The Current State of Relativistic Cosmology

M. P. Bronstein, Leningrad

From the Editors

The cosmological problem is one of the most difficult, if not the most difficult, problems of modern physics. Classical physics scarcely posed this problem. The most recent development of physics, and chiefly the general theory of relativity, make it possible to approach the formulation of this problem. However, it would be wrong to think that the present state of physics and astronomy provides sufficient material for solving the cosmological problem. The solutions to the cosmological problem that are being put forward at the present time can by no means claim even an approximate solution of the problem. They are all based on arbitrary assumptions, without which the theory cannot be constructed (for example, the assumption concerning the quantity of matter in the universe and its distribution), but for the confirmation of which we have almost no astrophysical data.

At the first stage of its development, the general theory of relativity advanced the proposition that the structure of space is determined by the arrangement of gravitating masses. In modern theories radiant energy is also taken into account. However, this proposition of the theory of relativity is still insufficient for solving the problem of the structure of the world. For this it is necessary to make a definite assumption about the density of the distribution of matter and radiation, and, in addition, it is necessary to choose a definite form for \(ds\), i.e. to introduce an agreement concerning the properties of the metric. The multiplicity of constructions of the cosmological problem in modern physics is due to the indefiniteness of these assumptions.

However, all modern constructions possess two

general features: the finitude of the world in space (a finite radius of the world) and the irreversibility of the process by which gravitating masses pass into radiation.

Even the dynamical theories that assume an unlimited growth of the radius of the world nevertheless, in essence, presuppose the finitude of the world in space, since at any given moment of time the radius of the world is always finite.

It would, however, be entirely wrong to present this conclusion as an achievement of physics as exact and unshakable as, for example, the radius of the atom or the value of Planck’s constant. Ardent defenders of the finite world are inclined, once the conclusion has been obtained, to forget the assumptions on the basis of which alone it can be obtained. Thus what is only a consequence of definite assumptions is elevated into an absolute and regarded as proved. It would therefore be entirely wrong to assert that modern physics and astronomy provide a proof of the finitude of the world.

The unsatisfactoriness and inadmissibility of theories that assert the finitude of the world consist precisely in the fact that they take the standpoint of finitude entirely, rejecting infinity and thereby creating the specific limitation characteristic of all theories that base their constructions on an abstract concept.

An objection to theories of the finitude of the world by no means presupposes a renunciation of the cognition of the infinite world in its wholeness. The whole point is to understand the true, dialectical relation between the finite and the infinite.

In a separate act of cognition, the finite is always given to us. But “all real, exhaustive cognition consists only in this: that in thought we extract the singular from its singularity and translate it into universality; it consists in this: that we find the infinite in the finite, the eternal in the transient. But the form of universality is a form closed in itself, and consequently a form of infinity. It is the union of many finite things into the infinite” (Engels).

These general propositions cannot, of course, be regarded as the solution of the physical problem of cosmology, but they provide the most general methodological orientation, one fundamentally opposed to that orientation which underlies modern cosmological theories. With still greater clarity the difference between these orientations is seen in the problem of the transition of matter into radiation. Modern cosmology proceeds from the premise of irreversible processes, i.e., in essence, in a new aspect returns to the standpoint

Claudius. The world tends toward an inevitable end, i.e., it is finite not only in space, but also in time.

Unlike Einstein’s first cosmological theories, at present cosmological theories cannot give an answer to the question of the development of matter.

However, the recognition of the absolute irreversibility of processes inevitably leads to a non-material act of creation that produced the initial deviation from the state of equilibrium. Thus the premise of the irreversibility of world processes as a whole is in insuperable contradiction with materialism.

We see that the present state of the cosmological problem not only cannot be considered in any measure final, but that all the theories suffer from limitedness and contain a whole series of methodologically unacceptable and incorrect premises. We are not even speaking of the physical and astronomical inconsistencies noted in the article. Our physical and astronomical knowledge is still too imperfect for it to be possible to give any satisfactory solution of the question. Moreover, the methodological orientation must be changed at the root: from a one-sided recognition of the finiteness of the world it is necessary to pose dialectically the problem of the relation between the finite and the infinite. Despite the indicated shortcomings of the theory, the editors consider it necessary to publish M. P. Bronshtein’s article, which gives an account of the treatment of the cosmological problem in contemporary physics.

§ 1. Introduction

One of the most remarkable consequences of the general theory of relativity was the possibility of a new approach to the cosmological problem, i.e., to the problem of the world as a whole. Formerly the universe was imagined as a chorus of seven luminaries revolving around the earthly sphere; this entire construction was enclosed in an impenetrable crystal sphere, covered on the inner side with the radiant hieroglyphs of the constellations. But the conception of the universe expanded in a remarkable way when astronomy freed itself from the fetters of the “mathematical syntax” of Claudius Ptolemy; the crystal sphere was shattered, and the riddle of the Milky Way was resolved, at that

the very evening when Galileo first directed his telescope toward the mysterious world of the fixed stars, and the universe suddenly appeared to the astronomer in the form of a galactic system, i.e. as a vast accumulation of stars, similar to our Sun but removed from us by colossal (in comparison with the Sun) distances. Even these distances proved possible to measure: in 1838 Struve, Bessel, and Henderson, working quite independently of one another, succeeded in measuring the distances to three comparatively nearby stars (α Lyrae, α Centauri, and 61 Cygni), and thus “at once in three places broke down the wall before which astronomy had so long stood in powerless rage” (the words of John Herschel). At the present time astronomers regard the galactic system, of which our Sun too is a member, as an assemblage of several tens of billions of stars, distributed with nonuniform density throughout the volume of a very flattened spheroid, whose major diameter is equal to several tens of thousands of parsecs (the parsec is the unit of length commonly used in stellar astronomy, \(3.08 \times 10^{18}\) cm).

But the universe of the modern astronomer is not limited to a single galactic system: alongside our galactic system there exists an enormous number of other similar “island universes,” consisting of tens of billions of stars and visible in our telescopes as faintly luminous nebulae, for the most part of regular form (spiral forms are encountered most often, but round, elliptical, and spindle-shaped nebulae also occur, as well as nebulae of irregular form). The distances to some of these objects were measured by Hubble, Lundmark, and Shapley, who made use of the fact that modern powerful astronomical instruments make it possible to discern, in the “island universes” nearest to us, the brightest individual stars; among these stars there are also variable stars of the type δ Cephei; for variable stars of this type (the so-called Cepheids) a relation between ...

their absolute luminosity (the luminosity they would have from the point of view of an observer located from them at one and the same standard distance of 10 parsecs) and the period of variation of their brightness; applying this relation to Cepheids discovered in “island universes” makes it possible to determine the absolute luminosity of these Cepheids, while comparison with their observed (apparent) luminosity gives the distance to these stars, and consequently also to the star clusters of which they are members. By this method Hubble found, for example, that the distance to the large nebula N.G.C. 2241 in the constellation Andromeda is 285 thousand parsecs, while the distance to the nebula N.G.C. 598 in the constellation Triangulum is equal to 263 thousand parsecs. Since the diameter of our galaxy according to Shapley does not exceed 90 thousand parsecs, it is clear that these objects are clusters of stars lying far beyond the limits of our Milky Way system, i.e. separate “island universes” similar to our own. For brevity we shall call these “island universes” galaxies.

Not all distances to galaxies could be measured by so direct a method as the distances to N.G.C. 598 and N.G.C. 224; in most cases one has to be guided by more indirect considerations, based on an estimate of the apparent diameters of these objects and their observed luminosities. The most powerful of modern astronomical instruments is the famous hundred-inch reflector of the Solar Observatory (Mount Wilson, Southern California); a spiral nebula of the 18th stellar magnitude, still visible in this telescope, turns out to be a stellar system at a distance of \(1.5 \times 10^{26}\) cm from us—this, in Jeans’s words, is the greatest distance with which practical astronomy has to deal. In a sphere of such a diameter, in the opinion

Hubble, there are about two million galaxies. Distances of this order are such that even the parsec, equal to \(3.08 \times 10^{18}\) cm, turns out to be too small a unit of length for it to be conveniently used; in questions of this kind it is far more convenient to use the so-called unit \(A\), proposed by de Sitter, which is equal to \(10^{+24}\) cm (the distance that light waves traverse in 1 \(A\) is one million years). Thus, according to Hubble’s estimate, two million galaxies are enclosed in a sphere whose volume is equal to 2 million cubic \(A\)’s; in other words, the island universes are distributed in space with an approximate mean density equal to one galaxy per cubic \(A\).

Such is the universe as it appears to the contemporary astronomer: star clusters containing several tens of billions of stars and separated from one another by colossal empty spaces (the mean distance between two neighboring galactic systems is about 1 \(A\)); this is what an observer equipped with the most powerful astronomical instruments can say about the universe.

But he can say nothing about what lies beyond the limits of a sphere of radius 150 \(A\); he does not know whether the creative fantasy of nature continues there, too, tirelessly to heap galaxy upon galaxy with the same density, equal to one galaxy per cubic \(A\), or whether perhaps this density becomes smaller and smaller with distance from some center of a cluster of galaxies to which our Milky Way also belongs. Even if the “range of vision” of astronomical instruments were increased many times over, the observer would still be able only to speak of the distribution of galaxies within that sphere (although expanding with the improvement of astronomical technology) which is accessible to his telescopic vision; he will not be able to say what is located in the unobservable spaces beyond the limits of this sphere; he will never know anything about the world as a whole. Therefore it may seem that the cosmological problem

is an impregnable fortress, the conquest of which cannot be the province of empirical science.

