Gravitational redshift

In Einstein's general theory of relativity, the gravitational redshiftis the phenomenon that clocks in a gravitational field tick slower when observed by a distant observer. More specifically the term refers to the shift of wavelength of a photon to longer wavelength (the red side in an optical spectrum) when observed from a point in a lower gravitational field. In the latter case the 'clock' is the frequency of the photon and a lower frequency is the same as a longer ('redder') wavelength. In Einstein's general theory of relativity, the gravitational redshiftis the phenomenon that clocks in a gravitational field tick slower when observed by a distant observer. More specifically the term refers to the shift of wavelength of a photon to longer wavelength (the red side in an optical spectrum) when observed from a point in a lower gravitational field. In the latter case the 'clock' is the frequency of the photon and a lower frequency is the same as a longer ('redder') wavelength. The gravitational redshift is a simple consequence ofEinstein's equivalence principle ('all bodies fall with the same acceleration, independent of their composition') and was found by Einstein eight years beforethe 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. 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 Δ λ λ ≈ g Δ y c 2 , {displaystyle {frac {Delta lambda }{lambda }}approx {frac {gDelta y}{c^{2}}},} where Δ y {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. 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 τ 2 = ( 1 − r s / R ) d t 2 + … {displaystyle d au ^{2}=(1-r_{s}/R)dt^{2}+ldots } , where d τ {displaystyle d au } is the clock time of an observer at distance R from the center, d t {displaystyle dt} is the time measured by an observer at infinity, r s {displaystyle r_{s}} is the Schwarzschild radius 2 G M / c 2 {displaystyle 2GM/c^{2}} , '...' represents terms that vanish if the observer is at rest, G {displaystyle G} is Newton's gravitational constant, M {displaystyle M} the mass of the gravitating body, and c {displaystyle c} the speed of light. The result is that frequencies and wavelengths are shifted according to the ratio λ ∞ λ e = ( 1 − r s / R e ) − 1 / 2 , {displaystyle {frac {lambda _{infty }}{lambda _{e}}}=(1-r_{s}/R_{e})^{-1/2},} where λ ∞ {displaystyle lambda _{infty },} is the wavelength of the light as measured by the observer at infinity, λ e {displaystyle lambda _{e},} is the wavelength measured at the source of emission, and R e {displaystyle R_{e}} radius at which the photon is emitted. This can be related to the redshift parameter conventionally defined as z = λ ∞ / λ e − 1 {displaystyle z=lambda _{infty }/lambda _{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 λ 1 / λ 2 = [ ( 1 − r s / R 1 ) / ( 1 − r s / R 2 ) ] 1 / 2 {displaystyle lambda _{1}/lambda _{2}=^{1/2}} . The redshift formula for the frequency ν = c / λ {displaystyle u =c/lambda } is ν o / ν e = λ e / λ o {displaystyle u _{o}/ u _{e}=lambda _{e}/lambda _{o}} . When R 1 − R 2 {displaystyle R_{1}-R_{2}} is small, these results are consistent with the equation given above based on the equivalence principle.