EVIDENCE OF A CONFINED UNIVERSE (SPHERICAL OR ELLIPTICAL), AS DEDUCTED BY EINSTEIN:

(Fragments)

As presented in two articles
included in the anthology “*The Principle of Relativity*”:

"Cosmological
Considerations On The General Theory of Relativity" [Translated from
"Kosmologische Betrachtungen Zur Allgemeinen Relativitätstheorie," *Sitzungsberichte
der Preussischen Akademie der Wissenschaften*, 1917]

§ 1. The Newtonian Theory

…We
imagine that there may be a place in universal space round about which the
gravitational field of matter viewed on a large scale, possesses spherical
symmetry… In this sense, therefore, the
universe according to Newton is finite… There is a finite ratio of densities
corresponding to the finite difference of potential…

§ 2. The
Boundary Conditions According to the General Theory of Relativity

...The
universal continuum in respect of its spatial dimensions is to be viewed as a
self-contained continuum of finite spatial (three-dimensional) volume…
Inertia would indeed be *influenced* but would not be *conditioned*
by matter (present in finite space)… if it were possible to regard the universe as a continuum
which is *finite* (*closed*) *with respect to its spatial
dimensions*, we should have no need at all of any such boundary
conditions. We shall proceed to show that both the
general postulate of relativity and the fact of the small stellar velocities
are compatible with the hypothesis of a spatially finite universe;
through certainly, in order to carry through this idea, we need a generalizing
modification of the field equations of gravitation.

§ 3. The Spatially
Finite Universe with a Uniform Distribution of Matter

According to the general theory
of relativity the metrical character (curvature)
of the four-dimensional space-time continuum is defined at every point by the
matter at that point and the state of that matter. Therefore, on account of the
lack of uniformity in the distribution of matter, the metrical structure of
this continuum must necessarily be extremely complicated. But if we are concerned with the structure
only on a large scale, we may represent matter to ourselves as being uniformly
distributed over enormous spaces, so that its density of distribution is a
variable function which varies extremely slowly. Thus our procedure will
somewhat resemble that of the geodesists who, by means of an ellipsoid,
approximate to the shape of the earth’s surface, which on a small scale is
extremely complicated… If we assume the universe to
be spatially finite… From our assumption as to the uniformity of
distribution of the masses generating the field, it
follows that the curvature of the required space must be constant. With
this distribution of mass, therefore, the required
finite continuum of the *x*_{1}, *x*_{2},* x*_{3},
with constant* x*_{4} (the time co-ordinate, independent for all
magnitudes), will be a spherical space.

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 *d**s* ; let, therefore,

*d**s *^{2} = *dξ*_{1}^{2} + *dξ*_{2}^{2} + *dξ*_{3}^{2} + *dξ*_{4}^{2} … (9)

In this space we consider the
hyper-surface

R^{2} = *ξ*_{1}^{2} + *ξ*_{2}^{2} + *ξ*_{3}^{2} + *ξ*_{4}^{2} … (10)

The four-dimensional Euclidean
space with which we started serves only for a convenient definition of our
hyper-surface. Only those points of the hyper-surface 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 three-dimensional
continuum we may employ the co-ordinates *ξ*_{1}, *ξ*_{2},
*ξ*_{3}^{ }(the projection upon the hyper-plane *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

*d**s *^{2} = g_{mn
}*dξ*_{m }*dξ*_{n}_{}

... (11)

*g _{mn}* = d

where d* _{mn}*
= 1, if

Now the linear element of the
required four-dimensional space-time universe is also given us. For the
potential *g** _{mn}*, both indices of which differ
from 4, we have to set

*g** _{mn}* = - (d

which equation, in combination
with (7) and (8), perfectly defines the behaviour of measuring-rods, clocks,
and light-rays [(7) is: *g** _{44}* = 1; (8)
is:

§ 4. On an Additional Term for
the Field Equations of Gravitation

My proposed field equations of
gravitation for any chosen system of co-ordinates runs as follows: -

G* _{mn}* = -

(13)

G* _{mn}* =
-(∂

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) energy-tensor 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 left-hand 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

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* - l*f* = 4*pkr* , 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 co-ordinates

*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}* =
(∂

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 quasi-static 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]

§ 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}* = – ½

§ 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}* = – ¼

… We now write the field
equations (1a) in the form

(G* _{mn}* – ½

On the other hand, we transform
the equations supplied with the cosmological term as already given

G* _{mn}* – l

Subtracting the scalar equation
multiplied by ½, we next obtain

(G* _{mn
}* – ½

Now in regions where only
electrical and gravitational fields are present, the right-hand 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}* – ½

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

§ 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}* +

Then if P* _{ik}* and
P are, respectively, the curvature tensor of
the second rank and the curvature scalar in the three-dimensional space, we have

G* _{ik}^{ }- *½

G*i*_{4} = G_{4i}
= G_{44} = 0

G = - P

- *g* = *g*.