But where the observing astronomer has come to despair of his own helplessness, the physicist approaches the solution of the hopeless problem. He undertakes to consider the universe from so general and so elevated a point of view that the very existence of the galactic systems themselves appears as an accidental inhomogeneity; just as, in a whole series of disciplines that consider bodies composed of atoms, the granular (atomic) structure of matter is replaced by a continuous one (for example, in hydrodynamics liquids are treated as media with a density continuously varying from point to point),—in the same way the relativist abstracts from the atomic structure of the world, in which the atoms are the galactic systems of stars, and assumes that matter is distributed in the universe with some uniform mean density. If it is indeed true that the approximately uniform distribution of galaxies in space, observed in the hundred-inch reflector of Mount Wilson, continues throughout all the rest of the universe, then this mean density is finite (assuming that the mass of a galaxy is on average \(10^{11}\) times greater than the mass of the Sun, as Oort asserts, we find that the assumed mean density of matter in the universe is equal to \(10^{11}\) \(\odot\)\(^{1}\) per one \(\Delta^{3}\), or, what is the same, \(2{,}10^{-28}\) g/cm\(^3\). One may be astonished at the negligible magnitude of this density, which corresponds approximately to the mass of one hydrogen atom in ten cubic decimeters! So colossal are the empty spaces between galactic systems in comparison with their own dimensions). But if the uniform distribution of galaxies in the region of space accessible to the observations of the modern astronomer has nothing in common with their distribution throughout all the rest of the universe, and if we have only by chance been fortunate enough to live in such a place in space where there are especially many island universes, then the mean density of matter in

\(^{1}\) The symbol \(\odot\) is used in astronomy to denote the mass of the Sun.

in space can be considerably less than this already astonishingly negligible figure; it can be arbitrarily close to zero (strictly speaking, it could be equal to zero only in the case where infinite space contains a finite number of stars, or, in general, a finite mass). In order, together with the relativist, to climb to that dizzying height from which it is possible to regard the distribution of matter in the universe as uniform, and to treat the distances between galactic systems as the distances between the atoms of bodies are treated in “phenomenological” theories based on the idea of a continuous distribution of matter, one must equip oneself with the usual relativistic mathematics, namely Riemannian geometry. For the reader who does not have the corresponding courses at hand, the following small digression may prove useful, in which we shall set out all the geometrical formulae we need.

§ 2. Excursus into the Domain of Mathematics.

The geometry of Riemann, which is used by the theory of relativity, is nothing other than a generalization of the ordinary geometry of Euclid. Let us consider, in ordinary Euclidean space, two points \(A\) and \(B\), and introduce a rectangular coordinate system. Suppose the coordinates of the point \(A\) in this system are \(x_A, y_A, z_A\), and let the coordinates of the point \(B\) be \(x_B, y_B, z_B\). Then, by the formulae of analytic geometry, the distance \(AB\) between the two points is equal to

\[ \overline{AB}=\sqrt{(x_A-x_B)^2+(y_A-y_B)^2+(z_A-z_B)^2}. \tag{1} \]

If, instead of rectangular coordinates, we introduce any other coordinates (for example, spherical coordinates, ellipsoidal coordinates, etc.), then the formula for calculating the distance between two points will turn out to be much more complicated. The matter is simplified, however, if we agree always to take two points infinitely close to one another. *

to each other. Let \(x_A=x,\ y_A=y,\ z_A=z,\ x_B=x+dx,\ y_B=y+dy,\ z_B=z+dz\), where \(dx,\ dy,\ dz\) are infinitely small quantities. If the distance between the infinitely close points \(A\) and \(B\) is denoted by \(ds\), then formula (1) gives

\[ ds=\sqrt{dx^2+dy^2+dz^2}. \tag{2} \]

Let us now suppose that, instead of the coordinates \(x,\ y,\ z\), some new coordinates \(\lambda,\ \mu,\ \nu\) have been introduced (not necessarily rectangular, but arbitrary); this means that from the coordinates \(x,\ y,\ z\) one can compute \(\lambda,\ \mu,\ \nu\), and conversely, from \(\lambda,\ \mu,\ \nu\) one can compute \(x,\ y,\ z\), i.e. that each of the quantities \(x,y,z\) is given as a function of \(\lambda,\ \mu,\ \nu\):

\[ x=f_1(\lambda,\mu,\nu), \]

\[ y=f_2(\lambda,\mu,\nu), \]

\[ z=f_3(\lambda,\mu,\nu). \]

Hence it follows that

\[ dx=\frac{\partial f_1}{\partial \lambda}\,d\lambda +\frac{\partial f_1}{\partial \mu}\,d\mu +\frac{\partial f_1}{\partial \nu}\,d\nu, \]

\[ dy=\frac{\partial f_2}{\partial \lambda}\,d\lambda +\frac{\partial f_2}{\partial \mu}\,d\mu +\frac{\partial f_2}{\partial \nu}\,d\nu, \]

\[ dz=\frac{\partial f_3}{\partial \lambda}\,d\lambda +\frac{\partial f_3}{\partial \mu}\,d\mu +\frac{\partial f_3}{\partial \nu}\,d\nu. \]

Substituting this into (2) and introducing the notations:

\[ g_{11}= \left(\frac{\partial f_1}{\partial \lambda}\right)^2 +\left(\frac{\partial f_2}{\partial \lambda}\right)^2 +\left(\frac{\partial f_3}{\partial \lambda}\right)^2, \]

\[ g_{22}= \left(\frac{\partial f_1}{\partial \mu}\right)^2 +\left(\frac{\partial f_2}{\partial \mu}\right)^2 +\left(\frac{\partial f_3}{\partial \mu}\right)^2, \]

\[ g_{33}= \left(\frac{\partial f_1}{\partial \nu}\right)^2 +\left(\frac{\partial f_2}{\partial \nu}\right)^2 +\left(\frac{\partial f_3}{\partial \nu}\right)^2, \]

\[ g_{23}= \frac{\partial f_1}{\partial \mu}\frac{\partial f_1}{\partial \nu} +\frac{\partial f_2}{\partial \mu}\frac{\partial f_2}{\partial \nu} +\frac{\partial f_3}{\partial \mu}\frac{\partial f_3}{\partial \nu} =g_{32}, \]

\[ g_{31}= \frac{\partial f_1}{\partial \nu}\frac{\partial f_1}{\partial \lambda} +\frac{\partial f_2}{\partial \nu}\frac{\partial f_2}{\partial \lambda} +\frac{\partial f_3}{\partial \nu}\frac{\partial f_3}{\partial \lambda} =g_{13}, \]

\[ g_{12}= \frac{\partial f_1}{\partial \lambda}\frac{\partial f_1}{\partial \mu} +\frac{\partial f_2}{\partial \lambda}\frac{\partial f_2}{\partial \mu} +\frac{\partial f_3}{\partial \lambda}\frac{\partial f_3}{\partial \mu} =g_{21}, \]

we obtain

\[ ds=\sqrt{ g_{11}d\lambda^2+g_{22}d\mu^2+g_{33}d\nu^2 +2g_{23}d\mu d\nu+2g_{31}d\nu d\lambda+2g_{12}d\lambda d\mu }. \]

If, for convenience, instead of \(\lambda, \mu, \nu\) we introduce new notations \(x_1, x_2, x_3\), then we may write

\[ ds=\sqrt{\sum_{i,k=1}^{3} g_{ik}\,dx_i\,dx_k}. \tag{3} \]

Expression (3) for the differential of length makes it possible to compute also the length of any line (straight or curved) drawn between two points \(A\) and \(B\) situated at a finite distance from one another. This length will be represented in the form of the integral

\[ \int_A^B \sqrt{\sum_{i,k=1}^{3} g_{ik}\,dx_i\,dx_k}, \]

taken along the curve connecting \(A\) and \(B\) (in order that this curve be specified, for example, in parametric form, three functions \(x_1(t), x_2(t), x_3(t)\) of the variable \(t\) are needed, which for some value \(t=t_0\) give the coordinates \(x_1, x_2, x_3\) of the point \(A\), for another value \(t=t_1\) give the coordinates of the point \(B\), and for intermediate values of \(t\) give the intermediate points of the curve. The coefficients \(g_{ik}\), which are, generally speaking, functions of \(x_1, x_2, x_3\), become functions of \(t\), which we shall denote by \(g_{ik}(t)\); instead of \(dx_i\) one must write \(\dfrac{dx_i(t)}{dt}\,dt\), and the length of the line from \(A\) to \(B\) is expressed by the definite integral

\[ \int_{t_0}^{t_1} \varphi(t)\,dt,\quad \text{where } \varphi(t)= \sqrt{\sum_{i,k=1}^{3} g_{ik}(t)\,\frac{dx_i(t)}{dt}\,\frac{dx_k(t)}{dt}}. \]

