Appendix III

the general theory of relativity. From a systematic theoretical point of view, we may imagine the process of evolution of an empirical science to be a continuous process of induction. Theories are evolved and are expressed in short compass as statements of a large number of individual object in the form of empirical laws from which the general laws can be ascertained by comparison. Regarded in this way, the development of a science bears some resemblance to the compilation of

a classified catalog. It is, as it were, a purely empirical enterprise. But this point of view by no means embraces the whole of the actual process, for it slurs over the important part played by intuition and deductive thought in the development of an exact science. As soon as a science has emerged from its initial stages, theoretical advances are no longer achieved merely by a process of arrangement

guided by empirical data. The investigator rather develops a system of thought which in general is built up logically from a small number of fundamental assumptions, the so called axioms. We call such a system of thought a theory. The theory finds the justification for its existence in the fact that it correlates a large number of single observations, and it is just here that the truth of the theory

lies corresponding to the same complex of empirical data. There may be several theories which differ from one another to a considerable extent, but as regards the deduction from the theories which are capable of being tested, the agreement between the theories may be so complete that it becomes difficult to find such deductions in which the two theories differ from

each other. As an example, a case of general interest is available in the province of biology, in the Darwinian theory of the development of species by selection, in the struggle for existence, and in the theory of development which is based on the hypothesis of the hereditary transmission of acquired characters. We have another instance of far reaching agreement between the deductions from the two theories and Newtonian mechanics on the one hand, and the general theory of relativity

on the other. This agreement goes so far that up to the present we have been able to find only a few deductions from the general theory of relativity which are capable of investigation and to which the physics of pre relativity days does not also lead, and this despite the profound difference in the fundamental assumption of the two theories. In what follows, we shall again consider these important deductions, and we shall also discuss the empirical evidence appertaining to them,

which has hitherto been obtained. A motion of the perihelion of Mercury. According to Newtonian mechanics and Newton's law of gravitation, a planet which is revolving around the Sun would describe an ellipse around the latter, or more correct, around the common center of gravity of the Sun and the planet.

In such a system, the Sun or the common center of gravity lies in one of the foci of the orbital lips in such a manner as that, in the course of a planet year, the distant Sun planet grows from a minimum to a maximum and then decreases again to a minimum. If instead of Newton's law, we insert a somewhat different law of attraction into the calculation, we find that according to this new law, the motion would still take place in such a manner that the distant

Sun planet exhibits periodic variations. But in this case, the angle described by the line joining Sun and planet during such a period preentzes from perihelium closest proximity to the Sun to perihelion, and prencees would differ from three hundred

and sixty degrees. The line of the orbit would not then be a closed one, but in the course of time it would fill up an an annular part of the orbital plane, that is, between the circle of least and the circle of greatest distance of the planet from the Sun. According also to the general theory of relativity, which differs, of course from the theory of Newton, a small variation from the Newton Kepler motion of a planet in its orbit should take place, and in such a

way that the angle described by the radius Sun planet between one perihelion and the next should exceed that corresponding to one complete revolution by an amount given by the formula plus twenty four pi cubed a square divided by t squared c squared times to quantity one minus e

squared nb. One complete revolution corresponds to the angle two to the pie power in the absolute angle measure customary in physics, and the above expression gives the amount by which the radius un planet exceeds this angle during the interval between one perihelium and the next close priens. In this expression, A represents the major semi axis of the ellipse e its eccentricity see 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 around the Sun in the same sense as the orbital motion of the planet. Theory requires that this rotation should amount the forty three 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. Footnote, especially since the next planet Venus has an orbit that is almost an exact circle, which makes it more difficult to locate the perihelium with precision end footnote. In point of fact, astronomers have found that the theory of Newton does not suffice to calculate the observed motion of Mercury with an exactness corresponding to that of the delicacy of observation attainable at the present time.

After taking account of all the disturbing influences exerted on Mercury by the remaining planets, it was found Prentzes Leverer eighteen fifty nine. In Newcome eighteen ninety five close prenzes that an unexplained perihelial movement of the orbit of Mercury remained over the amount of which does not differ sensibly from the above mentioned plus forty three seconds of arc per century. The uncertainty of the empirical results amounts to a few seconds only. B deflection of light by a

gravitational field. In section twenty two, it has been already mentioned that according to the general theory of relativity, a ray of light will experience the curvature of its path when passing through a gravitational field, this curvature being similar to that experience 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 delta sun radii from its center, the angle of deflection Prence's alpha closed brend should amount to alpha equals one point seven seconds

of arc divided by delta. 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 prences quote curvature end quote end preentheses of space caused by the Sun. This result admits of an experimental test by means of the photographic registration of stars during a total eclipse of the Sun. The 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 Reader's annotation Fig. Five. The Earth is shown as a dot at the bottom of the diagram. A straight line proceeding from there, labeled D sub one, proceeds upward and slightly to the right, passing the Sun at a

tangent Sun being represented by a circle. A second line, label D sub two, starts at the Earth proceeds at a relatively smaller angle, which results in its passing the Sun at a greater distance than the the initial line D one, which is signified by the symbol delta. After passing the Sun, the line becomes parallel to D sub

one end of Reader's annotation. If the Sun prencees, s, and preentzes were not present, a star which is practically infinitely distant would be seen in the direction D sub one 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 D sub two, that is, at a somewhat greater distance from the center of the Sun, then corresponds to its real position. In practice,

