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  (A) MOTION OF THE PERIHELION OF MERCURY

  According to Newtonian mechanics and Newton’s law of gravitation, a planet which is revolving round the sun would describe an ellipse round the latter, or, more correctly, round the common centre of gravity of the sun and the planet. In such a system, the sun, or the common centre of gravity, lies in one of the foci of the orbital ellipse in such a manner that, in the course of a planet-year, the distance 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 distance sun-planet exhibits periodic variations; but in this case the angle described by the line joining sun and planet during such a period (from perihelion—closest proximity to the sun—to perihelion) would differ from 360°. The line of the orbit would not then be a closed one but in the course of time it would fill up an annular part of the orbital plane, viz. 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

  (N.B.—One complete revolution corresponds to the angle 2π 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

  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 (Leverrier—1859—and Newcomb—1895) that an unexplained perihelial movement of the orbit of Mercury remained over, the amount of which does not differ sensibly from the above-mentioned + 43 seconds of arc per century. The uncertainty of the empirical result amounts to a few seconds only.

  (B) DEFLECTION OF LIGHT BY A GRAVITATIONAL FIELD

  In Section 12 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 Δ sun-radii from its centre, the angle of deflection (a) 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.

  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 that corresponds to its real position.

  FIG. 5.

  In practice, the question is tested in the following way. The stars in the neighbourhood 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, i.e. 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 (away from the centre of the sun) by an amount corresponding to the angle a.

  We are indebted to the Royal society and to the 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 discrepancies 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.

  The results of the measurements confirmed the theory in a thoroughly satisfactory manner. The rectangular components of the observed and of the calculated deviations of the stars (in seconds of arc) are set forth in the following table of results:

  (C) DISPLACEMENT OF SPECTRAL LINES TOWARDS THE RED

  In Section 23 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 γ from the centre of the disc, has a velocity relative to K which is given by

  where ω represents the angular 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 12, 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 ϕ, 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

  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 also valid from the standpoint of an observer who is rotating with the disc.

  Now, as judged from the disc, the latter is in a gravitational field of potential ϕ, 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 th
e 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

  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 γ 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, 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 certainly revealed by statistical investigations 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 “Zur Prüfung der aligemeinen Relativitäts-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 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.

  NOTE.—The displacement of spectral lines towards the red end of the spectrum was definitely established by Adams in 1924, by observations on the dense companion of Sirius, for which the effect is about thirty times greater than for the sun.

  R. W. L.

  1 Especially since the next planet Venus has an orbit that is almost an exact circle, which makes it more difficult to locate the perihelion with precision.

  APPENDIX FOUR

  THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY

  [SUPPLEMENTARY TO SECTION 32]

  Since the publication of the first edition of this little book, our knowledge about the structure of space in the large (“cosmo-logical problem”) has had an important development, which ought to be mentioned 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 same and 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, but only after a hypothetical term was added to the field equations, a term which was not required by the theory as such nor did it seem natural from a theoretical point of view (“cosmological term of the field equations”).

  Hypothesis (2) appeared unavoidable to me at the time, since I thought that one would get into bottomless speculations if one departed from it.

  However, already in the ’twenties, the Russian mathematician Friedman showed that a different hypothesis was natural from a purely theoretical point of view. He realized that it was possible to preserve hypothesis (1) without introducing the less natural cosmological term into the field equations of gravitation, if one was ready to drop hypothesis (2). Namely, the original field equations admit a solution in which the “world-radius” depends on time (expanding space). In that sense one can say, according to Friedman, that the theory demands an expansion of space.

  A few years later Hubble showed, by a special investigation of the extra-galactic nebulae (“milky ways”), that the spectral lines emitted showed a red shift which increased regularly with the distance of the nebulae. This can be interpreted in regard to our present knowledge only in the sense of Doppler’s principle, as an expansive motion of the system of stars in the large—as required, according to Friedman, by the field equations of gravitation. Hubble’s discovery can, therefore, be considered to some extent as a confirmation of the theory.

  There does arise, however, a stranger difficulty. The interpretation of the galactic line-shift discovered by Hubble as an expansion (which can hardly be doubted from a theoretical point of view), leads to an origin of this expansion which lies “only” about 109 years ago, while physical astronomy makes it appear likely that the development of individual stars and systems of stars takes considerably longer. It is in no way known how this incongruity is to be overcome.