In the particular case when \(g_{11}=g_{22}=g_{33}=1\), and all the remaining \(g_{ik}\) are equal to zero, formula (3) gives

\[ ds=\sqrt{dx_1^{\,2}+dx_2^{\,2}+dx_3^{\,2}}, \]

i.e. the same thing as formula (2). It is always possible to introduce such a coordinate system that the coefficients \(g_{ik}\) have this property, i.e. that for \(i=k\) they are equal to unity, and for \(i\ne k\) to zero; for this it is necessary and suffi-

to take exactly one of the possible systems of rectangular coordinates, which is never and by no one forbidden in Euclidean space. If, from the coordinates \(x_1, x_2, x_3\), we pass to new coordinates \(x'_1, x'_2, x'_3\), which are functions of the old ones and of which, in turn, the old coordinates are functions, then it is enough to introduce the functions

\[ g_{ik}'=\sum_{j,l=1}^{3} g_{jl}\frac{\partial x_j}{\partial x_i'}\frac{\partial x_l}{\partial x_k'}, \tag{4} \]

so that

\[ \sum_{i,k=1}^{3} g_{ik}'\, dx_i'\, dx_k' = \sum_{j,l,i,k=1}^{3} g_{jl}\frac{\partial x_j}{\partial x_i'}\frac{\partial x_l}{\partial x_k'}\, dx_i'\, dx_k' = \]

\[ = \sum_{j,l=1}^{3} g_{jl} \sum_{i=1}^{3}\frac{\partial x_j}{\partial x_i'}\, dx_i' \sum_{k=1}^{3}\frac{\partial x_l}{\partial x_k'}\, dx_k' = \sum_{j,l=1}^{3} g_{jl}\, dx_j\, dx_l = ds^2. \]

The functions \(g'_{ik}\) in the new system of coordinates play exactly the same role as the functions \(g_{ik}\) played in the old one. Formula (4), consequently, represents the law of transformation of the functions \(g_{ik}\). In Euclidean space, as has been said, we can always choose a new coordinate system in such a way that formula (4) gives \(g'_{ik}=1\) for \(i=k\) and \(g'_{ik}=0\) for \(i\ne k\). If a coordinate system is given, i.e. a way is given of denoting any point by a system of three numbers \(x_1, x_2, x_3\), and coefficients \(g_{ik}\) are given, representing functions of these variables \(x_1, x_2, x_3\), then in Euclidean space it is always possible to find such functions \(x_i(x'_1,x'_2,x'_3)\), \((i=1,2,3)\), that formula (4) gives the required \(g_{ik}\). What will happen in the case when the \(g_{ik}\) are such that it is impossible to find such functions? It is clear that this will not be Euclidean space. Thus, quite naturally and simply, we arrive at the idea of non-Euclidean geometry. It remains only to discard the wholly inessential limitation on the number of dimensions of space, imposed by our poor spatial imagination, in order to arrive at the formulation of Riemannian geometry.

A Riemann space, or continuum, is the set of points characterized by the assignment of \(n\) coordinates \(x_1, x_2, \ldots x_n\), by varying which within known limits (in the particular case from \(-\infty\) to \(+\infty\)) we obtain all the points of the continuum, and, in addition, by the assignment of \(n^2\) functions \(g_{ik}\) \((i, k = 1, 2, \ldots, n)\) of these \(n\) coordinates, under the condition that \(g_{ik}=g_{ki}\) for all \(i\) and \(k\), and that the determinant

\[ g= \begin{vmatrix} g_{11} & g_{12} & \cdots & g_{1n}\\ g_{21} & g_{22} & \cdots & g_{2n}\\ \cdots & \cdots & \cdots & \cdots\\ g_{n1} & g_{n2} & \cdots & g_{nn} \end{vmatrix} \tag{5} \]

does not vanish at any point of the continuum. A continuous sequence of points joining two points \(A\) and \(B\) is called a line; the length of the line is computed in the form of the integral taken along it

\[ \int_A^B \sqrt{\sum_{i,k=1}^{n} g_{ik}\, dx_i\, dx_k}. \]

Thus the distance \(ds\) between two infinitely close points \((x_1, x_2, \ldots, x_n)\) and \((x_1+dx_1,\ x_2+dx_2,\ \ldots,\ x_n+dx_n)\) is, by definition, given by the formula (metric)

\[ ds^2=\sum_{i,k=1}^{n} g_{ik}\, dx_i\, dx_k, \tag{6} \]

whereby the formula

\[ g'_{ik}=\sum_{j,l=1}^{n} g_{jl}\, \frac{\partial x_j}{\partial x'_i} \frac{\partial x_l}{\partial x'_k} \]

determines those coefficients \(g'_{ik}\) which take the place of the coefficients \(g_{ik}\) when, from the coordinates \(x_1, x_2, \ldots x_n\), we pass to another system of coordinates \(x'_1, x'_2, \ldots, x'_n\). In the particular case when it is possible to introduce such a transformation of coordinates that \(g'_{ik}=0\) for \(i\ne k\) and \(g'_{ik}=1\) for \(i=k\), the space with the metric (6) is called \(n\)-dimensional Euclidean space. But this is not always possible, and Euclidean geometry turns out to be

only a small special case of the much more comprehensive geometry of Riemann.

Before proceeding to the formulation of the laws of the theory of gravitation and the law of the propagation of light in the general theory of relativity, we shall derive one theorem, which we shall have to use later. To this end let us consider the problem: in an \(n\)-dimensional Riemannian space with metric determination (6), two points \(A\) and \(B\) are given; it is required to join them by the shortest line (shortest from the point of view of the laws of the geometry of this space, i.e. by such a line that the integral

\[ \int_A^B ds=\int_A^B \sqrt{\sum_{i,k=1}^{n} g_{ik}\,dx_i\,dx_k} \]

take the smallest possible value for it). This condition will be satisfied only by those lines for which

\[ \delta \int_A^B ds=0, \tag{7} \]

i.e. whose length (to within infinitesimals of the second order) is equal to the length of any other line infinitely close to them and joining the same points \(A\) and \(B\). Lines satisfying condition (7) are called geodesic lines. Let us derive the differential equations of a geodesic line. We have

\[ \delta(ds)=\delta\sqrt{\sum_{i,k=1}^{n} g_{ik}\,dx_i\,dx_k} = \frac{1}{2}\,\delta\, \frac{\sum_{i,k=1}^{n} g_{ik}\,dx_i\,dx_k} {\sqrt{\sum_{i,k=1}^{n} g_{ik}\,dx_i\,dx_k}} = \]

\[ = \frac{1}{2ds}\sum_{i,k=1}^{n} dx_i\,dx_k\,\delta g_{ik} + \frac{1}{2ds}\sum_{i,k=1}^{n} g_{ik}\,dx_i\,\delta dx_k + \frac{1}{2ds}\sum_{i,k=1}^{n} g_{ik}\,dx_k\,\delta dx_i = \]

\[ = \frac{1}{2}\sum_{i,k=1}^{n} \left\{ \frac{dx_i}{ds}\frac{dx_k}{ds}\,\delta g_{ik} + g_{ik}\frac{dx_i}{ds}\frac{d}{ds}(\delta x_k) + \right. \]

\[ +g_{ik}\frac{dx_k}{ds}\frac{d}{ds}(\delta x_i)\biggr\}\,ds = \]

\[ = \frac12 \left\{ \sum_{i,j,k=1}^{n}\frac{dx_i}{ds}\frac{dx_k}{ds}\frac{\partial g_{ik}}{\partial x_j}\,\delta x_j +\sum_{k=1}^{n}\left(\sum_{i=1}^{n} g_{ik}\frac{dx_i}{ds} +\sum_{j=1}^{n} g_{jk}\frac{dx_j}{ds}\right)\frac{d}{ds}(\delta x_k) \right\}\,ds . \]

Therefore the condition for a geodesic line will take the form

\[ 0=\delta\int_A^B ds=\int_A^B \delta(ds) =\frac12\int_A^B \sum_{i,j,k=1}^{n}\frac{dx_i}{ds}\frac{dx_k}{ds}\frac{\partial g_{ik}}{\partial x_j}\,\delta x_j\,ds+ \]

\[ +\frac12\int_A^B \sum_{j=1}^{n}\left(\sum_{i=1}^{n} g_{ij}\frac{dx_i}{ds} +\sum_{k=1}^{n} g_{jk}\frac{dx_k}{ds}\right)\frac{d}{ds}(\delta x_j)\,ds . \]

Let us compute the last integral by parts, keeping in mind that the curve being sought is compared with curves infinitely close to it that pass through the same points \(A\) and \(B\); in other words, \(\delta x_j\) vanishes at the beginning and at the end of the path of integration. Therefore integration by parts gives

\[ 0=\delta\int_A^B ds=\frac12\int_A^B \sum_{j=1}^{n}\left\{ \sum_{i,k=1}^{n}\frac{dx_i}{ds}\frac{dx_k}{ds}\frac{\partial g_{ik}}{\partial x_j} - \frac{d}{ds}\left(\sum_{i=1}^{n} g_{ij}\frac{dx_i}{ds} +\sum_{k=1}^{n} g_{jk}\frac{dx_k}{ds}\right) \right\}\delta x_j\,ds . \]

Since the variations \(\delta x_j\) are independent of one another, for the integral to vanish it is required that the coefficients of the variations \(\delta x_j\) vanish at all points of the path of integration; in other words, for all values of \(j\) it must be

\[ \sum_{i,k=1}^{n}\frac{dx_i}{ds}\frac{dx_k}{ds}\frac{\partial g_{ik}}{\partial x_j} -\sum_{i=1}^{n}\frac{dg_{ij}}{ds}\frac{dx_i}{ds} -\sum_{i=1}^{n} g_{ij}\frac{d^2x_i}{ds^2} - \]

\[ -\sum_{k=1}^{n}\frac{dg_{jk}}{ds}\frac{dx_k}{ds} -\sum_{k=1}^{n} g_{jk}\frac{d^2x_k}{ds^2}=0 . \]

