Gravitational Red Shift

According to the principle of equivalence from general relativity, any frequency shift which can be shown to arise from acceleration of a radiating source could also be produced by the appropriate gravitational field. Thus the expected shift in radiation frequency in a gravitational field can be related to the relativistic doppler shift experienced from an accelerating light source.


Gravity and the photon

Applications:
Harvard tower experiment
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Harvard Tower Experiment

In just 22.6 meters, the fractional gravitational red shift given by
is just 4.92 x 10-15 , but the Mossbauer effect with the 14.4 keV gamma ray from iron-57 has a high enough resolution to detect that difference. In the early 60's physicists Pound, Rebka,and Snyder at the Jefferson Physical Laboratory at Harvard measured the shift to within 1% of the predicted shift.

By just using the expression for gravitational potential energy near the Earth, and using the m in the relativistic energy expression, the gain in energy for a photon which falls distance h is

Comparing the energy shifts on the upward and downward paths gives a predicted difference

The measured difference was

The success of this experiment owed much to the care of Pound and Rebka in preparing the source. They electroplated cobalt-57 onto the surface of a thin sheet of iron and then heated the combination at 1220 K for an hour. The heat treatment caused the cobalt to diffuse into the iron to a depth of about 300 nm or 1000 atomic spacings. The source was then mounted on the cone of a loudspeaker driven at 10Hz to sweep the source velocity in a sinusoidal variation. The detector was a thin sheet of iron about 14 micrometers thick which was also annealed. The heat treatments were found to be crucial in obtaining high resolution.

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Scout Rocket Experiment

In 1976 the Smithsonian Astrophysical Observatory sent aloft a Scout rocket to a height of 10,000 km. At this height, a clock should run 4.5 parts in 1010 faster than one on the Earth. During two hours of free fall from its maximum height, the rocket transmitted timing pulses from a maser oscillator which acted as a clock and which was compared with a similar clock on the ground. This result confirmed the gravitational time dilation relationship to within 0.01%.

Gravity Probe A

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Gravitational Time Dilation

A clock in a gravitational field runs more slowly according to the gravitational time dilation relationship from general relativity
This is distinct from the time dilation from relative motion
where T is the time interval measured by a clock far away from the mass. For a clock on the surface of the Earth, this expression becomes

This time dilation is about 1 part in 109. .....Show

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Commercial airline experimentScout rocket experiment
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Time Dilation on Earth

The gravitational time dilation expression

has such a small second term in the denominator that excessive numerical accuracy is required to evaluate it directly. Using a binomial expansion :


so that the first approximation to the time expression is

The numerical values were calculated using g = 9.8 m/s2, R = 6.38 x 106m (mean radius), and c= 3 x 108m/s.

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