Speaker 1: Correction for this chapter in mathematical formulae instead of I
Speaker 1: here one. This is a LibriVox recording. All LibriVox recordings
Speaker 1: are in the public domain. For more information or to volunteer,
Speaker 1: please visit LibriVox dot org. Recording by mL Cullen, Cleveland, Ohio,
Speaker 1: March two thousand and seven. Relativity the Special and General
Speaker 1: Theory by Albert Einstein, Appendix three The experimental confirmation of
Speaker 1: the general theory of relativity. From a systematic theoretical point
Speaker 1: of view, we may imagine the process of evolution of
Speaker 1: an empirical science to be a continuous process of induction.
Speaker 1: Theories are evolved and are expressed in short compass as
Speaker 1: statements of a large number of individual object in the
Speaker 1: form of empirical laws from which the general laws can
Speaker 1: be ascertained by comparison. Regarded in this way, the development
Speaker 1: of a science bears some resemblance to the compilation of
Speaker 1: a classified catalog. It is, as it were, a purely
Speaker 1: empirical enterprise. But this point of view by no means
Speaker 1: embraces the whole of the actual process, for it slurs
Speaker 1: over the important part played by intuition and deductive thought
Speaker 1: in the development of an exact science. As soon as
Speaker 1: a science has emerged from its initial stages, theoretical advances
Speaker 1: are no longer achieved merely by a process of arrangement
Speaker 1: guided by empirical data. The investigator rather develops a system
Speaker 1: of thought which in general is built up logically from
Speaker 1: a small number of fundamental assumptions, the so called axioms.
Speaker 1: We call such a system of thought a theory. The
Speaker 1: theory finds the justification for its existence in the fact
Speaker 1: that it correlates a large number of single observations, and
Speaker 1: it is just here that the truth of the theory
Speaker 1: lies corresponding to the same complex of empirical data. There
Speaker 1: may be several theories which differ from one another to
Speaker 1: a considerable extent, but as regards the deduction from the
Speaker 1: theories which are capable of being tested, the agreement between
Speaker 1: the theories may be so complete that it becomes difficult to
Speaker 1: find such deductions in which the two theories differ from
Speaker 1: each other. As an example, a case of general interest
Speaker 1: is available in the province of biology, in the Darwinian
Speaker 1: theory of the development of species by selection, in the
Speaker 1: struggle for existence, and in the theory of development which
Speaker 1: is based on the hypothesis of the hereditary transmission of
Speaker 1: acquired characters. We have another instance of far reaching agreement
Speaker 1: between the deductions from the two theories and Newtonian mechanics
Speaker 1: on the one hand, and the general theory of relativity
Speaker 1: on the other. This agreement goes so far that up
Speaker 1: to the present we have been able to find only
Speaker 1: a few deductions from the general theory of relativity which
Speaker 1: are capable of investigation and to which the physics of
Speaker 1: pre relativity days does not also lead, and this despite
Speaker 1: the profound difference in the fundamental assumption of the two theories.
Speaker 1: In what follows, we shall again consider these important deductions,
Speaker 1: and we shall also discuss the empirical evidence appertaining to them,
Speaker 1: which has hitherto been obtained. A motion of the perihelion
Speaker 1: of Mercury. According to Newtonian mechanics and Newton's law of gravitation,
Speaker 1: a planet which is revolving around the Sun would describe
Speaker 1: an ellipse around the latter, or more correct, around the
Speaker 1: common center of gravity of the Sun and the planet.
