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.Thus great accuracy wasnecessary in making the adjustments required for the taking of the photographs, and in theirsubsequent measurement.The results of the measurements confirmed the theory in a thoroughly satisfactory manner.Therectangular components of the observed and of the calculated deviations of the stars (in seconds ofarc) are set forth in the following table of results :79 Relativity: The Special and General Theory(c) Displacement of Spectral Lines Towards the RedIn Section 23 it has been shown that in a system K1 which is in rotation with regard to a Galileiansystem K, clocks of identical construction, and which are considered at rest with respect to therotating reference-body, go at rates which are dependent on the positions of the clocks.We shallnow examine this dependence quantitatively.A clock, which is situated at a distance r from thecentre of the disc, has a velocity relative to K which is given byV = wrwhere w represents the angular velocity of rotation of the disc K1 with respect to K.If v0, representsthe number of ticks of the clock per unit time (" rate " of the clock) relative to K when the clock is atrest, then the " rate " of the clock (v) when it is moving relative to K with a velocity V, but at rest withrespect to the disc, will, in accordance with Section 12, be given byor with sufficient accuracy byThis 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 clockand the centre of the disc by , i.e.the work, considered negatively, which must be performed onthe unit of mass against the centrifugal force in order to transport it from the position of the clock onthe rotating disc to the centre of the disc, then we haveFrom this it follows thatIn the first place, we see from this expression that two clocks of identical construction will go atdifferent rates when situated at different distances from the centre of the disc.This result is aisovalid 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 , hence the result wehave obtained will hold quite generally for gravitational fields.Furthermore, we can regard an atomwhich 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 thegravitational field in which it is situated.80 Relativity: The Special and General TheoryThe frequency of an atom situated on the surface of a heavenly body will be somewhat less thanthe frequency of an atom of the same element which is situated in free space (or on the surface ofa smaller celestial body).Now = - K (M/r), where K is Newton's constant of gravitation, and M is the mass of the heavenlybody.Thus a displacement towards the red ought to take place for spectral lines produced at thesurface of stars as compared with the spectral lines of the same element produced at the surface ofthe earth, the amount of this displacement beingFor the sun, the displacement towards the red predicted by theory amounts to about two millionthsof the wave-length.A trustworthy calculation is not possible in the case of the stars, because ingeneral 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) astronomersare working with great zeal towards the solution.Owing to the smallness of the effect in the case ofthe sun, it is difficult to form an opinion as to its existence.Whereas Grebe and Bachem (Bonn), asa result of their own measurements and those of Evershed and Schwarzschild on the cyanogenbands, 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.Mean displacements of lines towards the less refrangible end of the spectrum are certainlyrevealed by statistical investigations of the fixed stars ; but up to the present the examination of theavailable data does not allow of any definite decision being arrived at, as to whether or not thesedisplacements are to be referred in reality to the effect of gravitation.The results of observationhave been collected together, and discussed in detail from the standpoint of the question which hasbeen engaging our attention here, in a paper by E.Freundlich entitled "Zur Prfung derallgemeinen Relativitts-Theorie" (Die Naturwissenschaften, 1919, No.35, p.520: Julius Springer,Berlin).At all events, a definite decision will be reached during the next few years.If the displacement ofspectral lines towards the red by the gravitational potential does not exist, then the general theoryof relativity will be untenable.On the other hand, if the cause of the displacement of spectral linesbe definitely traced to the gravitational potential, then the study of this displacement will furnish uswith important information as to the mass of the heavenly bodies.[A]Next: Appendix IV: The Structure of Space According to the General Theory of RelativityFootnotes1)Especially since the next planet Venus has an orbit that is almost an exact circle, which makes itmore difficult to locate the perihelion with precision.[A]The displacentent of spectral lines towards the red end of the spectrum was definitelyestablished by Adams in 1924, by observations on the dense companion of Sirius, for which theeffect is about thirty times greater than for the Sun.R.W.L. translator81 Relativity: The Special and General TheoryRelativity: The Special and General Theory82 Relativity: The Special and General TheoryAlbert Einstein: RelativityAppendixAppendix IVThe Structure of Space According to the General Theory ofRelativity(Supplementary to Section 32)Since the publication of the first edition of this little book, our knowledge about the structure ofspace in the large (" cosmological problem ") has had an important development, which ought to bementioned even in a popular presentation of the subject.My original considerations on the subject were based on two hypotheses:(1) There exists an average density of matter in the whole of space which is everywhere the sameand different from zero.(2) The magnitude (" radius ") of space is independent of time.Both these hypotheses proved to be consistent, according to the general theory of relativity, butonly after a hypothetical term was added to the field equations, a term which was not required bythe theory as such nor did it seem natural from a theoretical point of view (" cosmological term ofthe field equations ").Hypothesis (2) appeared unavoidable to me at the time, since I thought that one would get intobottomless speculations if one departed from it.However, already in the 'twenties, the Russian mathematician Friedman showed that a differenthypothesis was natural from a purely theoretical point of view
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