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. If a photon of frequency υ₀ 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 υ₀ 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.

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. 在22.6米的范围内，由公式给出的分数引力红移仅为4.92×10⁻¹⁵，但铁-57的14.4 keV伽马射线的莫塞莱效应具有足够的分辨率，可以检测到这一差异。20世纪60年代初，哈佛大学的Jefferson物理实验室的Pound、Rebka和Snyder测量了该红移，其精度达到预测值的1%。

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.

Scout Rocket Experiment

Gravitational Time Dilation

This is distinct from the time dilation from relative motion 这与相对运动引起的时间膨胀不同

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

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.