The last sum, as is not difficult to see, is equal to one third of the end; therefore one may write

\[ -\frac{1}{2}\sum_{i,k=1}^{n}\frac{dx_i}{ds}\frac{dx_k}{ds} \left( \frac{\partial g_{ik}}{\partial x_j} -\frac{\partial g_{ij}}{\partial x_k} -\frac{\partial g_{jk}}{\partial x_i} \right) + \sum_{i=1}^{n} g_{ij}\frac{d^2 x_i}{ds^2}=0, \tag{8} \]

if one makes use of the obvious equalities

\[ \frac{d g_{ij}}{ds} = \sum_{k=1}^{n}\frac{\partial g_{ij}}{\partial x_k}\frac{dx_k}{ds}, \qquad \frac{d g_{jk}}{ds} = \sum_{i=1}^{n}\frac{\partial g_{jk}}{\partial x_i}\frac{dx_i}{ds}. \]

Let us now introduce the following notation: denote by \(g^{ik}\) (with the indices \(i\) and \(k\) above) the algebraic cofactor of the element \(g_{ik}\) in the determinant \(g\) (formula 5), divided by this same determinant. From elementary properties of determinants it follows that

\[ \sum_{j=1}^{n} g^{lj} g_{ij} = \begin{cases} 1 & \text{when } i=l,\\ 0 & \text{when } i\ne l. \end{cases} \]

If, as is customary in physics, we denote by the symbol \(\delta_{il}\) the number equal to one when \(i=l\) and to zero when \(i\ne l\), then we obtain

\[ \sum_{j=1}^{n} g^{lj} g_{ij}=\delta_{il}. \tag{9} \]

Multiply both sides of formula (8) by \(g^{lj}\) and sum over \(j\) \((=1,2,\ldots,n)\). With the aid of formula (9) we obtain

\[ -\frac{1}{2}\sum_{i,k=1}^{n}\frac{dx_i}{ds}\frac{dx_k}{ds} \sum_{j=1}^{n} g^{lj} \left( \frac{\partial g_{ik}}{\partial x_j} -\frac{\partial g_{ij}}{\partial x_k} -\frac{\partial g_{jk}}{\partial x_i} \right) + \sum_{i=1}^{n}\delta_{il}\frac{d^2x_i}{ds^2}=0. \]

In the last sum all terms vanish except the one in which \(i=l\) (by the property of the symbol \(\delta_{il}\)). Taking this into account and introducing the notation

\[ \left\{ {ik \atop l} \right\} = \frac{1}{2}\sum_{j=1}^{n} g^{lj} \left( \frac{\partial g_{jk}}{\partial x_i} + \frac{\partial g_{ij}}{\partial x_k} - \frac{\partial g_{ik}}{\partial x_j} \right), \tag{10} \]

we obtain the final equations of a geodesic line

\[ \frac{d^2 x_l}{ds^2}+\sum_{i,k=1}^{n}\left\{\begin{array}{c} ik\\ l \end{array}\right\}\frac{dx_i}{ds}\frac{dx_k}{ds}=0, \tag{11} \]

in which the symbols \(\left\{\begin{array}{c}ik\\ l\end{array}\right\}\), called Christoffel brackets, are determined by formula (10). The Christoffel brackets are, as is evident from (10), functions of the coordinates \(x_1, x_2,\ldots,x_n\), which can be calculated if only the functions \(g_{ik}\) of the same coordinates are known. Since the equality

\[ \left\{\begin{array}{c} ik\\ l \end{array}\right\} = \left\{\begin{array}{c} ki\\ l \end{array}\right\} \]

must always hold, which is not difficult to verify from formula (10), the number of mutually independent Christoffel brackets in a space of \(n\) dimensions must be equal to \(\frac{n^2(n+1)}{2}\) (the product of the number of possible values of the index \(l\) by the number \(n+\frac{n(n-1)}{2}\) of possible combinations of the numbers \(1,2,\ldots,n\) with repetitions). Thus, for example, in the case \(n=4\), which is significant for the general theory of relativity (the four-dimensional continuum), we have \(\frac{4^2\cdot 5}{2}=40\) independent Christoffel brackets.

§ 3. The fundamental equations and the cosmological problem. The solution of Einstein and de Sitter.

Having armed ourselves with this minimum of necessary information from differential geometry, let us again turn away from mathematical abstractions in order to come into contact with reality. The world of things and events with which the physicist deals is a four-dimensional continuum. Indeed, in order to characterize any physical phenomenon it is necessary to indicate what exactly occurred at each point of space and at each moment of time.

Each point of space is characterized by three coordinates—space is three-dimensional, as everyday experience teaches us; each moment of time is characterized by one coordinate (the reading of a clock, or, what is the same,

by a time interval elapsed from some quite definite moment, e.g., from noon). The totality of space and time is the form of existence of matter; in order to characterize any physical phenomenon, all the physical quantities characterizing it must be known at every point of this four-dimensional background, i.e., as functions of the four coordinates \(x_1, x_2, x_3, x_4\), of which one is time, measured by some quite definite observer, and the other three are the spatial coordinates measured by the same observer (by an observer* in the theory of relativity is meant a definite coordinate system, i.e., a definite way of measuring these four coordinates). At the same time the theory of relativity postulates: there also exist (in the given coordinate system) quantities \(g_{ik}\) \((i, k = 1, 2, 3, 4)\), satisfying the condition of symmetry \((g_{ik} = g_{ki})\) and of non-vanishing determinant (5) (where \(n = 4\)), which possess the following properties: \(1^\circ\), if in the expression

\[ ds^2 = \sum_{i,k=1}^{n} g_{ik}\,dx_i\,dx_k \]

one sets equal to zero the differential of that coordinate (e.g., \(x_4\)) which, for the given observer, denotes time, and, moreover, changes the sign, then one obtains the expression

\[ ds^2 = - \sum_{i,k=1}^{3} g_{ik}\,dx_i\,dx_k, \]

which determines the geometry of space from the point of view of this observer (i.e., determines for the given observer the behavior of the scales used for determining lengths, volumes, angles, etc.); \(2^\circ\), if a light signal, sent from the point with spatial coordinates \(x_1, x_2, x_3\) at the time \(x_4\), was received at the time \(x_4 + dx_4\) at the point \(x_1 + dx_1\), \(x_2 + dx_2\), \(x_3 + dx_3\), then

\[ ds^2 = \sum_{i,k=1}^{4} g_{ik}\,dx_i\,dx_k = 0; \]

\(3^\circ\), upon passing from one observer to another, i.e., upon transition

\[ \text{* See the notes at the end of the article. Ed.} \]

from one coordinate system \((x_1, x_2, x_3, x_4)\) to another \((x_1', x_2', x_3', x_4')\), the functions \(g_{ik}\) are replaced by such \(g'_{ik}\) that

\[ \sum_{i,k=1}^{4} g_{ik}\,dx_i\,dx_k = \sum_{i,k=1}^{4} g'_{ik}\,dx_i'\,dx_k', \]

i.e. the quantity \(ds^2\) remains unchanged (the quantity \(ds\), representing the most important invariant of the theory of relativity, is called the interval between the events \(x_1, x_2, x_3, x_4\) and \(x_1+dx_1, x_2+dx_2, x_3+dx_3, x_4+dx_4\)). \(4^\circ\), if an observer using material coordinates has established such a coordinate system in which, at a given point, all \(g_{ik}\) with \(i \ne k\) vanish, then of the remaining \(g_{ik}\), i.e. of the numbers \(g_{11}, g_{22}, g_{33}, g_{44}\), three will have one sign, and the fourth (which is the coefficient of the square of the time differential) the other sign; \(5^\circ\), bodies not charged with electricity (or, more precisely, bodies not subject to the action of electromagnetic forces) move in such a way that the line connecting their successive positions on the four-dimensional space-time background turns out to be a geodesic line \(\delta\int ds=0\); the same applies also to the motion of light particles (photons), i.e. to the propagation of electromagnetic oscillations along the lines of light rays.