  I further want to remark that the theory of expanding space, together with the empirical data of astronomy, permit no decision to be reached about the finite or infinite character of (three-dimensional) space, while the original “static” hypothesis of space yielded the closure (finiteness) of space.

  Sidelights on Relativity

  What really happens when one billiard ball strikes another? Prior to the twentieth century, it was understood that the cue ball and target ball only interacted during the brief moment when they were in contact with one another, as is dictated by common sense. This is all well and good for billiard balls, but what about the forces of gravity and electromagnetism, which appear to act at a distance? Scientists had hypothesized that those forces must propagate through some ponderable medium, known as the “luminiferous ether,” much as a shock wave propagates through the air.

  The ether, however, did not stand up to serious scientific scrutiny. In 1887, Albert Michelson and Edward Morley showed that, whatever the ether might be, it did not behave like normal matter. For example, a water wave traveling along a flowing river will propagate faster in the direction of the water’s motion than against it. In the case of light, however, Michelson and Morley showed that the speed of propagation was the same regardless of the relative motion of the observer and the hypothetical ether.

  In “Ether and the Theory of Relativity,” Einstein notes that special relativity is predicated on the observational fact that light travels at a constant speed for all observers, and thus ether, whatever it is, cannot be like ordinary matter. His theory of general relativity further complicates this matter by proposing that gravity gives rise to the structure of space itself. To put this plainly, gravity is defined even in “empty” space, and thus, there must be something.

  That “something” is the ether, or, in modern language, a field. General relativity and Maxwell’s theory of electromagnetism represented the first field theories: descriptions of how the world works in terms of omnipresent fields, rather than tiny particles. In many respects, this is one of the most important contributions of relativity to physics. In the modern view, all forces arise from fields. The billiard balls described above don’t really collide at all, but their electromagnetic fields repel each other on very small scales. In quantum field theory, developed in the mid-twentieth century, about forty years after the present work, not only do the forces, but the particles themselves arise from the field. Consider this work, then, as a transitional commentary between Isaac Newton’s classical particle picture and the modern picture in which the universe is comprised fundamentally of fields.

  ETHER AND THE THEORY

  OF RELATIVITY

  An Address delivered on May 5th, 1920, in the University o
f Leyden

  How does it come about that alongside of the idea of ponderable matter, which is derived by abstraction from everyday life, the physicists set the idea of the existence of another kind of matter, the ether? The explanation is probably to be sought in those phenomena which have given rise to the theory of action at a distance, and in the properties of light which have led to the undulatory theory. Let us devote a little while to the consideration of these two subjects.

  Outside of physics we know nothing of action at a distance. When we try to connect cause and effect in the experiences which natural objects afford us, it seems at first as if there were no other mutual actions than those of immediate contact, e.g. the communication of motion by impact, push and pull, heating or inducing combustion by means of a flame, etc. It is true that even in everyday experience weight, which is in a sense action at a distance, plays a very important part. But since in daily experience the weight of bodies meets us as something constant, something not linked to any cause which is variable in time or place, we do not in everyday life speculate as to the cause of gravity, and therefore do not become conscious of its character as action at a distance. It was Newton’s theory of gravitation that first assigned a cause for gravity by interpreting it as action at a distance, proceeding from masses. Newton’s theory is probably the greatest stride ever made in the effort towards the causal nexus of natural phenomena. And yet this theory evoked a lively sense of discomfort among Newton’s contemporaries, because it seemed to be in conflict with the principle springing from the rest of experience, that there can be reciprocal action only through contact, and not through immediate action at a distance. It is only with reluctance that man’s desire for knowledge endures a dualism of this kind. How was unity to be preserved in his comprehension of the forces of nature? Either by trying to look upon contact forces as being themselves distant forces which admittedly are observable only at a very small distance—and this was the road which Newton’s followers, who were entirely under the spell of his doctrine, mostly preferred to take; or by assuming that the Newtonian action at a distance is only apparently immediate action at a distance, but in truth is conveyed by a medium permeating space, whether by movements or by elastic deformation of this medium. Thus the endeavour toward a unified view of the nature of forces leads to the hypothesis of an ether. This hypothesis, to be sure, did not at first bring with it any advance in the theory of gravitation or in physics generally, so that it became customary to treat Newton’s law of force as an axiom not further reducible. But the ether hypothesis was bound always to play some part in physical science, even if at first only a latent part.