It therefore follows for our case
that

G* _{ik}^{ }- *½

G_{44} *- *½*g*_{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 energy-tensor of the electro-magnetic 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 (three-dimensional)
spherical universe…

§ 4. Concluding Remarks

The above reflexions show the possibility
of a theoretical construction of matter out of gravitational fiend and
electro-magnetic 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

---------------------------------

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
quasi-Euclidean. 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 quasi-spherical. But it will be necessarily finite. In fact, the theory supplies us with a simple connection ** ^{1)}** between the space-expanse of the universe and the
average density of matter in it.

*R*^{2} = (2/*k**p*)

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 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 [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 (three-dimensional) space, while the original "static" hypothesis of space yielded the closure
(finiteness) of space.

---------------------------------

As presented in the book “*The
Meaning of Relativity*”

“In 1921, five years
after the appearance of his comprehensive paper on general relativity and
twelve years before he left Europe permanently to join the *Institute for
Advanced Study*, Albert Einstein visited *Princeton University*, where
he delivered the *Stafford Little Lectures* for that year. These four
lectures constituted an overview of his then controversial theory of
relativity. *Princeton University Press* made the lectures available under
the title *The Meaning of Relativity*, the first book by Einstein to be
produced by an American publisher”.

“The General Theory of Relativity (*Continued*)” [*Finiteness of the Universe*]

… [I] shall give a brief
discussion of the so-called cosmological problem… our previous considerations,
based upon the field equations (96), had for a foundation the conception that
space on the whole is Galilean-Euclidean, and that this character is disturbed
only by asses embedded in it. This conception was certainly justified as long
as we were dealing with spaces of the order of magnitude of those that
astronomy has to do with. But whether portions of the universe, however large
they may be, are quasi-Euclidean, is a wholly different question. We can make
this clear by using an example from the theory of surfaces which we have
employed many times. If a portion of a surface is
observed by the eye to be practically plane, it does not at all follow that the
whole surface has the form of a plane; the
surface might just as well be a sphere, for example, of sufficiently large
radius. The question as to whether the universe as a whole is
non-Euclidean was much discussed from the geometrical point of view before the
development of the theory of relativity. But with the theory of relativity,
this problem has entered upon a new stage, for according to this theory the
geometrical properties of bodies are not independent, but depend upon the
distribution of masses [the *field equation* is represented as (96): *R** _{mn}* – ½

…The possibility seems to be particularly
satisfying that the universe is spatially bounded and… is of constant
curvature, being either spherical or elliptical; for then the boundary
conditions at infinity which are so inconvenient from the standpoint of the
general theory of relativity, may be replaced by the much more natural
conditions for a closed surface…

Thus we may present the following
arguments against the conception of a space-infinite,
and for the conception of a space-bounded, universe:-

1. From the standpoint of the theory of relativity, the condition for a
closed surface is very much simpler than the corresponding boundary condition
at infinity of the quasi-Euclidean structure of the universe.

2. The idea
that Mach expressed, that inertia depends upon the mutual action of bodies, is
contained, to a first approximation, in the equations of the theory of
relativity; it follows from these equations that inertia depends, at least in
part, upon mutual actions between masses. As it is an unsatisfactory assumption
to make that inertia depends in part upon mutual actions, and in part upon an
independent property of space, Mach’s idea gains in probability. But this idea of Match’s corresponds only to a finite
universe, bounded in space, and not to a
quasi-Euclidean, infinite universe. From the
standpoint of epistemology it is more satisfying to have the mechanical
properties of space completely determined by matter, and this is the case only
in a space-bounded universe.

3. An
infinite universe is possible only if the mean density of matter in the
universe vanishes. Although such an assumption is logically possible, it is
less probable than the assumption that there is a
finite mean density of matter in the universe.

Appendix for the Second Edition.