The claims of all these postulates, as the reader can see, are very great; they determine the geometry of space (postulate 1), the velocity of propagation of light signals (postulate 2), and thereby also the behavior of clocks (since in physics, for the ideal comparison of two clocks located in different places, the following condition is used: if a light signal is sent from point A to point B, is reflected at point B, and returns to point A, then the time of reception of the signal at point B, as measured by the clock at point B, must be the arithmetic mean of the instants of sending the signal from A and of its return reception, measured by the clock at point A). In addition, the 5th of the listed postulates, determining the laws of motion of bodies on which no forces act, consequently gives the field of inertia of the four-dimensional background under consideration. The simplest case of application of the listed postulates

takes place when it is possible to introduce such a coordinate system \(x, y, z, t\) that

\[ ds^{2}=dx^{2}-dy^{2}-dz^{2}+c^{2}dt^{2}, \tag{12} \]

where \(c\) is a positive constant. In this case \(ds^{2}\), obtained for space according to the 1st postulate, has the form \(dx^{2}+dy^{2}+dz^{2}\); consequently, the behavior of measuring rods in this space corresponds to the ordinary geometry of Euclid; according to the 2nd postulate the speed of light is equal to \(c\), since, in order that a light signal sent from the point \(x,y,z\) reach the point \(x+dx, y+dy, z+dz\) in the time interval \(dt\), the condition is required

\[ \sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}+\left(\frac{dz}{dt}\right)^{2}}=c; \]

the 4th postulate is satisfied; the 5th postulate requires that the trajectory of a body on which no forces act be determined by the equations

\[ \frac{d^{2}x}{ds^{2}}=\frac{d^{2}y}{ds^{2}}=\frac{d^{2}z}{ds^{2}}=\frac{d^{2}t}{ds^{2}}=0 \]

(since all Christoffel brackets vanish, see formulas 10 and 11), whence it follows that \(x,y,z\) and \(t\) are linear functions of \(s\); eliminating \(s\), we find that \(x,y\) and \(z\) are linear functions of \(t\), i.e. a body not subjected to the action of forces moves uniformly and rectilinearly (Newton’s first law); finally, from the same 5th postulate it follows that rays of light are straight lines. Thus, everything is remarkably simple in the case when the four-dimensional continuum (the four-dimensional world, as physicists say) possesses, at least in some part of it, the metric (12); but there is no necessity that the metric should actually be such; in particular, experience shows that near large masses (e.g., near the earth or celestial bodies) the metric of the four-dimensional world differs essentially from (12); bodies on which no forces act no longer obey Newton’s first law in the form

\[ \frac{d^{2}x}{ds^{2}}=\frac{d^{2}y}{ds^{2}}=\frac{d^{2}z}{ds^{2}}=\frac{d^{2}t}{ds^{2}}=0; \]

instead, they move according to law (11), where one must put \(n=4\), \(x_{1}=x\), \(x_{2}=y\), \(x_{3}=z\), \(x_{4}=t\). In this case physics said that a “force of gravitation” acts on these bodies, proceed—

...depending on the aforementioned large masses; in the terminology of relativistic physics, these bodies move by inertia, but the field of inertia is modified by the nearby presence of massive bodies. The law satisfied by the field of inertia \(g_{ik}\) in the presence of material bodies and radiant energy is called the law of gravitation; we shall now proceed to formulate this law, noting in advance that, although it follows from this law that the measure-determination (12) is possible only in the absence of material bodies, the converse conclusion by no means follows—namely, that in the absence of material bodies the measure-determination (12) must necessarily take place. This result, according to which gravitation [and by gravitation we usually mean the non-coincidence of the measure-determination of the four-dimensional world with its simplest form (12)] may manifest itself even in the absence of massive “attracting” bodies, is of enormous significance for relativistic cosmology.

In order to write the law of gravitation in the form in which it was discovered by Einstein, let us introduce the quantities \(G_{ik}\) \((i,k=1,2,3,4)\), called the “components of the contracted Riemann–Christoffel tensor.” The definition of these quantities is as follows:

\[ G_{ik} = -\sum_{j=1}^{4} \frac{\partial}{\partial x_j} \left\{ \begin{matrix} ik\\ j \end{matrix} \right\} + \sum_{j,l=1}^{4} \left\{ \begin{matrix} ij\\ l \end{matrix} \right\} \left\{ \begin{matrix} kl\\ j \end{matrix} \right\} + \frac{\partial^2}{\partial x_i \partial x_k} \log \sqrt{-g} - \sum_{j=1}^{4} \left\{ \begin{matrix} ik\\ j \end{matrix} \right\} \frac{\partial}{\partial x_j} \log \sqrt{-g}, \tag{13} \]

whence, incidentally, it is seen that \(G_{ik}=G_{ki}\). We shall not here enter into a detailed explanation of why precisely this complicated expression, and not some other, plays a role in the formulation of the law of gravitation; let us note only that an important property of this formulation is connected with it, namely that it does not depend on the special coordinate system chosen by one observer or another. Let us denote by the letter \(G\) the quantity

\[ G=\sum_{i,k=1}^{4} g^{ik}G_{ik}. \tag{14} \]

If we denote by the letter \(\rho\) the density at a given point of the four-dimensional world (including here not only the number of grams of matter in a cubic centimeter, but also the number of grams of radiant energy in the same unit of volume),** and by the letter \(p\) the pressure of the radiant energy, then Einstein’s law of gravitation, by which the dependence of the geometry of the world on the presence in it of material bodies and radiant energy is determined, will be expressed in the following form:

\[ G_{ik} - \frac{1}{2}Gg_{ik} + \]

\[ + \lambda g_{ik} + \varkappa\left( \sum_{\alpha=1}^{4}\sum_{\beta=1}^{4} g_{i\alpha}g_{k\beta}\left(\rho+\frac{p}{c^{2}}\right) \frac{dx_{\alpha}}{ds}\frac{dx_{\beta}}{ds} - g_{ik}\frac{p}{c^{2}} \right)=0, \tag{15} \]

where the quantities \(\dfrac{dx_{\alpha}}{ds}\) \((\alpha=1, 2, 3, 4)\) determine the velocity of motion of matter at the given point (if in the chosen coordinate system \(x_{1}=x,\ x_{2}=y,\ x_{3}=z,\ x_{4}=t\), then \(\dfrac{dx_{1}}{ds}:\dfrac{dx_{2}}{ds}:\dfrac{dx_{3}}{ds}:\dfrac{dx_{4}}{ds}=v_{x}:v_{y}:v_{z}:1\), where \(v_{x},v_{y},v_{z}\) are the components of the velocity of matter, and moreover \(\sum_{i,k=1}^{4} g_{ik}\dfrac{dx_i}{ds}\dfrac{dx_k}{ds}=1\)), the quantity \(\lambda\) is a certain universal constant whose value is still unknown,\(^1\) and by the letter \(\varkappa\) is denoted Newton’s gravitational constant multiplied by \(8\pi\) and divided by the square of the velocity of light \((\varkappa=1.87\times10^{-27}\ \mathrm{g}^{-1}\ \mathrm{cm})\). We shall not explain here why the law of gravitation (15) has precisely this form and not some other;\(^2\) all the details—

\(^1\) The constant \(\lambda\) is in any case very small: its probable value is of the order of magnitude \(10^{-54}\ \mathrm{cm}^{-2}\). If one assumes that \(\lambda\) is not zero, then the quantity \(\lambda^{-1/2}\) will represent a “natural” unit of length; in Eddington’s opinion (see his Mathematical Theory of Relativity), the existence of such a unit of length is necessary, since otherwise “the electron would not know what size it should be.”

\(^2\) The reader who compares formula (15) with that formula for the law of gravitation which is given, for example, by Einstein in his Kosmologische Betrachtungen (see the bibliography) will notice the difference between them, which is explained by the fact that Einstein considers the case in which radiant energy is absent, but the particles of matter move with the very same

information on this question the reader will find in any good course on the theory of relativity (for example, in Eddington, Weyl, or Laue). If we recall that in an isotropic field of radiant energy the density of the radiant energy is equal to three times its pressure, then we find that the quantity \(\rho\) can be represented in the form

\[ \rho=\rho_0+\frac{3p}{c^2}; \tag{16} \]

where \(\rho_0\) is the material density (the number of units of mass of matter in a unit volume), and \(\frac{3p}{c^2}\) is the number of units of mass of radiant energy in a unit volume. The quantities \(\rho\) and \(p\) at a given point of the four-dimensional world turn out to be the same for all possible observers, i.e. they do not depend on the coordinate system; then, as is proved in courses on the theory of relativity, the law of gravitation (15) likewise does not depend on the coordinate system; in other words, if it is satisfied in the form (15) for one observer, then in exactly the same form it will be satisfied for another observer as well, although the coordinates \(x_i\), and consequently also \(\frac{dx_i}{ds}\), and also all \(g_{ik}\) and \(G_{ik}\), will already be different (in this consists the well-known “requirement of covariance,” the despot of modern physics, to which every physical law claiming to be considered true must submit).

We are now in a position to approach directly the cosmological problem and those of its solutions which were known until recently. If we adopt that lofty point of view which was mentioned in § 1,

with varied and chaotically distributed velocities (the molecules of a liquid or gas): \(p\) in his formulas denotes the pressure of the liquid or gas (and not the pressure of radiant energy), while \(\frac{dx_\alpha}{ds}\) refers not to the actual velocities of the particles, but to the velocity of the center of gravity of a very small volume of matter, which nevertheless contains an enormous number of molecules; it should be noted, however, that concerning certain details of formula (15), as applied to the case where radiant energy is present, final agreement among physicists has not yet been reached.

and we shall assume that the world is filled with matter and radiation of uniform density, then our problem will consist in finding sixteen functions \(g_{ik}\) of four coordinates, satisfying the condition \(g_{ik}=g_{ki}\), the condition that the determinant \(g=\|g_{ik}\|\) is not equal to zero, and the fourth postulate, and which at the same time are solutions of equation (15) for certain constants \(\rho\) and \(p\). In the history of relativistic cosmology, the chief role has been played above all by three solutions, which de Sitter called solutions \(A\), \(B\), and \(C\), and of which the last is to some extent “trivial.” Let us consider these three solutions in order.

