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The wavelengths of photons change as they pass through a gravitational field. This effect is called the gravitational redshift (Fig. 8.9). It is really a consequence of the principle of equivalence.

We can make a plausibility argument to estimate the magnitude of the effect. We have already seen in the previous section that the gravitational effect of some mass is to alter the trajectories of photons (i.e. they follow geodesies that are not straight lines). This makes it plausible that the gravitational field can do work on the photon, changing its energy. In order to estimate the gravitational potential energy of a photon (-GMm/r) we assign an "effective mass", E/c2, and since E = hc/l, this effective mass is h/cl. So if a photon moves from r1 out to r2 conservation of energy would give us

Solving for the ratio of the wavelengths gives

There is an interesting way to measure the gravitational redshift on Earth. It utilizes a phenomenon known as the Mossbauer effect.

This involves the emission of a gamma-ray by a nucleus held firmly in place by a solid crystal. In a free nucleus, the gamma-ray would lose a little energy due to the recoil of the nucleus. (The recoil is to conserve momentum.) When the nucleus is in a crystal, the whole crystal takes up the recoil. It moves very little because of its large mass, and the energy loss by the gamma-ray is small. This means that the gamma-ray energy is well defined.

If the gamma-ray is emitted by a nucleus in one crystal, it can be absorbed by a nucleus in an identical crystal, as long as there is no wavelength shift while the photon is in motion. A group of physicists tried an arrangement in which the gamma-rays were emitted in the basement and absorbed on the roof. The small gravitational redshift was enough for the gamma-rays to arrive at the roof with the wrong wavelength to be absorbed. The gamma-rays could be blueshifted back to the right wavelength by moving the crystal on the roof towards that in the basement. By seeing what Doppler shift is necessary to offset the gravitational redshift, the size of the gravitational redshift can be measured. The result agrees with the theoretical prediction..

A phenomenon related to the gravitational redshift is gravitational time dilation. All oscillators or clocks run slower in a strong gravitational field than they do in a weaker field. If we have two clocks at r1 and r2, the times they keep will be related by the same expression as the gravitational redshift.

From this we see that t2 > t1. This effect has been tested by taking identical clocks, leaving one on the ground and placing the other in an airplane. (Of course, you must first correct for the special relativistic effect due to the motion of the airplane.) The airplane experiments have yielded results that agree with theory. Even more recently, tests on rockets have yielded even more accurate results.