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In physics and general relativity, gravitational redshift (known as Einstein shift in older literature)^[1][2] is the phenomenon that electromagnetic waves or photons travelling out of a gravitational well (seem to) 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 (seem to) gain energy when travelling into a gravitational well, is known as a gravitational blueshift (a type of blueshift). The effect was first described by Einstein in 1907,^[3][4] eight years before his publication of the full theory of relativity.

Gravitational redshift can be interpreted as a consequence of the equivalence principle (that gravity and acceleration are equivalent and the redshift is caused by the Doppler effect)^[5] or as a consequence of the mass–energy equivalence and conservation of energy ('falling' photons gain energy),^[6][7] though there are numerous subtleties that complicate a rigorous derivation.^[5][8] A gravitational redshift can also equivalently be interpreted as gravitational time dilation at the source of the radiation:^[8][2] if two oscillators (attached to transmitters producing electromagnetic radiation) are operating at different gravitational potentials, the oscillator at the higher gravitational potential (farther from the attracting body) will seem to ‘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, ${\displaystyle z=\Delta U/c^{2}}$, 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 2 ppm or 2 × 10⁻⁶.^[9] Navigational signals from GPS satellites orbiting at 20,000 km altitude are perceived blueshifted by approximately 0.5 ppb or 5 × 10⁻¹⁰,^[10] corresponding to a (negligible) increase of less than 1 Hz in the frequency of a 1.5 GHz GPS radio signal (however, the accompanying gravitational time dilation affecting the atomic clock in the satellite is crucially important for accurate navigation^[11]). On the surface of the Earth the gravitational potential is proportional to height, ${\displaystyle \Delta U=g\Delta h}$, and the corresponding redshift is roughly 10⁻¹⁶ (0.1 part 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 m/s receding velocity, roughly of the same magnitude as convective motions in the Sun, thus complicating the measurement.^[9] 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 a white dwarf is gravitationally redshifted on average by around 50 km/s/c (around 170 ppm).^[12]

Observing the gravitational redshift in the Solar System is one of the classical tests of general relativity.^[13] 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

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

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

where ${\displaystyle \Delta y}$ 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 1.1 × 10⁻¹⁶, the equivalent of a 3.3 × 10⁻⁸ m/s Doppler shift, for every meter of height differential.

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

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

where

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

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

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 ${\displaystyle {\mathrm{\lambda <!--\; \lambda \; -->}}_1/{\mathrm{\lambda <!--\; \lambda \; -->}}_2={}^{}}$Zdroj:https://en.wikipedia.org?pojem=Gravitational_redshift
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