(say). This gives the work done in moving unit mass from the source of light to the observer (the source of light is here the point to which the potential energy is referred in the field).
Therefore, if we transform the accelerational field of the observer into the gravitational field, we get the result:
This means that a spectral line of frequency
will appear to a distant observer to be displaced, if compared with the position of the same line, when produced by a source at a different point in the field. Each of these lines, produced by vibrating electrons, may be regarded as a clock, and this simple calculation shows how time-measurements are affected by the state of the gravitational field. This effect amounts to 0·008 Ångstroms, for a wavelength of 4000 Å. The same displacement would be produced as a Doppler effect by a velocity of 0·6 kms. per sec. When this test was put into practice, it was found difficult to discriminate it from the various superposed effects due to other causes such as the radial velocities of the stars, proper velocities of the gaseous envelopes, pressure, etc. The conditions of the emission of light by the sun have not been fully ascertained, nor is the light of the arc lamp free from disturbing elements. Dr. Erwin Freundlich, of the Neubabelsberg Observatory, has discussed, in conjunction with Professor Einstein, the possibility of recognizing this effect in spite of these obscuring influences. He points out three ways of establishing the result quantitatively. They may be briefly classified as being based on (1) statistical methods; (2) nebular spectra; (3) calcium lines in the spectra of the atmosphere surrounding double-stars.
I. If we consider a great number of stars of about the same mass evenly distributed over the heavens, and represent the spectral shift due to radial velocities (i.e. velocities in the line of sight) graphically, we should expect these velocities to be distributed according to the law of probability about the value zero, i.e. as depicted by Gauss's Error Curve, which resembles a vertical section of a bell. If, however, Einstein's gravitational effect really exists, we should expect these velocities to group themselves symmetrically about a positive velocity which would be that corresponding to this spectral shift. Gauss's Error Curve would thus appear displaced by precisely the amount of the radial velocity corresponding to this shift, as all the radial velocities would be falsified by just this amount.
The values of the radial velocities have been plotted in the case of