Solution \(A\) was given by Einstein [1]. Put \(x_1=\chi\), \(x_2=\vartheta\), \(x_3=\varphi\), \(x_4=t\), and consider the four-dimensional continuum in which

\[ g_{ik}=0 \quad \text{for } i\ne k,\quad g_{11}=-R^2,\quad g_{22}=-R^2\sin^2\chi, \]
\[ g_{33}=-R^2\sin^2\chi\sin^2\vartheta,\quad g_{44}=c^2, \]

where \(c\) is the velocity of light \((3\times10^{10}\ \text{cm/sec})\), and \(R\) is a certain constant quantity having the dimension of length, and where \(\chi\) and \(\vartheta\) vary from \(0\) to \(\pi\), and \(\varphi\) from \(0\) to \(2\pi\). In such a four-dimensional world \(ds^2\) is expressed by the formula

\[ ds^2=-R^2\left[d\chi^2+\sin^2\chi\left(d\vartheta^2+\sin^2\vartheta\,d\varphi^2\right)\right]+c^2dt^2. \tag{17} \]

Anyone who can, by straining his spatial imagination, think in terms of many-dimensional spaces will obtain a “visual” representation of the geometry of this four-dimensional world if he considers a five-dimensional Euclidean space with rectangular coordinates \(x_1,x_2,x_3,x_4,x_5\), and in it a four-dimensional right circular cylinder of radius \(R\), whose axis coincides with the \(x_1\)-axis. The equations of such a cylinder are expressed, by analogy with a two-dimensional cylindrical surface situated in three-dimensional space, as follows:

\[ \begin{aligned} x_1&=\xi_1\\ x_2&=R\cos\xi_2\\ x_3&=R\sin\xi_2\cos\xi_3\\ x_4&=R\sin\xi_2\sin\xi_3\cos\xi_4\\ x_5&=R\sin\xi_2\sin\xi_3\sin\xi_4, \end{aligned} \tag{18} \]

where \(\xi_1, \xi_2, \xi_3, \xi_4\) are four parameters, or coordinates, determining the position of a point on the cylinder. In order to obtain all possible points of the section of the cylinder by the plane \(x_1=\mathrm{const}\), it is necessary to vary \(\xi_2\) and \(\xi_3\) from 0 to \(\pi\), and \(\xi_4\) from 0 to \(2\pi\). Since the five-dimensional space was assumed to be Euclidean, we have

\[ ds^2=dx_1^2+dx_2^2+dx_3^2+dx_4^2+dx_5^2. \]

Substituting here the differentials of the coordinates \(x_i\), computed from formula (18), we shall see that \(ds^2\) on the surface of the cylinder is equal to

\[ ds^2=d\xi_1^2+R^2\left[d\xi_2^2+\sin^2\xi_2\left(d\xi_3^2+\sin^2\xi_3 d\xi_4^2\right)\right]. \]

If now we replace \(R\) by \(iR\) (where \(i\) is the imaginary unit) and put \(\xi_1=ct,\ \xi_2=\chi,\ \xi_3=\vartheta,\ \xi_4=\varphi\), then from this expression for \(ds^2\) we obtain formula (17). Thus (if we disregard the imaginariness) the four-dimensional world with metric (17) is a cylindrical world of radius \(R\). Here for the first time we encounter the radius of the world.

How must Einstein’s cylindrical world appear to an observer compelled to live in this world? First of all it turns out that if the radius of the world, i.e. \(R\), is very large, then the cylindrical form of the world makes itself felt in phenomena occurring in comparatively small parts of this world just as little as the spherical form of the earth makes itself felt in phenomena occurring within the limits of one room. Indeed, let us introduce, instead of the coordinate \(\chi\), another coordinate \(r\) by means of the relation

\[ \chi=\frac{r}{R}. \tag{19} \]

We obtain

\[ ds^2=-dr^2-r^2\left(\frac{\sin \frac{r}{R}}{\frac{r}{R}}\right)^2(d\vartheta^2+\sin^2\vartheta d\varphi^2)+c^2dt^2, \]

and if \(R\) is very large in comparison with \(r\), then one may approximately write

\[ ds^2=-dr^2-r^2(d\vartheta^2+\sin^2\vartheta d\varphi^2)+c^2dt^2. \]

\(*\)

But this last expression for \(ds^2\) can, by introducing new coordinates \(x=r\cos\vartheta,\ y=r\sin\vartheta\cos\varphi,\ z=r\sin\vartheta\sin\varphi\), be transformed to the form

\[ ds^2=-(dx^2+dy^2+dz^2)+c^2dt^2, \]

which, as we have seen, indicates that in the world everything is arranged very simply: light propagates along straight lines with velocity \(c\), bodies not subject to forces move rectilinearly and uniformly, and so on. Here \(x,y,z\) are rectilinear coordinates of space, and consequently \(r,\vartheta,\varphi\) are the ordinary polar coordinates familiar to everyone from analytic geometry. This explains the physical meaning of the coordinates \(\chi,\vartheta,\varphi\) in formulas (17): \(\vartheta\) and \(\varphi\) have their usual meaning, while \(\chi\) is the distance from the origin of coordinates, measured in units of length \(R\).

However, the matter is not so simple if the observer does not confine himself to measurements in the immediate neighborhood of the origin of coordinates, but considers more extensive parts of the world, or even the world as a whole. Then he will have to take into account that its three-dimensional space possesses not the line element

\[ ds^2=dr^2+r^2(\sin^2\vartheta\,d\varphi^2+d\vartheta^2), \]

but the more complicated one

\[ ds^2=R^2\{d\chi^2+\sin^2\chi\,(d\vartheta^2+\sin^2\vartheta\,d\varphi^2)\}. \]

If from the origin of coordinates, as from a center, a sphere of radius \(r=R\chi\) is drawn, then the magnitude of its surface will turn out to be not \(4\pi r^2\), but

\[ 4\pi R^2\sin^2\chi = 4\pi r^2 \left( \frac{\sin \frac{r}{R}}{\frac{r}{R}} \right)^2 . \]

The volume enclosed between spheres of radii \(R\chi\) and \(R(\chi+d\chi)\) will be equal to

\[ 4\pi R^2\sin^2\chi\cdot R\,d\chi . \]

Since, as \(\chi\) varies from \(0\) to \(\pi\), we exhaust all points of space, the total volume of three-dimensional space for an observer whose geometry of four-dimensional pro-

of the space-time world is described by formula (17), will be equal to

\[ V=\int_{0}^{\pi} 4\pi R^3 \sin \chi \, d\chi^2 = 2\pi^2 R^3 \tag{20} \]

This volume is therefore finite. The non-Euclidean geometry of such a three-dimensional space possesses very unusual properties in the eyes of anyone accustomed to imagine three-dimensional space as necessarily Euclidean and infinite; but in the case where the radius \(R\) is very large, paradoxes will arise only when considering phenomena on a very large scale: sufficiently small parts of the space with metric (17) practically do not differ in any way from parts of ordinary Euclidean space. An important property of solution \(A\) (which, of course, solutions \(B\) and \(C\) also possess) is that all points of the four-dimensional world with metric (17) are completely equivalent: from each of its points this world appears exactly the same as from every other point.

Let us now verify that the metric (17) indeed solves the cosmological problem: if we, successively applying formulas (10), (13), and (14), compute the expressions

\[ G_{ik} - \frac{1}{2} G g_{ik} + \lambda g_{ik} \]

for various combinations of \(i\) and \(k\), it turns out that for \(i\ne k\) they all vanish, while for \(i=k=1\) one obtains \(1-\lambda R^2\); for \(i=k=2\) to this there is added the further factor \(\sin^2\chi\); for \(i=k=3\), besides this, another factor \(\sin^2\vartheta\); and for \(i=k=4\) we obtain \(\lambda c^2-\dfrac{3c^2}{R^2}\). If we introduce the notation

\[ T_{ik}=-\frac{1}{\varkappa}G_{ik}+\frac{1}{2\varkappa}Gg_{ik}-\frac{\lambda}{\varkappa}g_{ik}, \tag{21} \]

then we obtain, accordingly, in Einstein’s cylindrical world,

\[ T_{11}=\frac{\lambda R^2-1}{\varkappa}, \qquad T_{22}=\frac{\lambda R^2-1}{\varkappa}\sin^2\chi, \]

\[ T_{33}=\frac{\lambda R^2-1}{\varkappa}\sin^2\chi\sin^2\vartheta,\quad T_{44}=\frac{3c^2}{\varkappa R^2}-\frac{\lambda c^2}{\varkappa},\quad T_{ik}=0 \text{ for } i\ne k. \]

Comparing with the law of gravitation (15), we find that it is indeed satisfied under the condition

\[ p=\frac{c^2}{\varkappa R^2}-\frac{\lambda c^2}{\varkappa},\quad \rho=\frac{3}{\varkappa R^2}-\frac{\lambda}{\varkappa},\quad \frac{dr}{ds}=\frac{d\vartheta}{ds}=\frac{d\varphi}{ds}=0, \]

\[ \frac{dt}{ds}=\frac{1}{c}, \tag{22} \]

whence, among other things, it follows that

\[ \rho_0=\frac{2\lambda}{\varkappa},\quad \rho-\rho_0=\frac{3p}{c^2} =\frac{3}{\varkappa}\left(\frac{1}{R^2}-\lambda\right). \tag{23} \]

We see from this that the four-dimensional world with line element (17) can indeed exist; moreover, matter and radiant energy are distributed in it with uniform density, and the matter is in relative rest. Multiplying the density \(\rho\) by the volume of the world \(V\), computed by formula (20), we find that the mass of the world is equal to

\[ M=\frac{6\pi^2 R}{\varkappa}-\frac{2\pi^2\lambda}{\varkappa}R^3, \tag{24} \]

and of this whole mass the share belonging to matter is

\[ M_0=\frac{4\lambda}{\varkappa}\pi^2 R^3, \]

while the share belonging to radiant energy is

\[ M-M_0=\frac{6\pi^2}{\varkappa}(R-\lambda R^3). \]

We see that, for given values of the universal constants \(\lambda\) and \(\varkappa\), the properties of the world are determined by its radius \(R\), and we can choose this radius so that the mass of the world is greatest.

