Appendix three, the experimental confirmation of 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 observation 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 deduction to thought and 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 deduction 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 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 round the Sun would describe an ellipse round the latter,
or more correctly, round 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 orbit ellipse, in such a manner
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 princes from perihelian closest proximity to the Sun to Perihelian and
Princes 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 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 perihelian and the next
should exceed that corresponding to one complete revolution by an amount given by the formula
plus twenty four pie cubed a square divided by t squared c squared times The
quantity one e squared n b one complete revolution corresponds to the angle two to
the pie power in the absolute angle room 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 perihelian and the next close prends. 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 the forty three seconds of arc percentury 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, end 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 Prenzes Leverier eighteen fifty nine. In
Newcome eighteen ninety five closed Prenzes that an unexplained perihelium 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 art 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 a 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 prens these alpha closed brand 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 princes quote curvature end quote end princes 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 star is 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 preceding 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 labeled 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 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 Rena's annotation. If the Sun,
prencees, s, and princes 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 action 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
than 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. Pren
sees 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 a material and a psychological nature aroused by the war, these
societies equipped to expeditions to Sobral, Brazil and to the island of Principe,
West Africa, and sent several of Britain's most celebrated astronomers Prinzes, Eddington,
Cottingham, Cromelin, 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 observing during the eclipse and a 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 and seconds of
an arc and pren. These are set forth in the following table of results.
Reader's annotation detable 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 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 to five. First coordinate observed plus
zero point two nine calculated plus zero point three one. Second coordinate observed negative
zero point four six, calculated minus zero point four 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 point two
seven calculated 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 Galilean system, K, clocks of
identicals 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 v sub zero represents
the number of ticks of a clock per unit time prints ces quote rate end
quote of the clock closed brands relative decay. When the clock is at rest,
then the quote rate end quote of the clock prints c v close brands.
When it is moving relative to k with a velocity V, but at
rest with respect at a disc will, in accordance with Section twelve, be
given by V equals v sub zero times the square root of one V squared
over C squared, or with sufficient accurately, by V equals v zero times
the quantity one one half V squared over C squared. This expression may be
also stated in the following form V equals v sub zero times to quantity one
one oversea squared times omega squared are squared over two. If we represent the
difference of potential of the centrifical force between the position of the clock and the
center of the disc by fi, that is, the work considered negatively which
must be performed on the unit of mass against the centrifical 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 five equals omega squared are squared divided by
two. From this, it follows that V equals v sub zero times to
quantity one plus five over c 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 disk. This result is also
valid from the standpoint of an observer who is rotating with the disc. Now,
as judge from the disk, the latter is in a gravitational field of
potential FI. 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 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 prencs or on the surface
of a smaller celestial body closed princes period Now FI equals 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 all 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 subzero V divided by V subzero 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 Grebb and Bachum Prentz's bond and prinds as a result of their own
measurements and those of ever Shed and schwartz Child 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. Feindlich and Ti sirperfundur alamingun rabatats theory Princes die Exosenshaften,
nineteen nineteen, number thirty five, Page five twenty Julia Springer, Berlin,
Close Prince 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.