}{c^{2}}}

where $g {\displaystyle g}$ is thegravitational acceleration at $R_{\text{e}}$ . For Earth's surface with respect to infinity,z is approximately7×10⁻¹⁰ (the equivalent of a 0.2 m/s radial Doppler shift); for the Moon it is approximately3×10⁻¹¹ (about 1 cm/s). The value for the surface of the Sun is about2×10⁻⁶, corresponding to 0.64 km/s. (For non-relativistic velocities, the radialDoppler equivalent velocity can be approximated by multiplyingz with the speed of light.)

The z-value can be expressed succinctly in terms of theescape velocity at $R_{\text{e}}$ , since thegravitational potential is equal to half the square of theescape velocity, thus:

z\approx {\frac {1}{2}}\left({\frac {v_{\text{e}}}{c}}\right)^{2}

where $v_{\text{e}}$ is the escape velocity at $R_{\text{e}}$ .

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

z\approx \left({\frac {v_{\text{o}}}{c}}\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 thetransverse Doppler effect, $z\approx {\tfrac {1}{2}}\beta ^{2}$ whereβ =v/c, while both are much smaller than theradial Doppler effect, for which $z\approx \beta$ .

Prediction of the Newtonian limit using the properties of photons

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The formula for the gravitational red shift in the Newtonian limit can also be derived using the properties of a photon:^[14]

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

{\frac {\mathrm {d} E}{\mathrm {d} t}}=m{\vec {g}}\cdot {\vec {v}}={\vec {g}}\cdot {\vec {p}}

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

{\frac {\mathrm {d} \omega }{\mathrm {d} t}}={\vec {g}}\cdot {\vec {k}}

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

{\vec {g}}=-GM{\frac {\vec {r}}{r^{3}}}

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

{\vec {k}}={\frac {\omega }{c}}{\frac {\vec {r}}{r}}

the energy equation becomes

{\frac {\mathrm {d} \omega }{\mathrm {d} t}}=-{\frac {GM}{c}}{\frac {\omega }{r^{2}}}.

Using $\mathrm {d} r=c\,\mathrm {d} t$ an ordinary differential equation which is only dependent on the radial distance $r {\displaystyle r}$ is obtained:

{\frac {\mathrm {d} \omega }{\mathrm {d} r}}=-{\frac {GM}{c^{2}}}{\frac {\omega }{r^{2}}}

For a photon starting at the surface of a spherical body with a Radius $R_{e}$ with a frequency $\omega _{0}=2\pi \nu _{0}$ the analytical solution is:

{\frac {\mathrm {d} \omega }{\mathrm {d} r}}=-{\frac {GM}{c^{2}}}{\frac {\omega }{r^{2}}}\quad \Rightarrow \quad \omega (r)=\omega _{0}\exp \left(-{\frac {GM}{c^{2}}}\left({\frac {1}{R_{e}}}-{\frac {1}{r}}\right)\right)

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

\omega _{\text{obs}}=\omega _{0}\exp \left(-{\frac {GM}{c^{2}}}\left({\frac {1}{R_{e}}}\right)\right)\simeq \omega _{0}\left(1-{\frac {GM}{R_{e}c^{2}}}+{\frac {1}{2}}{\frac {G^{2}M^{2}}{R_{e}^{2}c^{4}}}-\ldots \right).

Therefore, the red shift is:

z={\frac {\omega _{0}-\omega _{\text{obs}}}{\omega _{\text{obs}}}}={\frac {1-\exp \left(-{\frac {GM}{R_{e}c^{2}}}\right)}{\exp \left(-{\frac {GM}{R_{e}c^{2}}}\right)}}={\frac {1-\exp \left(-{\frac {r_{S}}{2R_{e}}}\right)}{\exp \left(-{\frac {r_{S}}{2R_{e}}}\right)}}

In the linear approximation

z={\frac {{\frac {GM}{R_{e}c^{2}}}-{\frac {1}{2}}{\frac {G^{2}M^{2}}{R_{e}^{2}c^{4}}}+\dots }{1-{\frac {GM}{R_{e}c^{2}}}+{\frac {1}{2}}{\frac {G^{2}M^{2}}{R_{e}^{2}c^{4}}}-\ldots }}\simeq {\frac {\frac {GM}{R_{e}c^{2}}}{1-{\frac {GM}{R_{e}c^{2}}}+{\frac {1}{2}}{\frac {G^{2}M^{2}}{R_{e}^{2}c^{4}}}-\dots }}\simeq {\frac {GM}{c^{2}R_{e}}}

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

History

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The gravitational weakening of light from high-gravity stars was predicted byJohn Michell in 1783 andPierre-Simon Laplace in 1796, usingIsaac 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 byJohann 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 bygeneral 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.^[15] A confusing mix of complex and subtle issues plague even famous textbook descriptions of the phenomenon.^[16]

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.^[17] He considered an accelerating box, and noted that according to thespecial 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 $x {\displaystyle x}$ direction) with velocity $v {\displaystyle v}$ relative to the rest frame, the clocks at a nearby position $d x {\displaystyle dx}$ are ahead by $(dx/c)(v/c)$ (to the first order); so an acceleration $g {\displaystyle g}$ (that changes speed by $g/dt$ per time $d t {\displaystyle dt}$ ) makes clocks at the position $d x {\displaystyle dx}$ 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 $g {\displaystyle g}$ 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 $V {\displaystyle V}$ is $-dV/dx$ , we get

{dR \over dx}=g/c^{2}=-{dV/c^{2} \over dx}

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

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 themass–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 theirSchwarzschild radius.

Astronomical observations

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Terrestrial tests

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For experiments measuring the slowing of clocks, seeGravitational time dilation § Experimental confirmation.

Between 1925 and 1955, very few attempts were made to measure the gravitational redshift.^[33]The effect is now considered to have been definitively verified by the experiments ofPound, Rebka and Snider between 1959 and 1965. ThePound–Rebka experiment of 1959 measured the gravitational redshift in spectral lines using a terrestrial⁵⁷Fe gamma source over a vertical height of 22.5 metres.^[34] This paper was the first determination of the gravitational redshift which used measurements of the change in wavelength of gamma-ray photons generated with theMössbauer 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.^[35]

A very accurate gravitational redshift experiment was performed in 1976,^[36] where ahydrogen maser clock on a rocket was launched to a height of10000 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 theGlobal 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.^[37]

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.^[38]^[39]

In 2020, a group at theUniversity of Tokyo measured the gravitational redshift of two strontium-87optical lattice clocks.^[40] The measurement took place atTokyo Skytree where the clocks were separated by approximately 450 m and connected by telecom fibers. The gravitational redshift can be expressed as

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

where $\Delta \nu =\nu _{2}-\nu _{1}$ is the gravitational redshift, $\nu _{1}$ is the optical clock transition frequency, $\Delta U=U_{2}-U_{1}$ is the difference in gravitational potential, and $\alpha$ denotes the violation from general relativity. ByRamsey 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 az-value of approximately 5 × 10⁻¹⁴. 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.^[41]^[42]

In October 2021, a group atJILA led by physicistJun Ye reported a measurement of gravitational redshift in the submillimeter scale. The measurement is done on the⁸⁷Sr clock transition between the top and the bottom of a millimeter-tall ultracold cloud of 100,000strontium atoms in anoptical lattice.^[43]^[44]