the question is tested in the following way. The stars in the neighborhood of the Sun are photographed during a solar eclipse. In addition, a second photograph of the same stars is taken when the Sun is situated at another position in the sky, that is a few months earlier

or later. As compared with the standard photograph, the positions of the stars on the eclipse photograph ought to appear displaced radially outwards prenzes away from the center of the Sun close friends by an amount corresponding to the angle a. 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 the material and a psychological nature aroused by the war, these societies equipped two expeditions to Sobral, Brazil and to the island of princeip West Africa, and sent several of Britain's most celebrated astronomers Prenzies Eddington, Cottingham, Crommelin, Davidson and Prinz in order to obtain photographs of the solar eclipse of twenty

ninth May nineteen nineteen. The relative discrepancies to be expected between the stellar photographs of jeting during the eclipse, and the comparison photographs amounted to a few hundreds of a millimeter only. Thus, great accuracy was necessary in making the adjustments required for taking of the photographs and in their subsequent measurement. The results of the measurements confirmed the theory

in a thoroughly satisfactory manner. The rectangular components of the observed and of the calculated deviation of the stars, prenzes and seconds of an arc and prenzes are set forth in the following table of results Reader's annotation. The table consists of measurements on seven stars, which are then tabulated in four additional columns, which are entitled first coordinate and second coordinate, and then for each of those the observed

and calculated measurements are given end reader's annotation. Number of the star eleven first coordinate observed minus zero point one nine calculated minus zero point two two. Second coordinate observed plus zero point one six, calculated plus zero point zero two. Star number five first coordinate observed plus zero point twenty nine calculated plus zero point three to one. Second coordinate observed negative zero point four to six calculated minus zero

point four to three. Star number four observed zero point one one calculated zero point one zero. Second coordinate observed zero point eight three calculated plus zero point seven four. Star number three observed plus zero point two zero calculated plus zero point one two, second coordinate observed plus one

point zero zero calculated plus zero point eight seven. Star number six observed at the first coordinate plus zero point one zero calculated plus zero point zero four, second coordinate observed plus zero point five seven calculated plus zero point

four zero. Number the star ten observed minus zero point zero eight calculated plus zero point zero nine, second coordinate observed plus zero point three five calculated plus zero point three two number of the star two observed plus zero point nine five calculated plus zero point eight five, and at the second coordinate observed minus point two seven calculated minus zero point zero nine c displacement of the spectral

line towards the red. In section twenty three, it has been shown that in a system K prime which is in rotation with regard to a Galalian 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 position of the clocks. We shall

now examine this dependence quantitatively. A clock which is situated at a distance R from the center of the disc has a velocity relative k, which is given by V equals omega R, where omega represents the angle of velocity of rotation of the disc k prime with respect to k. If the sub zero represents the number of ticks of a clock per unit time prints these quote rate end quote of the clock close brands relative to k. When the clock is at rest, then the quote rate end

quote of the clock prince thes v close priends when it is moving relative to kay with a velocity v but at rest with respect to the disc, will, in accordance with section twelve, be given by V equals v sub zero times the square root of one minus V squared over C squared, or with sufficient accuracy, buy v equals v zero times the quantity one minus one half

V squared over C squared. This expression may be also stated in the following form V equals v sub zero times the quantity one minus one over C squared times omega squared are squared over two. If we represent a difference of potential of the centrifugal force between the position of the clock and the center of the disc y five.

That is, the work considered negatively which must be performed on the unit of mass against this trifical force in order to transport it from the position of the clock on the rotating disc to the center of the disc. Then we have Pi equals minus omega squared r square divided by two. From this, it follows that v equals v sub zero times to quantity one plus five overse squared.

In the first place. We see from this expression that the two clocks of identical construction will go at different rates when situated at different distances from the center of the disc. This result is also valid from the standpoint of an observer who is rotating with the disc. Now, as judge from the disc, the latter is in a gravitational field of potential five. 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 that the following statement will hold. An atom absorbs or amidst 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 prenzes or on the surface of a smaller celestial body close Prentzes period now five equals minus k times m over R, where K is Newton's constant 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 v sub zero minus v divided by v sub zero equals k over c squared times m over R. For the Sun, the displacement towards the red predicted by theory amounts to about two millions of the wavelength.

A trustworthy calculation is not possible in the case of the stars, because in general, neither the mass M nor the radius R is known. It is an open question whether or not this effect exists, and at the present time 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 Greb and Bacham, Prenzes, Bond and Priends, as a result of their own measurements and those of ever Sad and Schwartzchild on the cyanogen bands, have placed the existence of the effect almost beyond doubt. Other investigators, particularly Saint 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 certainly revealed by statistical investigation of the fixed stars, but up to the present the examination of the available data does not allow of any definite decision being arrived at as to whether or not these displacements are to be

referred in reality to the effect of gravitation. The results of observation have been collected together and discussed in detail from the standpoint of the question which has been engaging our attention here in a paper by E. Freundlich entitled Sir Profunder alamingun rebtats Theori Prentzes dionito Zenshoften, nineteen nineteen, Number thirty five, page five point twenty Julia Springer, Berlin, Close Pranz period. At all events, a definite decision will

be reached during the next few years. If the displacement of spectral lines towards the red by the gravitational potential does not exist, then the general theory of relativity will be untenable. On the other hand, if the cause of the displacement of spectral lines be definitely traced to the gravitational potential, then the study of this displacement will furnish us with important information as to the mass of the heavenly bodies. End of Appendix three