# Gravitational redshift

In the general theory of relativity by Albert Einstein, the gravitational redshift or Einstein shift is the effect 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 of the Einstein equivalence principle ("all bodies fall with the same acceleration, independent of their composition") 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.

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## First experimental verification

Experimental verification of the gravitational redshift requires good clocks since at Earth the effect is small. The first experimental confirmation came as late as in 1960, in the Pound-Rebka experiment (R.V. Pound, G.A. Rebka, Phys. Rev. Lett. 4, p.337) later improved by Pound and Snider. The famous experiment is generally called the Pound-Rebka-Snider experiment. They used a very well-defined "clock" in the form of an atomic transition which results in a very narrow line of electromagnetic radiation (a photon of well-defined energy). A narrow line implies a very well defined frequency. The line is in gamma ray range and emitted from the isotope Fe57 at 14.4 keV. The narrowness of the line is caused by the so called Mossbauer effect.

The emitter and absorber were placed in a tower of only 22 meter height at the bottom and top respectively. The observed gravitational redshift z, defined as the relative change in wavelength, the ratio

[itex] z = \frac{\Delta\lambda}{\lambda_e}[itex]

with [itex] \Delta\lambda = \lambda_{\rm o} - \lambda_{\rm e}[itex] the difference between the observed [itex] \lambda_{\rm o}[itex] and emitted [itex] \lambda_{\rm e} [itex] wavelength. z is proportional to the difference in gravitational potential. With the gravitational acceleration g of the Earth, c the velocity of light and with a height h=22 m, the prediction

[itex]
\Delta\lambda/\lambda = \frac{gh}{c^2} = 2.5\times 10^{-15}

[itex] was obtained with a 1% accuracy. Nowadays the accuracy is measured up to 0.02%

Note from the formula above that the loss of energy of the photon is just equal to the difference in potential energy [itex]gh[itex]). You can't make a perpetuum mobile by having photons going up and down in a gravitational field, something that was, strictly speaking, possible within Newton's theory of gravity.

## Gravitational redshift in stars

Photons emitted from a stellar surface on a star of mass M and radius R are expected to have a redshift equal to the difference in gravitational potential. With G the gravitational constant, this potential at the stellar surface is [itex]-GM/R[itex] and zero at infinity, so

[itex]\frac{\Delta\lambda}{\lambda} = \frac{G}{c^2} \cdot \frac{M}{R}[itex]

where [itex]c[itex] is the speed of light. The coefficient G/c2 = 7.414×10-29cm/g. For the Sun, M = 2.0×1033g and R = 6.955×1011cm, so Δλ/λ = 2.12×10-6. In other words, each spectral line should be shifted towards the red end of the spectrum by a little over one millionth of its original wavelength. This effect was measured for the first time on the Sun in 1962.

In addition, observation of much more massive and compact stars such as white dwarfs have shown that Einstein shift does occur and is within the correct order of magnitude. Recently also the gravitational redshift of a neutron star has been measured from spectral lines in the x-ray range. The result gives the quantity M/R, the mass M and radius R of the neutron star. If the mass is obtained by other means (for example from the motion of the neutron star around a campanion star), one can measure the radius of a neutron star in this way.

## Black holes have infinite gravitational redshift

The gravitational redshift increases to infinity around a black hole when an object approaches the event horizon of the black hole which is situated at the so-called Schwarzschild radius. In fact a black hole can best be defined as a massive compact object surrounded by an area at which the redshift (as observed from a large distance) is infinitely large.

When a star is imploding to form a black hole, one never observes the star to pass the Schwarzschild radius. As the star approaches this radius it will appear increasingly redder and dimmer in a very short time. In the past such a star was called a frozen star instead of a black hole. However, in a very short time the collapsing star emits its "last photon" and the object thereafter is black indeed. The terminology black hole is preferred above frozen star.

In general the gravitational redshift z for a spherical mass M with radius R is given

[itex]

1+z = \frac{1}{\sqrt{1-\frac{2GM}{c^2 R} }} [itex] (where G is the gravitational constant and c the velocity of light). This formula reduces to the one used above for the Sun for large R. Note also that this formula reduces to the one used at Earth for a gravitational acceleration [itex]g = GM/R^2[itex] and a difference in gravitational potential between [itex]R[itex] and [itex]R+h[itex] for small [itex]h[itex].

For [itex]r[itex] approaching [itex]2GM/c^2[itex] the redshift [itex] z\rightarrow\infty[itex]. The quantity [itex]2GM/c^2[itex] is called the Schwarzschild radius.

## Gravitational redshift, the "applied side of general relativity"

Corrections for gravitational redshift are nowadays common practice in many situations. We could almost call it "the applied side of General Relativity". With present-day accuracies, clocks in orbit around the Earth must be corrected for this effect. This is in particular the case with satellite-based navigational systems such as the Global Positioning System (GPS). To get accuracies of order 10 m, light travel times with an accuracy of order 30 ns (nanoseconds) have to be measured. Special-relativistic time dilatation (caused by the velocity) and gravitational redshift corrections in these satellites are of order 30000 ns per day.

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