Since

\[ \frac{dM}{dR}=\frac{6\pi^2}{\varkappa}(1-\lambda R^2),\quad \frac{d^2M}{dR^2}=-\frac{12\pi^2\lambda}{\varkappa}R, \]

then, for positive \(\lambda\) and \(\varkappa\), this problem is solved by the radius

\[ R=\frac{1}{\sqrt{\lambda}}, \tag{25} \]

which gives

\[ p=0,\quad \rho=\rho_0=\frac{2}{\varkappa R^2},\quad M=M_0=\frac{4\pi^2}{\varkappa}R, \tag{26} \]

in other words, in this case we have a world filled

matter with density \(\dfrac{2}{\chi R^2}\) and completely free of radiation. This special case, too, was considered by Einstein in his 1917 paper; but we see that, alongside this special case, the requirements of the cosmological problem are satisfied if we put \(R<\dfrac{1}{\sqrt{\lambda}}\) (if we put \(R>\dfrac{1}{\sqrt{\lambda}}\), then the pressure and density of radiant energy become negative, which, so far as we can judge, is impossible). The world will then contain, besides matter, also radiant energy, and its total mass will be determined by formula (24).

Let us suppose for a moment that the world, considered from the “high” point of view mentioned more than once, is in fact such as solution \(A\) depicts it. Taking

\[ \rho_0=\rho=2\times 10^{-28}\ \mathrm{g/cm^3}, \]

(see § 1), we shall find that

\[ \begin{aligned} R&=2.3\times 10^{27}\ \mathrm{cm}=2300\ A,\\ \lambda&=1.9\times 10^{-55}\ \mathrm{cm^{-2}},\\ M&=4.88\times 10^{55}\ \mathrm{g}. \end{aligned} \]

This, therefore, is the greatest mass that the universe can possess if it is constructed according to the type of solution \(A\). This mass exceeds the mass of the Sun by \(2.46\times 10^{22}\) times; assuming that each galaxy contains on average \(10^{11}\odot\), this gives \(2.5\times 10^{11}\) galaxies in the universe, of which only \(0.003\%\) lie within the visibility limits of our most powerful telescopes. Let us also note that if the universe is constructed according to type \(A\) and at the same time \(\rho=\rho_0=2\times 10^{-28}\ \mathrm{g/cm^3}\), then it cannot contain more than \(3\times 10^{79}\) protons.

Such was the first solution of the cosmological problem—

\(^1\) The figures given by us in the text are borrowed from contemporary conceptions of the universe as a system of galactic systems (§ 1). In Einstein’s paper [1] there were no numerical data; the founder of the theory of relativity avoided them, either considering all possible estimates of the dimensions of the world too flimsy speculations, or leaving this part of the task to specialist astronomers. But in de Sitter’s work—

... which was given by the theory of relativity; irrespective of the question of whether this solution may be considered acceptable from the astronomical point of view or not, it will forever remain an unsurpassed monument to the boundless boldness of the human mind.

In the same year 1917, in which Einstein gave the just-described solution \(A\) of the cosmological problem, the Dutch astronomer de Sitter showed [2] that there also exists another solution, called by him solution \(B\), which also satisfies the fundamental equations (15) under the condition that \(\rho\) and \(p\) are constant in space.

De Sitter’s solution is a world with the metric

\[ ds^{2}=c^{2}\cos^{2}\chi\,dt^{2}-R^{2}\left[d\chi^{2}+\sin^{2}\chi\left(d\vartheta^{2}+\sin\vartheta\,d\varphi^{2}\right)\right]. \tag{27} \]

In order to see the “visual” geometrical meaning of this metric, let us consider in five-dimensional Euclidean space with rectangular coordinates \(x_1, x_2, x_3, x_4, x_5\) a four-dimensional spherical surface of radius \(R\). Its equations in parametric form will be

\[ \begin{aligned} x_1&=R\cos\xi_1,\\ x_2&=R\sin\xi_1\cos\xi_2,\\ x_3&=R\sin\xi_1\sin\xi_2\cos\xi_3,\\ x_4&=R\sin\xi_1\sin\xi_2\sin\xi_3\cos\xi_4,\\ x_5&=R\sin\xi_1\sin\xi_2\sin\xi_3\sin\xi_4. \end{aligned} \]

* In the de Sitter paper written in that same year, 1917, [2], one can find estimates of the radius of the world which correspond to solution \(A\): thus, for example, taking \(\rho=\rho_0=6\times10^{-24}\ \mathrm{g/cm^3}\), which corresponds to the density of stars in the center of the galactic system found by Kapteyn (80 stars per 1000 cubic parsecs), he obtains \(R=1.35\times10^{25}\ \mathrm{cm}\). Making the hypothesis, which he regarded as only “very probable,” that beyond our galaxy there is a large number of other galaxies, and assuming that the mass of each of them is equal to \(\frac{1}{3}\times10^{10}\odot\), and that the mean distance between two neighboring galaxies is \(1.5\times10^{23}\ \mathrm{cm}\), de Sitter obtains the mean density of matter in the universe \(2\times10^{-27}\ \mathrm{g/cm^3}\) and the corresponding radius \(7.5\times10^{26}\ \mathrm{cm}\). In doing so he makes the reservation that this radius should be regarded as an upper limit of the possible radius of the world, since it is quite possible that, besides the masses concentrated in galaxies, there are also other masses in the universe, the inclusion of which should give a larger mean density and hence a smaller radius of the world.

From them it follows that

\[ ds^2=dx_1^2+dx_2^2+dx_3^2+dx_4^2+dx_5^2= \]
\[ =R^2\{d\xi_1^2+\sin^2\xi_1[d\xi_2^2+\sin^2\xi_2(d\xi_3^2+\sin^2\xi_3d\xi_4^2)]\}. \]

Let us introduce

\[ \sin\chi=\sin\xi_1\sin\xi_2, \]
\[ \operatorname{tgh}\frac{ct}{R}=i\tg\xi_1\cos\xi_2,\quad \vartheta=\xi_3,\quad \varphi=\xi_4, \]

then

\[ d\chi=\frac{\cos\xi_1\sin\xi_2d\xi_1+\sin\xi_1\cos\xi_2d\xi_2} {\sqrt{1-\sin^2\xi_1\sin^2\xi_2}}, \]

\[ \frac{c}{R}dt=i\,\frac{\cos\xi_2d\xi_1-\sin\xi_1\cos\xi_1\sin\xi_2d\xi_2} {\cos^2\xi_1+\sin^2\xi_1\cos^2\xi_2}. \]

Elementary calculations, which any of our readers can carry out, give

\[ R^2d\chi^2-c^2\cos^2\chi\,dt^2 =R^2(d\xi_1^2+\sin^2\xi_1d\xi_2^2). \]

Replacing \(R\) by \(iR\) and \(c\) by \(ic\), we obtain precisely the line element (27). Thus the four-dimensional world with the line element (27) is a spherical world.

Since on the surface of the sphere \(\xi_1,\xi_2,\xi_3\) varied from \(0\) to \(\pi\), and \(\xi_4\) from \(0\) to \(2\pi\), \(\chi\) and \(\vartheta\) must vary from \(0\) to \(\pi\), \(\varphi\) from \(0\) to \(2\pi\), and \(t\) from \(-\infty\) to \(+\infty\).

If we put \(r=\chi R\), then the line element (27) assumes the form

\[ ds^2=c^2\cos^2\frac{r}{R}\,dt^2-dr^2-R^2\sin^2\frac{r}{R}(d\vartheta^2+\sin^2\vartheta d\varphi^2), \]

from which it is clear that the de Sitter world too, like the Einstein world, at distances from the origin very small in comparison with the radius \(R\), is practically indistinguishable from the world with line element \(ds^2=c^2dt^2-(dx^2+dy^2+dz^2)\). The coordinates \(\chi,\vartheta,\varphi,t\) have the same physical meaning as in the Einstein world (\(\chi\) is the distance from the origin, measured in units of \(R\); \(\vartheta\) and \(\varphi\) are polar angles, \(t\) is time).

