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§ 1. Defects of the Present View
…As I
have shown in the previous paper, the general theory of relativity requires
that the universe be spatially finite. But this view of the universe
necessitated an extension of equations (1), with the introduction of a new
universal constant l, standing in a fixed
relation to the total mass of the universe (or, respectively, to the
equilibrium density of matter). This is gravely detrimental to the formal
beauty of the theory [(1) is: G_{mn} = – ½g_{mn
}G =  kT_{mn
}]
§ 2. The Field Equations Freed of
Scalars
The difficulties set forth above
are removed by setting in place of field equations (1) the field equations
G_{mn}
= – ¼g_{mn }G =  kT_{mn
}... (1a)
… We now write the field
equations (1a) in the form
(G_{mn} – ½g_{mn
}G) + ¼g_{mn }G_{0 }=
 k [T_{mn} + (1/4k)g_{mn
}(G  G_{0})] ... (9)
On the other hand, we transform
the equations supplied with the cosmological term as already given
G_{mn} – lg_{mn
} =
 k (T_{mn}  ½g_{mn }T).
Subtracting the scalar equation
multiplied by ½, we next obtain
(G_{mn
} – ½g_{mn
}G) + g_{mn }l =  kT_{mn
}
Now in regions where only
electrical and gravitational fields are present, the righthand side of this
equation vanishes. For such regions we obtain, by forming the scalar,
 G + 4l = 0.
In such regions, therefore, the
scalar of curvature is consistent, so that l may be replaced by ¼G_{0}. Thus we may
write the earlier field equation (1) in the form
G_{mn} – ½g_{mn
}G + ¼g_{mn }G_{0 =
} kT_{mn
} ... (10)
Comparing (9) with (10), we see
that there is no difference between the new field equations and the earlier
ones, except that instead of T_{mn} as
tensor of "gravitational mass” there now occurs T_{mn
}+ (1/4k)g_{mn
}(G  G_{0}) which is independent of the scalar of curvature. But the new formulation has this great
advantage, that the quantity l appears in the fundamental
equations as a constant of integration, and no longer as a universal constant
peculiar to the fundamental law.
§ 3. On the Cosmological Question
The last result already permits
the surmise that with our new formulation the
universe may be regarded as spatially finite, without any necessity for
an additional hypothesis. As in the preceding paper
I shall again show that with a uniform distribution of matter, a spherical
world is compatible with the equations.
In the first place we set
ds^{2} =  g_{ik}dx_{i}dx_{k} + dx_{4}^{2}
(i, k = 1, 2, 3) … (11)
Then if P_{ik} and
P are, respectively, the curvature tensor of
the second rank and the curvature scalar in the threedimensional space, we have
G_{ik}^{ } ½g_{ik
}G = P_{ik} (i, k = 1, 2, 3)
Gi_{4} = G_{4i}
= G_{44} = 0
G =  P
 g = g.
It therefore follows for our case
that
G_{ik}^{ } ½g_{ik
}G = P_{ik } ½g_{ ik }P (i,
k = 1, 2, 3)
G_{44}  ½g_{44}G
= ½P.
We pursue our reflexions, from
this point on, in two ways. Firstly, with the support of equation (1a). Here T_{mn
}denotes the energytensor of the electromagnetic field,
arising from the electrical particles constituting matter… our fundamental equations permit the idea of a spherical
universe … it is known (Cf. H. Weyl, "Raum,
Zeit, Materie,” § 33) that this system is satisfied by a (threedimensional)
spherical universe…
§ 4. Concluding Remarks
The above reflexions show the
possibility of a theoretical construction of matter out of gravitational fiend
and electromagnetic field alone, without the introduction of hypothetical
supplementary terms on the lines of Mie’s theory. This possibility appears
particularly promising in that it frees us from the necessity of introducing a
special constant l for the solution of the
cosmological problem. On the other hand, there is a peculiar difficulty. For,
if we specialize (1) for the spherically symmetrical
static case we obtain one equation too few for defining the g_{mn
}and f_{mn
}, with the result that any spherically
symmetrical distribution of electricity appears capable of remaining
in equilibrium. Thus the problem of the constitution of the elementary quanta
cannot yet be solved on the immediate basis of the given field equations.

Albert Einstein: Relativity
Relativity: The Special and General Theory © 1920, Publisher:
Methuen & Co Ltd. First Published: December, 1916. Translated: Robert W.
Lawson (Authorized translation).
Part III: Considerations on the Universe as a Whole
The Structure of Space According to the General Theory of
Relativity
According to the general
theory of relativity, the geometrical properties of space are not independent,
but they are determined by matter…
If we are to have in the universe an average density of matter which differs
from zero, however small may be that difference, then the universe cannot be
quasiEuclidean. On the contrary, the results of calculation indicate that if matter be distributed
uniformly, the universe would necessarily be spherical (or elliptical). Since in reality the detailed distribution of matter is
not uniform, the real universe will deviate in individual parts from the
spherical, i.e. the universe will be quasispherical. But it will be necessarily finite. In fact, the theory supplies us with a simple connection ^{1)} between the spaceexpanse of the universe and the
average density of matter in it.
Footnote:
^{1)} For the
radius R of the universe we obtain the
equation
R^{2} = (2/kp)
The use of the C.G.S. system in this equation gives 2/k = 1^{.}08.10^{27}; p is the average density of the matter and k is a constant connected with the Newtonian constant of
gravitation.
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 extragalactic 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 lineshift
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 [see below], 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 (threedimensional) space, while the original "static" hypothesis of space yielded the closure
(finiteness) of space. To go to Einstein 6: http://fdocc.ucoz.com/index/091