Speaker 1: In such a system, the Sun or the common center
Speaker 1: of gravity lies in one of the foci of the
Speaker 1: orbital lips in such a manner as that, in the
Speaker 1: course of a planet year, the distant Sun planet grows
Speaker 1: from a minimum to a maximum and then decreases again
Speaker 1: to a minimum. If instead of Newton's law, we insert
Speaker 1: a somewhat different law of attraction into the calculation, we
Speaker 1: find that according to this new law, the motion would
Speaker 1: still take place in such a manner that the distant
Speaker 1: Sun planet exhibits periodic variations. But in this case, the
Speaker 1: angle described by the line joining Sun and planet during
Speaker 1: such a period preentzes from perihelium closest proximity to the
Speaker 1: Sun to perihelion, and prencees would differ from three hundred
Speaker 1: and sixty degrees. The line of the orbit would not
Speaker 1: then be a closed one, but in the course of
Speaker 1: time it would fill up an an annular part of
Speaker 1: the orbital plane, that is, between the circle of least
Speaker 1: and the circle of greatest distance of the planet from
Speaker 1: the Sun. According also to the general theory of relativity,
Speaker 1: which differs, of course from the theory of Newton, a
Speaker 1: small variation from the Newton Kepler motion of a planet
Speaker 1: in its orbit should take place, and in such a
Speaker 1: way that the angle described by the radius Sun planet
Speaker 1: between one perihelion and the next should exceed that corresponding
Speaker 1: to one complete revolution by an amount given by the
Speaker 1: formula plus twenty four pi cubed a square divided by
Speaker 1: t squared c squared times to quantity one minus e
Speaker 1: squared nb. One complete revolution corresponds to the angle two
Speaker 1: to the pie power in the absolute angle measure customary
Speaker 1: in physics, and the above expression gives the amount by
Speaker 1: which the radius un planet exceeds this angle during the
Speaker 1: interval between one perihelium and the next close priens. In
Speaker 1: this expression, A represents the major semi axis of the
Speaker 1: ellipse e its eccentricity see the velocity of light, and
Speaker 1: T the period of revolution of the planet. Our result
Speaker 1: may also be stated as follows. According to the general
Speaker 1: theory of relativity, the major axis of the ellipse rotates
Speaker 1: around the Sun in the same sense as the orbital
Speaker 1: motion of the planet. Theory requires that this rotation should
Speaker 1: amount the forty three seconds of arc per century for
Speaker 1: the planet Mercury, but for the other planets of our
Speaker 1: solar system, its magnitude should be so small that it
Speaker 1: would necessarily escape detection. Footnote, especially since the next planet
Speaker 1: Venus has an orbit that is almost an exact circle,
Speaker 1: which makes it more difficult to locate the perihelium with
Speaker 1: precision end footnote. In point of fact, astronomers have found
Speaker 1: that the theory of Newton does not suffice to calculate
Speaker 1: the observed motion of Mercury with an exactness corresponding to
Speaker 1: that of the delicacy of observation attainable at the present time.
Speaker 1: After taking account of all the disturbing influences exerted on
Speaker 1: Mercury by the remaining planets, it was found Prentzes Leverer
Speaker 1: eighteen fifty nine. In Newcome eighteen ninety five close prenzes
Speaker 1: that an unexplained perihelial movement of the orbit of Mercury
Speaker 1: remained over the amount of which does not differ sensibly
Speaker 1: from the above mentioned plus forty three seconds of arc
Speaker 1: per century. The uncertainty of the empirical results amounts to
Speaker 1: a few seconds only. B deflection of light by a
Speaker 1: gravitational field. In section twenty two, it has been already
Speaker 1: mentioned that according to the general theory of relativity, a
Speaker 1: ray of light will experience the curvature of its path
Speaker 1: when passing through a gravitational field, this curvature being similar
Speaker 1: to that experience by the path of a body which
Speaker 1: is projected through a gravitational field. As a result of
Speaker 1: this theory, we should expect that a ray of light
Speaker 1: which is passing close to a heavenly body would be
Speaker 1: deviated towards the latter. For a ray of light which
Speaker 1: passes the Sun at a distance of delta sun radii
Speaker 1: from its center, the angle of deflection Prence's alpha closed
Speaker 1: brend should amount to alpha equals one point seven seconds
Speaker 1: of arc divided by delta. It may be added that,
Speaker 1: according to the theory, half of this deflection is produced
Speaker 1: by the Newtonian field of attraction of the Sun and
Speaker 1: the other half by the geometrical modification prences quote curvature
Speaker 1: end quote end preentheses of space caused by the Sun.
Speaker 1: This result admits of an experimental test by means of
Speaker 1: the photographic registration of stars during a total eclipse of
Speaker 1: the Sun. The reason why we must wait for a
Speaker 1: total eclipse is because at every other time the atmosphere
Speaker 1: is so strongly illuminated by the light from the Sun
Speaker 1: that the stars situated near the Sun's disc are invisible,
Speaker 1: the predicted effect can be seen clearly from the accompanying
Speaker 1: diagram Reader's annotation Fig. Five. The Earth is shown as
Speaker 1: a dot at the bottom of the diagram. A straight
Speaker 1: line proceeding from there, labeled D sub one, proceeds upward
Speaker 1: and slightly to the right, passing the Sun at a
Speaker 1: tangent Sun being represented by a circle. A second line,
Speaker 1: label D sub two, starts at the Earth proceeds at
Speaker 1: a relatively smaller angle, which results in its passing the
Speaker 1: Sun at a greater distance than the the initial line
Speaker 1: D one, which is signified by the symbol delta. After
Speaker 1: passing the Sun, the line becomes parallel to D sub
Speaker 1: one end of Reader's annotation. If the Sun prencees, s,
Speaker 1: and preentzes were not present, a star which is practically
Speaker 1: infinitely distant would be seen in the direction D sub
Speaker 1: one as observed from the Earth, but as a consequence
Speaker 1: of the deflection of light from the star by the Sun,
Speaker 1: the star will be seen in the direction D sub two,
Speaker 1: that is, at a somewhat greater distance from the center
Speaker 1: of the Sun, then corresponds to its real position. In practice,
Speaker 1: the question is tested in the following way. The stars
Speaker 1: in the neighborhood of the Sun are photographed during a
Speaker 1: solar eclipse. In addition, a second photograph of the same
Speaker 1: stars is taken when the Sun is situated at another
Speaker 1: position in the sky, that is a few months earlier
Speaker 1: or later. As compared with the standard photograph, the positions
Speaker 1: of the stars on the eclipse photograph ought to appear
Speaker 1: displaced radially outwards prenzes away from the center of the
Speaker 1: Sun close friends by an amount corresponding to the angle a.
