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

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 x₁, x₂, x₃, with constant x₄ (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, ξ₁, ξ₂, ξ₃, ξ₄, with a linear element ds ; let, therefore,

ds ² = dξ₁² + dξ₂² + dξ₃² + dξ₄² … (9)

In this space we consider the hyper-surface

R² = ξ₁² + ξ₂² + ξ₃² + ξ₄² … (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 ξ₁, ξ₂, ξ₃(the projection upon the hyper-plane ξ₄ = 0) since, by reason of (10), ξ₄ can be expressed in terms of ξ₁, ξ₂, ξ₃. Eliminating ξ₄ from (9) we obtain for the linear element of the spherical space the expression

ds ² = g_mndξ_mdξ_n

... (11)

g_mn = d_mn + [(ξ_mξ_n)/(R² – r ²)]

where d_mn = 1, if m = n ; d_mn = 0, if m ≠ n , and r² = ξ₁² + ξ₂² + ξ₃². The coordinates chosen are convenient when it is a question of examining the environment of one of the two points ξ₁ = ξ₂ = ξ₃ = 0.

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_mn + [(x_mx_n)/(R² – (x₁² + x₂² + x₃²)] … (12)

which equation, in combination with (7) and (8), perfectly defines the behaviour of measuring-rods, clocks, and light-rays [(7) is: g₄₄ = 1; (8) is: g₁₄ = g₂₄ = g₃₄ = 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 co-ordinates runs as follows: -

G_mn = - k (T_mn – ½g_mnT),

(13)

G_mn = -(∂ /∂ x_a){mn, a} + {ma, b} {nb, a} + {[(∂ ²log√-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) 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 g_mn = - k (T_mn – ½g_mnT) … (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 ▼²f - 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 co-ordinates

x₁ = x₂= x₃= x₄= 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 = (∂ /∂ x₁){mn, 1} + (∂ /∂ x₂){mn, 2} + (∂ /∂ x₃){mn, 3}+ [(∂ ²log√-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/R²) + l = -(kr/2), - l = -(kr/2),

l = (kr/2) = (1/R²) … (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 2p²R³ of spherical space. The total mass M of the universe, according to our view, is finite, and is in fact

M = r . 2p²R³ = 4p²(R/k) = p²√(32/k³r) … (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.

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"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 = – ½g_mnG = - 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_mnG = - kT_mn... (1a)

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

(G_mn – ½g_mnG) + ¼g_mnG₀= - k [T_mn + (1/4k)g_mn(G - G₀)] ... (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_mnT).

Subtracting the scalar equation multiplied by ½, we next obtain

(G_mn – ½g_mnG) + g_mnl = - kT_mn

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 ¼G₀. Thus we may write the earlier field equation (1) in the form

G_mn – ½g_mnG + ¼g_mnG_{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 - G₀) 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

ds² = - g_ikdx_idx_k + dx₄² (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 three-dimensional space, we have

G_ik- ½g_ikG = P_ik (i, k = 1, 2, 3)

Gi₄ = G_4i = G₄₄ = 0

G = - P

- g = g.

It therefore follows for our case that

G_ik- ½g_ikG = P_ik- ½g_ikP (i, k = 1, 2, 3)

G₄₄ - ½g₄₄G = ½P.

We pursue our reflexions, from this point on, in two ways. Firstly, with the support of equation (1a). Here T_mndenotes 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_mnand 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.

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Albert Einstein: Relativity

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.

Footnote:

¹⁾ For the radius R of the universe we obtain the equation

R² = (2/kp)

The use of the C.G.S. system in this equation gives 2/k = 1^.08.10²⁷; 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 10⁹ 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.

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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 – ½g_mnR = -kT_mn]…

…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 10⁴. 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 + cr²); B = 4c

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+(r²/4)] = 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 x₄.

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 10⁶ 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” (10⁹). 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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Einstein, el cosmos y M = E/c^2.

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Footnote:

1) For the radius R of the universe we obtain the equation

¹⁾ For the radius R of the universe we obtain the equation