On the “Cosmologic Problem”

…The mathematician Friedman found
a way out of this dilemma (the introduction of l (a
universal constant, the “cosmologic constant”) he showed that it is possible,
according to the field equations, to have a finite
density in the whole (three-dimensional) space, without enlarging these
field equations *ad hoc*. Zeitschr. F. Phys. 10 (1922)). His result then
found a surprising confirmation by Hubble’s discovery of the expansion of the
stellar system (a red shift of the spectral lines which increases uniformly
with distance. The existence of the red shift of the spectral lines by the
(negative) gravitational potential of the place of origin. This demonstration
was made possible by the discovery of so-called “dwarf stars” whose average
density exceeds that of water by a factor of the order 10^{4}. For such
a star (e.g. the faint companion of Sirius), whose mass and radius can be
determined (the mass is derived from the reaction on Sirius by spectroscopic
means, using the Newtonian laws; the radius is derived from the total lightness
and from the intensity of radiation per unit area, which may be derived from
the temperature of its radiation), this red shift was expected, by the theory,
to be about 20 times as large as for the sun, and indeed it was demonstrated to
be within the expected range). The following is essentially nothing but an
exposition of Friedman’s idea:

FOUR-DIMENSIONAL SPACE

WHICH IS ISOTROPIC WITH RESPECT TO

THREE DIMENSIONS

…The
surfaces of constant radius are then surfaces of constant (positive) curvature
which are everywhere perpendicular to the (radial) geodesics… There
exists a family of surfaces orthogonal to the geodesics. Each of these surfaces is a surface of constant curvature…

CHOICE OF COORDINATES

(3c) *A* = (1/1 + *cr*^{2}); *B* = 4*c*

*c* > 0 (spherical space)

*c* < 0
(pseudospherical space)

*c* = 0
(Euclidean space)

…we can further get in the first
case *c* = ¼, in the second case *c* = -¼

… In
the spherical case the “circumference” of the unit space (*G *= 1) is

∫ [*dr*/1+(r^{2}/4)]
= 2p

[ ∫ going from infinite (^{∞})
to infinite(_{∞}) ]

the “radius” of the unit space is
1. In all three cases the function *G* of time is a measure for the change
with time of the distance of two points of matter (measured on a spatial
section). In the spherical case, *G* is the
radius of space at the time *x*_{4}.

THE FIELD EQUATIONS

… Since *G* is in all cases
a relative measure for the metric distance of two material particles as
function of time, *G*’/*G* expresses Hubble’s expansion…

THE SPECIAL CASE OF VANISHING
SPATIAL CURVATURE

…The relation between Hubble’s
expansion… and the average density…, is comparable to some extent with
experience, at least as far as the order of magnitude is concerned. The
expansion is given as 432 km/sec for the distance of 10^{6} parsec…

…Can
the present difficulty, which arouse under the assumption of a practically
negligible spatial curvature, be eliminated by the introduction of a suitable
spatial curvature?…

SUMMARY AND OTHER REMARKS

(1) The
introduction of the “cosmologic member” (l) into
the equation of gravity, though possible from the point of view of relativity,
is to be rejected from the point of view of logical economy. As Friedman was the first to show one can reconcile an
everywhere finite density of matter with the original form of the equations of
gravity if one admits the time variability of the metric distance of two mass
points [If Hubble’s expansion had been discovered at the time of the
conception of the “general theory of relativity”, the cosmologic member (l) would never have been introduced. It seems now so
much less justified to introduce such a member into the field equations, since
its introduction loses its sole original justification, - that of leading to a natural solution of the cosmologic problem].

(2) The demand for *spatial* isotropy of the universe alone leads to
Friedman’s form. It is therefore undoubtedly the general form, which fits the
cosmologic problem.

(3) Neglecting
the influence of spatial curvature, one
obtains a relation between the mean density and Hubble’s expansion which, as to
order of magnitude, is confirmed empirically…

(6) …It
seems to me… that the “theory of evolution” of the stars rests on weaker
foundations than the field equations.

…The
“beginning of the world” (the “beginning of the expansion”) really constitutes
a beginning, from the point of view of the development of the now existing
stars and systems of stars, at which those stars and systems of stars did not
yet exist as individual entities.

(8) For
the reasons given it seems that we have to take the idea of an expanding
universe seriously, in spite of the short “lifetime” (10^{9}). If one
does so, the main question becomes whether space has
positive (spherical case) or negative (pseudospherical case) spatial curvature…

… It is imaginable that the proof would be given that the
world is spherical (it is hardly imaginable
that one could prove it to be pseudospherical)…

---------------------------------

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