Speaker 1: We are indebted to the Royal Society and to Royal
Speaker 1: Astronomical Society for the investigation of this important deduction. Undaunted
Speaker 1: by the war and by difficulties of both the material
Speaker 1: and a psychological nature aroused by the war, these societies
Speaker 1: equipped two expeditions to Sobral, Brazil and to the island
Speaker 1: of princeip West Africa, and sent several of Britain's most
Speaker 1: celebrated astronomers Prenzies Eddington, Cottingham, Crommelin, Davidson and Prinz in
Speaker 1: order to obtain photographs of the solar eclipse of twenty
Speaker 1: ninth May nineteen nineteen. The relative discrepancies to be expected
Speaker 1: between the stellar photographs of jeting during the eclipse, and
Speaker 1: the comparison photographs amounted to a few hundreds of a
Speaker 1: millimeter only. Thus, great accuracy was necessary in making the
Speaker 1: adjustments required for taking of the photographs and in their
Speaker 1: subsequent measurement. The results of the measurements confirmed the theory
Speaker 1: in a thoroughly satisfactory manner. The rectangular components of the
Speaker 1: observed and of the calculated deviation of the stars, prenzes
Speaker 1: and seconds of an arc and prenzes are set forth
Speaker 1: in the following table of results Reader's annotation. The table
Speaker 1: consists of measurements on seven stars, which are then tabulated
Speaker 1: in four additional columns, which are entitled first coordinate and
Speaker 1: second coordinate, and then for each of those the observed
Speaker 1: and calculated measurements are given end reader's annotation. Number of
Speaker 1: the star eleven first coordinate observed minus zero point one
Speaker 1: nine calculated minus zero point two two. Second coordinate observed
Speaker 1: plus zero point one six, calculated plus zero point zero two.
Speaker 1: Star number five first coordinate observed plus zero point twenty
Speaker 1: nine calculated plus zero point three to one. Second coordinate
Speaker 1: observed negative zero point four to six calculated minus zero
Speaker 1: point four to three. Star number four observed zero point
Speaker 1: one one calculated zero point one zero. Second coordinate observed
Speaker 1: zero point eight three calculated plus zero point seven four.
Speaker 1: Star number three observed plus zero point two zero calculated
Speaker 1: plus zero point one two, second coordinate observed plus one
Speaker 1: point zero zero calculated plus zero point eight seven. Star
Speaker 1: number six observed at the first coordinate plus zero point
Speaker 1: one zero calculated plus zero point zero four, second coordinate
Speaker 1: observed plus zero point five seven calculated plus zero point
Speaker 1: four zero. Number the star ten observed minus zero point
Speaker 1: zero eight calculated plus zero point zero nine, second coordinate
Speaker 1: observed plus zero point three five calculated plus zero point
Speaker 1: three two number of the star two observed plus zero
Speaker 1: point nine five calculated plus zero point eight five, and
Speaker 1: at the second coordinate observed minus point two seven calculated
Speaker 1: minus zero point zero nine c displacement of the spectral
Speaker 1: line towards the red. In section twenty three, it has
Speaker 1: been shown that in a system K prime which is
Speaker 1: in rotation with regard to a Galalian system K clocks
Speaker 1: of identical construction and which are considered at rest with
Speaker 1: respect to the rotating reference body, go at rates which
Speaker 1: are dependent on the position of the clocks. We shall
Speaker 1: now examine this dependence quantitatively. A clock which is situated
Speaker 1: at a distance R from the center of the disc
Speaker 1: has a velocity relative k, which is given by V
Speaker 1: equals omega R, where omega represents the angle of velocity
Speaker 1: of rotation of the disc k prime with respect to k.
