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%.

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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
Applications:

Commercial
airline experiment

Scout
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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