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at the surface of the earth, the amount of this displacement being

                        eq. 48: file eq48.gif

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 r 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, 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 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 allgemeinen
Relativit&umlaut;ts-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. [5][A]


  Notes

*) 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.

The displacentent 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. -- translator



APPENDIX IV

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 (" cosmological
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 strange 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 10^9 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 rernark 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.


K = co-ordinate system
x, y = two-dimensional co-ordinates
x, y, z = three-dimensional co-ordinates