Speaker 1: If the sub zero represents the number of ticks of
Speaker 1: a clock per unit time prints these quote rate end
Speaker 1: quote of the clock close brands relative to k. When
Speaker 1: the clock is at rest, then the quote rate end
Speaker 1: quote of the clock prince thes v close priends when
Speaker 1: it is moving relative to kay with a velocity v
Speaker 1: but at rest with respect to the disc, will, in
Speaker 1: accordance with section twelve, be given by V equals v
Speaker 1: sub zero times the square root of one minus V
Speaker 1: squared over C squared, or with sufficient accuracy, buy v
Speaker 1: equals v zero times the quantity one minus one half
Speaker 1: V squared over C squared. This expression may be also
Speaker 1: stated in the following form V equals v sub zero
Speaker 1: times the quantity one minus one over C squared times
Speaker 1: omega squared are squared over two. If we represent a
Speaker 1: difference of potential of the centrifugal force between the position
Speaker 1: of the clock and the center of the disc y five.
Speaker 1: That is, the work considered negatively which must be performed
Speaker 1: on the unit of mass against this trifical force in
Speaker 1: order to transport it from the position of the clock
Speaker 1: on the rotating disc to the center of the disc.
Speaker 1: Then we have Pi equals minus omega squared r square
Speaker 1: divided by two. From this, it follows that v equals
Speaker 1: v sub zero times to quantity one plus five overse squared.
Speaker 1: In the first place. We see from this expression that
Speaker 1: the two clocks of identical construction will go at different
Speaker 1: rates when situated at different distances from the center of
Speaker 1: the disc. This result is also valid from the standpoint
Speaker 1: of an observer who is rotating with the disc. Now,
Speaker 1: as judge from the disc, the latter is in a
Speaker 1: gravitational field of potential five. Hence, the result we have
Speaker 1: obtained will hold quite generally for gravitational fields. Furthermore, we
Speaker 1: can regard an atom which is emitting spectral lines as
Speaker 1: a clock that the following statement will hold. An atom
Speaker 1: absorbs or amidst light of a frequency which is dependent
Speaker 1: on the potential of the gravitational field in which it
Speaker 1: is situated. The frequency of an atom situated on the
Speaker 1: surface of a heavenly body will be somewhat less than
Speaker 1: the frequency of an atom of the same element which
Speaker 1: is situated in free space prenzes or on the surface
Speaker 1: of a smaller celestial body close Prentzes period now five
Speaker 1: equals minus k times m over R, where K is
Speaker 1: Newton's constant gravitation and M is the mass of the
Speaker 1: heavenly body. Thus, a displacement towards the red ought to
Speaker 1: take place for spectral lines produced at the surface of stars,
Speaker 1: as compared with the spectral lines of the same element
Speaker 1: produced at the surface of the Earth, the amount of
Speaker 1: this displacement being v sub zero minus v divided by
Speaker 1: v sub zero equals k over c squared times m
Speaker 1: over R. For the Sun, the displacement towards the red
Speaker 1: predicted by theory amounts to about two millions of the wavelength.
Speaker 1: A trustworthy calculation is not possible in the case of
Speaker 1: the stars, because in general, neither the mass M nor
Speaker 1: the radius R is known. It is an open question
Speaker 1: whether or not this effect exists, and at the present
Speaker 1: time astronomers are working with great zeal towards the solution
Speaker 1: owing to the smallness of the effect in the case
Speaker 1: of the sun, it is difficult to form an opinion
Speaker 1: as to its existence. Whereas Greb and Bacham, Prenzes, Bond
Speaker 1: and Priends, as a result of their own measurements and
Speaker 1: those of ever Sad and Schwartzchild on the cyanogen bands,
Speaker 1: have placed the existence of the effect almost beyond doubt.
Speaker 1: Other investigators, particularly Saint John, have been led to the
Speaker 1: opposite opinion in consequence of their measurements. Mean displacements of
Speaker 1: lines towards the less refrangible end of the spectrum are
Speaker 1: certainly revealed by statistical investigation of the fixed stars, but
Speaker 1: up to the present the examination of the available data
Speaker 1: does not allow of any definite decision being arrived at
Speaker 1: as to whether or not these displacements are to be
Speaker 1: referred in reality to the effect of gravitation. The results
Speaker 1: of observation have been collected together and discussed in detail
Speaker 1: from the standpoint of the question which has been engaging
Speaker 1: our attention here in a paper by E. Freundlich entitled
Speaker 1: Sir Profunder alamingun rebtats Theori Prentzes dionito Zenshoften, nineteen nineteen,
Speaker 1: Number thirty five, page five point twenty Julia Springer, Berlin,
Speaker 1: Close Pranz period. At all events, a definite decision will
Speaker 1: be reached during the next few years. If the displacement
Speaker 1: of spectral lines towards the red by the gravitational potential
Speaker 1: does not exist, then the general theory of relativity will
Speaker 1: be untenable. On the other hand, if the cause of
Speaker 1: the displacement of spectral lines be definitely traced to the
Speaker 1: gravitational potential, then the study of this displacement will furnish
Speaker 1: us with important information as to the mass of the
Speaker 1: heavenly bodies. End of Appendix three