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We arrive at such a space, for
example, in the following way. We start from a Euclidean space of four
dimensions, ξ_{1}, ξ_{2}, ξ_{3},
ξ_{4}, with a linear element ds ; let, therefore,
ds ^{2} = dξ_{1}^{2} + dξ_{2}^{2} + dξ_{3}^{2} + dξ_{4}^{2} … (9)
In this space we consider the
hypersurface
R^{2} = ξ_{1}^{2} + ξ_{2}^{2} + ξ_{3}^{2} + ξ_{4}^{2} … (10)
The fourdimensional Euclidean
space with which we started serves only for a convenient definition of our
hypersurface. Only those points of the hypersurface are of interest to us
which have metrical properties in agreement with those of physical space with a
uniform distribution of matter. For the description of this threedimensional
continuum we may employ the coordinates ξ_{1}, ξ_{2},
ξ_{3}^{ }(the projection upon the hyperplane ξ_{4}
= 0) since, by reason of (10), ξ_{4} can be expressed in
terms of ξ_{1}, ξ_{2}, ξ_{3}.
Eliminating ξ_{4} from (9) we obtain for the linear element
of the spherical space the expression
ds ^{2} = g_{mn
}dξ_{m }dξ_{n}_{}
...
(11)
g_{mn} = d_{mn}
+ [(ξ_{m }ξ_{n}_{ })/(R^{2} – r ^{2})]
where d_{mn}
= 1, if m = n ; d_{mn}
= 0, if m ≠ n , and r^{2} = ξ_{1}^{2}
+ ξ_{2}^{2} + ξ_{3}^{2}. The
coordinates chosen are convenient when it is a question of examining the
environment of one of the two points ξ_{1} = ξ_{2} = ξ_{3} = 0.
Now the linear element of the
required fourdimensional spacetime universe is also given us. For the
potential g_{mn}, both indices of which differ
from 4, we have to set
g_{mn} =  (d_{mn}
+ [(x_{m }x_{n}_{ })/(R^{2} – (x_{1}^{2} + x_{2}^{2}
+ x_{3}^{2})] … (12)
which equation, in combination
with (7) and (8), perfectly defines the behaviour of
measuringrods, clocks, and lightrays [(7) is: g_{44} = 1; (8) is: g_{14} = g_{24} = g_{34} = 0; whereas (4) is: m√
g g_{ma}(dx_{a}/ds), but the text may be that it "differ from § 4” or
that "differ from the number 4” as the number 4 is introduced in equations (2)
and (15), see it below].
§ 4. On an Additional Term for
the Field Equations of Gravitation
My proposed field equations of
gravitation for any chosen system of coordinates runs as follows: 
G_{mn} =  k (T_{mn} – ½g_{mn
}T),
G_{mn} = (∂ /∂ x_{a }){mn, a} + {ma, b} {nb, a} + {[(∂
^{2}log√g)/(∂ x_{m }∂ x_{n}_{ })]  {mb, a}[(∂ log√g)/(∂
x_{ma })}
The system of equations (13) is
by no means satisfied when we insert for the g_{mn} the
values given in (7), (8), and (12), and for the (covariant) energytensor of
matter the values indicated in (6). It will be shown in the next paragraph how
this calculation may conveniently be made. So that, if it were certain that the
field equations (13) which I have hitherto employed were the only ones
compatible with the postulate of general relativity, we should probably have to
conclude that the theory of relativity does not admit the hypothesis of a spatially finite universe. [(6) is: 0 0 0 0
0 0 0 0
0 0 0 0 … (6) ]
0 0 0 r
However, the
system of equations (14) allows a readily suggested extension (to admit the
hypothesis of a spatially finite universe) which is compatible with the
relativity postulate, and is perfectly analogous to the extension of Poisson’s
equation given by equation (2). For on the lefthand side of field equation
(13) we may add the fundamental tensor g_{mn
}, multiplied by a universal constant  l, at present
unknown, without destroying the general covariance. In place of field equation
(13) we write
G_{mn}
=  l g_{mn} =  k (T_{mn} – ½g_{mn
}T) … (13a)
This field equation, with l sufficiently small, is in any case also compatible
with the facts of experience derived from the solar system. It also satisfies
laws of conservation of momentum and energy, because we arrive at (13a) in
place of (13) by introducing into Hamilton’s principle, instead of the scalar
of Riemann’s tensor, this scalar increased by a universal constant ; and
Hamilton’s principle, of course, guarantees the validity of laws of
conservation. It will be shown in § 5 that field equation (13a) is compatible
with our conjectures on field and matter [(2) is ▼^{2}f  lf = 4pkr , where l denotes a universal constant].
§ 5
Calculation and Result
Since all points of our continuum
are on an equal footing, it is sufficient to carry through the calculation for
one point, e.g. for one of the two points with the coordinates
x_{1} = x_{2
}= x_{3 }= x_{4 }= 0.
Then for the g_{mn} in (13a)
we have to insert the values
1
0 0 0
0
1 0 0
0 0 1 0
0 0 0 1
wherever they
appear differentiated only once or not at all. We thus obtain in the first
place
G_{mn} =
(∂ /∂ x_{1}){mn, 1} + (∂
/∂ x_{2}){mn, 2} + (∂ /∂ x_{3}){mn, 3}+ [(∂ ^{2}log√g)/(∂
x_{m }∂
x_{n})]
From this we readily discover,
taking (7), (8), and (13) into account, that all equations (13a) are satisfied
if the two relations
(2/R^{2})
+ l = (kr/2),  l = (kr/2),
or
l = (kr/2) = (1/R^{2})
… (14)
are
fulfilled.
Thus the newly introduced
universal constant l defines both the mean
density of distribution r which can
remain in equilibrium and also the radius R and the volume 2p^{2}R^{3} of spherical space. The
total mass M of the universe, according to our view, is finite, and is
in fact
M = r . 2p^{2}R^{3}
= 4p^{2}(R/k) = p^{2}√(32/k^{3}r) … (15)
Thus the theoretical view of the
actual universe, if it is in correspondence with our reasoning, is the
following. The curvature of space is variable in
time and place, according to the distribution of matter, but we may roughly
approximate to it by means of a spherical space. At any rate, this view
is logically consistent, and from the standpoint of the general theory of
relativity lies nearest at hand; whether, from the standpoint of present
astronomical knowledge, it is tenable, will not here be discussed. In order to
arrive at this consistent view, we admittedly had to introduce an extension of
the field equations of gravitation which is not justified by our actual
knowledge of gravitation. It is to be emphasized,
however, that a positive curvature of space is given by our results, even if
the supplementary term is not introduced. That term is necessary only
for the purpose of making possible a quasistatic distribution of matter, as
required by the fact of the small velocities of the stars.

"Do
Gravitational Fields Play An Essential Part In The Structure Of The Elementary
Particles Of Matter" [Translated from "Spielen
Gravitationsfelder im Aufber der materiellen
Elementarteilchen eine wesentliche Rolle?", Sitzungsberichte
der Preussischen Akademie der Wissenschaften,
1919] To go to Einstein 5: http://fdocc.ucoz.com/index/090
