Albert Einstein > Relativity: The Special and General Theory > Appendix 3


Appendix III

The Experimental Confirmation of the General Theory of Relativity




  (N.B.—One complete revolution corresponds to the angle 2 in the absolute angular measure customary in physics, and the above expression gives the amount by which the radius sun-planet exceeds this angle during the interval between one perihelion and the next.) In this expression a represents the major semi-axis of the ellipse, e its eccentricity, c the velocity of light, and T the period of revolution of the planet. Our result may also be stated as follows: According to the general theory of relativity, the major axis of the ellipse rotates round the sun in the same sense as the orbital motion of the planet. Theory requires that this rotation should amount to 43 seconds of arc per century for the planet Mercury, but for the other planets of our solar system its magnitude should be so small that it would necessarily escape detection. 1   7


  In Section XXII it has been already mentioned that, according to the general theory of relativity, a ray of light will experience a curvature of its path when passing through a gravitational field, this curvature being similar to that experienced by the path of a body which is projected through a gravitational field. As a result of this theory, we should expect that a ray of light which is passing close to a heavenly body would be deviated towards the latter. For a ray of light which passes the sun at a distance of sun-radii from its centre, the angle of deflection () should amount to

It may be added that, according to the theory, half of this deflection is produced by the Newtonian field of attraction of the sun, and the other half by the geometrical modification (“curvature”) of space caused by the sun.

FIG. 5.

  This result admits of an experimental test by means of the photographic registration of stars during a total eclipse of the sun. The only reason why we must wait for a total eclipse is because at every other time the atmosphere is so strongly illuminated by the light from the sun that the stars situated near the sun’s disc are invisible. The predicted effect can be seen clearly from the accompanying diagram. If the sun (S) were not present, a star which is practically infinitely distant would be seen in the direction D1, as observed from the earth. But as a consequence of the deflection of light from the star by the sun, the star will be seen in the direction D2, i.e. at a somewhat greater distance from the centre of the sun than corresponds to its real position.   10
  We are indebted to the Royal Society and to Royal Astronomical Society for the investigation of this important deduction. Undaunted by the war and by difficulties of both a material and a psychological nature aroused by the war, these societies equipped two expeditions—to Sobral (Brazil) and to the island of Principe (West Africa)—and sent several of Britain’s most celebrated astronomers (Eddington, Cottingham, Crommelin, Davidson), in order to obtain photographs of the solar eclipse of 29th May, 1919. The relative descrepancies to be expected between the stellar photographs obtained during the eclipse and the comparison photographs amounted to a few hundredths of a millimetre only. Thus great accuracy was necessary in making the adjustments required for the taking of the photographs, and in their subsequent measurement.   13
Number of
the Star.
First Co-ordinate.Second Co-ordinate.


  In Section XXIII it has been shown that in a system k' which is in rotation with regard to a Galileian system K, clocks of identical construction and which are considered at rest with respect to the rotating reference-body, go at rates which are dependent on the positions of the clocks. We shall now examine this dependence quantitatively. A clock, which is situated at a distance r from the centre of the disc, has a velocity relative to K which is given by
v = [omega]r,
where [omega] represents the velocity of rotation of the disc k' with respect to K. If v0 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 XII, be given by

or with sufficient accuracy by

This expression may also be stated in the following form:

If we represent the difference of potential of the centrifugal force between the position of the clock and the centre of the disc by [phi], 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

From this it follows that

  Now, as judged from the disc, the latter is in a gravitational field of potential [phi], 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:   16
  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 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



