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Figure 2.31The frequency of a photon emitted from the surface of a star decreases as it moves away

from the star.

R

mass = M

vv′

Gravitational Red Shift

An interesting astronomical effect is suggested by the gravitational behavior of light. If

the frequency associated with a photon moving toward the earth increases, then the

frequency of a photon moving away from it should decrease.

The earth’s gravitational field is not particularly strong, but the fields of many stars

are. Suppose a photon of initial frequencyis emitted by a star of mass Mand radius

R, as in Fig. 2.31. The potential energy of a mass mon the star’s surface is

PE

where the minus sign is required because the force between Mand mis attractive. The

potential energy of a photon of “mass” hc^2 on the star’s surface is therefore

PE

and its total energy E, the sum of PE and its quantum energy h, is

Ehh 1  

At a larger distance from the star, for instance at the earth, the photon is beyond

the star’s gravitational field but its total energy remains the same. The photon’s energy

is now entirely electromagnetic, and

Eh

where  is the frequency of the arriving photon. (The potential energy of the photon

in the earth’s gravitational field is negligible compared with that in the star’s field.)

Hence

hh 1  

 1 

GM



c^2 R







GM



c^2 R

GM



c^2 R

GMh



c^2 R

GMh



c^2 R

GMm



R

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