Gravitational Red ShiftAccording 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. If a photon of frequency υ_{0} is emitted radially outward from the surface of a gravitational mass M, then the photon energy observed at a distance from the mass will be observed to be lower, or "red shifted". If observed at a great distance, we could denote the observed frequency as υ_{∞}. The result of general relativity using the Schwarzschild metric is which is valid for even strong gravitational fields. For weak gravitational fields or small displacements in a gravity field, one can make use of the approximation: to express the frequency shift between two locations as Where υ_{0} represents a photon emitted closer to the gravitational mass. This form made practical some laboratory experiments using nuclear sources with the gravitational term expressed in terms of the gravitational acceleration g on the Earth's surface. For the Harvard Tower experiment, the shifts in radiation frequency in the gravitational field were related to the relativistic doppler shift experienced from an accelerating light source. Gravity and the photon

Index General relativity ideas References: Carroll & Ostlie Sec 17.1  

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Harvard Tower Experiment
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 cobalt57 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 ExperimentIn 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 10^{10} 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 DilationA clock in a gravitational field runs more slowly according to the gravitational time dilation relationship from general relativity
This time dilation is about 1 part in 10^{9}. .....Show Applications:

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Time Dilation on EarthThe 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/s^{2}, R = 6.38 x 10^{6}m (mean radius), and c= 3 x 10^{8}m/s. 
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