Mathematics of Physics and Engineering

118 Theory of Relativity

solution. On an intuitive level, this shift can be explained without using
the Schwarzschild solution or any general relativity. Indeed, let a photon
of wavelength A be emitted from a far-away star of mass M and radius
R. At the moment of the emission, the photon has radiation energy hu,
mass m = hv/c^2 , and potential energy —mMG/R. Far away from the
star, with no other gravitating objects around, the potential energy of the
photon is approximately zero. This increase in potential energy of the
photon must be compensated by a decrease of its radiation energy. Thus,
mMG/R = -hAu or Av/v = -MG/{Rc^2 ) = -Ra/{2R). Assuming that
the relative changes are small, and using v = c/A, we can write the result
as

EXERCISE 2.4.14. (a)c Estimate the gravitational red shift for the Sun.

(take M = 2 • 1030 kg, R — 7 • 109 m). (b)c Consider the radiation emitted

at the surface of the Earth and received H meters above the surface, with H

much smaller than the radius of the Earth. Denote by g the gravitational

acceleration on the Earth surface. Show that the observed red shift should

be

£ - f • <*".>

(c)A Explain how the slow-down of the clock and shrinking of length in

the gravitational field can be deduced from the Schwarzschild solution, and

connect these effects with the gravitational red shift. Hint: for example, with

dO = d<p = dt = 0 in (24.35), we have ds = (1 - {Ra/r))~1/2dr.

Einstein predicted the gravitational red shift eight years before formu-
lating the theory of general relativity, but, because of the need for very high
accuracy, both in the generation of the radiation and in the measurements,
it was only in the 1960s that the gravitational red shift was observed in an
experiment (for H = 90m, the right-hand side of (2.4.45) is about 10~^14 ).
The famous Pound-Rebka-Snider experiment was first conducted in 1960
at Harvard University by R. V. Pound and G. A. Rebka Jr.; in 1964, R.
V. Pound and J. L. Snider carried out a more accurate experiment; as of
2005, Robert V. Pound is Emeritus Professor of Physics at Harvard.

For an observer on the Earth, the clock on an orbiting satellite will
appear to be running slow because of the special relativity effects, see for-
mula (2.4.12) on page 101, and because of the gravitational red shift: the