V = wr

where w represents the angular velocity of rotation of the disc K1 with respect to K. If v[0], represents the number of ticks of the clock per unit time (” rate " of the clock) relative to K when the clock is at rest, then the " rate " of the clock (v) when it is moving relative to K with a velocity V, but at rest with respect to the disc, will, in accordance with Section 12, be given by

eq. 43:  file eq43.gif

or with sufficient accuracy by

eq. 44:  file eq44.gif

This expression may also be stated in the following form: 

eq. 45:  file eq45.gif

If we represent the difference of potential of the centrifugal force between the position of the clock and the centre of the disc by f, i.e. the work, considered negatively, which must be performed on the unit of mass against the centrifugal force in order to transport it from the position of the clock on the rotating disc to the centre of the disc, then we have

eq. 46:  file eq46.gif

From this it follows that

eq. 47:  file eq47.gif

In the first place, we see from this expression that two clocks of identical construction will go at different rates when situated at different distances from the centre of the disc.  This result is aiso valid from the standpoint of an observer who is rotating with the disc.

Now, as judged from the disc, the latter is in a gravititional field of potential f, hence the result we have obtained will hold quite generally for gravitational fields.  Furthermore, we can regard an atom which is emitting spectral lines as a clock, so that the following statement will hold: 

An atom absorbs or emits light of a frequency which is dependent on the potential of the gravitational field in which it is situated.

The frequency of an atom situated on the surface of a heavenly body will be somewhat less than the frequency of an atom of the same element which is situated in free space (or on the surface of a smaller celestial body).

Now f = — K (M/r), where K is Newton’s constant of gravitation, and M is the mass of the heavenly body.  Thus a displacement towards the red ought to take place for spectral lines produced at the surface of stars as compared with the spectral lines of the same element produced at the surface of the earth, the amount of this displacement being

eq. 48:  file eq48.gif

For the sun, the displacement towards the red predicted by theory amounts to about two millionths of the wave-length.  A trustworthy calculation is not possible in the case of the stars, because in general neither the mass M nor the radius r are known.

It is an open question whether or not this effect exists, and at the present time (1920) astronomers are working with great zeal towards the solution.  Owing to the smallness of the effect in the case of the sun, it is difficult to form an opinion as to its existence.  Whereas Grebe and Bachem (Bonn), as a result of their own measurements and those of Evershed and Schwarzschild on the cyanogen bands, have placed the existence of the effect almost beyond doubt, while other investigators, particularly St. John, have been led to the opposite opinion in consequence of their measurements.