At the distance \(r=\frac{\pi}{2}R\) from the origin, \(ds^2\) assumes the form

\[ ds^2=0\cdot dt^2-R^2[d\chi^2+d\vartheta^2+\sin^2\vartheta d\varphi], \]

i.e., the determinant \(g=\|g_{ik}\|\) takes the value zero, and in the expression \(ds^2\) not only the coefficients of \(dx^2\) do not disappear,

\(d\vartheta^2\) and \(d\varphi^2\) (as if the four-dimensional world had turned into a three-dimensional one). Other paradoxes are connected with this as well: if at a distance \(R\chi\) from an observer located at the origin of coordinates \((\chi=0)\) there are stationary \((d\chi=d\vartheta=d\varphi=0)\) clocks, then for these clocks we have

\[ ds=c\cdot\cos\chi\cdot dt, \]

and since, from the point of view of the theory of relativity, a clock is an instrument for measuring the interval, \(ds\), and its reading is equal to the interval divided by the constant factor \(c\), it follows that the time interval \(d\tau\), counted by these clocks, is equal to

\[ d\tau=\cos\chi\cdot dt, \]

where \(t\) is the time measured by the observer’s clock (i.e. by a clock located at the origin of coordinates). The farther the clock is from the observer, the more slowly it runs from his point of view! Finally, at \(\chi=\frac{\pi}{2}\), i.e. at a distance \(\frac{\pi}{2}R\) from the observer, we must have \(d\tau=0\), i.e. clocks located at such a distance from him appear to him to be stopped, and consequently all physical processes developing in time appear, from this distance, to have come to a halt. In fact, however, the observer will not be able to see anything from such a distance, no matter how powerful the optical means at his disposal: if we recall that for a ray of light directed toward the observer \((d\varphi=d\vartheta=0)\), the speed of light is determined by the condition \(ds^2 \equiv c^2\cos^2\chi dt^2-Rd^2\chi=0\), then it turns out that the interval of time necessary for a light signal to reach the observer from a distance \(r\) is equal to

\[ \int_{0}^{\frac{r}{R}}\frac{R}{c}\frac{d\chi}{\cos\chi} = \frac{R}{c}\log\sqrt{\frac{1+\operatorname{tg}\frac{r}{2R}}{1-\operatorname{tg}\frac{r}{2R}}}. \]

For \(r=\frac{\pi}{2}R\) this interval of time proves to be infinite: letters addressed to a point situated at a distance \(\frac{\pi}{2}R\) from the nearest post office, in the de Sitter world ...

Sitters never reach their destination, even if the mail carries them with the speed of light (i.e. carries them by radio). To an observer located at the origin of the coordinates, the universe appears bounded on all sides by a “horizon” (as Weyl called it), situated at a distance \(\frac{\pi}{2}R\). On this “horizon” time has ceased to flow; ancient Chronos, having stopped, has kept his scythe, and nature has frozen in eternal repose. Light rays, the swiftest messengers in the world, never manage to reach us from the horizon; still less do we have any possibility ever of learning what is happening beyond it: the second half of the de Sitter world is hidden from the observer still more secretly than the mysterious far side of the Moon is hidden from the curious gaze of the astronomer. Our physical world is bounded on all sides by the horizon \(\chi=\frac{\pi}{2}\), like the world of Homer’s heroes, passing into infinity as an ocean. But unlike Homer’s ocean, the horizon of the de Sitter world proves not only to be in principle unattainable, but even, in general, to be an illusion of each individual observer: all points of the spherical world are equivalent to one another; the space of each observer is surrounded by the same horizon, and if we move, wishing to get closer to the horizon, it will recede from us in exactly the same way as a rainbow recedes from a child trying to approach it. De Sitter’s space-time is completely homogeneous, like Euclidean space and like Einstein’s cylindrical space-time.

Let us now verify that de Sitter’s spherical world is indeed one of the solutions of the cosmological problem. For this, as before, we compute the expressions \(T_{ik}\) by formula (21). The calculation gives (with the numbering \(x_1=\chi,\ x_2=\vartheta,\ x_3=\varphi,\ x_4=t\))

\[ T_{11}=\frac{\lambda R^2-3}{\chi},\quad T_{22}=\sin^2\chi\,\frac{\lambda R^2-3}{\chi},\quad T_{33}=\sin^2\chi\sin^2\vartheta\,\frac{\lambda R^2-3}{\chi}, \]

\[ T_{44}=\frac{c^2\cos^2\chi}{\chi}\left(\frac{3}{R^2}-\lambda\right),\quad T_{ik}=0\ \text{for } i\ne k. \]

Comparing with formula (15), we find

\[ \frac{d\chi}{ds}=\frac{d\vartheta}{ds}=\frac{d\varphi}{ds}=0,\quad p=\frac{c^{2}}{\varkappa}\left(\lambda-\frac{3}{R^{2}}\right), \]

\[ \rho=\frac{p}{c^{2}}(\operatorname{tg}^{2}\chi-1),\quad c\cos\chi\,\frac{dt}{ds}=1, \]

therefore a world with the metric distribution (27) will solve the cosmological problem, i.e., will give uniform density and pressure, only under the condition

\[ R=\sqrt{\frac{3}{\lambda}}. \tag{28} \]

Then one obtains

\[ p=\rho=0, \]

i.e. the de Sitter world is free of matter and of radiant energy. Hence it is evident that the de Sitter solution can claim applicability to the real astronomical world only in the case where the average density of matter in the universe is not equal to \(2\times 10^{-28}\ \mathrm{g/cm^{3}}\), as it appears to an observer equipped with a hundred-inch telescope, but is practically zero (this will be true if the amount of matter in the universe is finite; for example, if beyond the part of the world accessible to present-day observations the cluster of galaxies soon comes to an end, and the radius of the world \(R\) is at the same time so large that the quotient obtained by dividing the mass of the world by its volume is a vanishingly small quantity). Thus the de Sitter world represents an ideal limiting case; the question can only be whether the properties of the de Sitter world may be used as a first approximation to the properties of the actual world. To what extent this can be done, the reader will learn from the following paragraph.

Let us compare certain properties of the Einstein world and of the de Sitter world: 1. The Einstein world admits an infinite multitude of variations; for the same values of the constants \(\varkappa\) and \(\lambda\) it can possess different radii, and in connection with this also different masses and different ratios of the material part of the universe to its radiant part.

1. The de Sitter world, for given \(x\) and \(\lambda\), can be only one; its radius is determined by formula (28); its mass is equal to zero; there is neither matter nor radiant energy in it.

2. The Einstein world, in contrast to the de Sitter world, cannot exist while empty: if (see formula 24) its mass \(M\) is equal to zero, then the radius \(R\) satisfies a cubic equation with roots \(0\), \(\sqrt{\dfrac{3}{\lambda}}\) and \(-\sqrt{\dfrac{3}{\lambda}}\). But since the radius of the Einstein world must necessarily lie in the interval from \(0\) to \(\dfrac{1}{\sqrt{\lambda}}\) (inclusive), only the root \(R=0\) remains, which corresponds to the nonexistence of space. In this, according to Laux, is contained “the sharpest formulation of the idea that all space-time relations, and hence all physical phenomena in general, are conditioned by the existence of material bodies.” In the de Sitter world a nonzero \(ds^2\), and consequently also the inertial field, exists without the presence of material bodies. This is why de Sitter pointed out that the basis of Einstein’s solution (solution \(A\)) is Mach’s “material postulate of inertia,” according to which the very existence of inertia is conditioned by the presence of material bodies, whereas in de Sitter’s solution (solution \(B\)) this Mach postulate is replaced by a “mathematical postulate of inertia.”

Above we mentioned the existence, alongside the hypotheses \(A\) and \(B\), of a third cosmological hypothesis \(C\), which we called “trivial.” This is the limiting case which is obtained both from Einstein’s solution

\[ ds^2 = c^2 dt^2 - dr^2 - r^2 \left( \frac{\sin \dfrac{r}{R}}{\dfrac{r}{R}} \right)^2 (d\vartheta^2 + \sin^2 \vartheta\, d\varphi^2) \]

and from de Sitter’s solution

\[ ds^2 = c^2 \cos^2 \frac{r}{R}\, dt^2 - r^2 \left( \frac{\sin \dfrac{r}{R}}{\dfrac{r}{R}} \right)^2 (d\vartheta^2 + \sin^2 \vartheta\, d\varphi^2), \]

if we put \(R=\infty\). The geometry corresponding to solution \(C\) is determined by the square of the element of length

\[ ds^2=c^2dt^2-(dx^2+dy^2+dz^2). \]

This is the ordinary space-time of the special theory of relativity, the four-dimensional world of Minkowski. Some of its properties have already been formulated above. If in formulas (22) we put \(R=\infty\), it turns out that

\[ p=-\frac{\lambda c^3}{\varkappa},\quad \rho=-\frac{\lambda}{\varkappa}; \]

and since \(\rho\) cannot be negative, one must admit that

\[ \lambda=0,\ ^1\quad p=0,\quad \rho=0; \]

the same also follows from the de Sitter solution if \(R\) is made infinite (see formula 28). We shall see below that solution \(C\), possessing none of the merits of solutions \(A\) and \(B\), possesses all of their shortcomings.

§ 4. Some astronomical consequences of the de Sitter and Einstein solutions.

Let us place in the Einstein world and in the de Sitter world a “test body,” i.e. a body of such small mass that its displacement from one place to another will not cause or disturb the homogeneity of the world. For brevity we shall call such a test body a comet. The equations of motion of the comet in the case when no forces act upon it are determined by formula (11). In the case of the Einstein world this will be

\[ \left. \begin{aligned} \frac{d^2\chi}{ds^2} &-\sin\chi\cos\chi\left[\left(\frac{d\vartheta}{ds}\right)^2+\sin^2\vartheta\left(\frac{d\varphi}{ds}\right)^2\right]=0,\\ \frac{d^2\vartheta}{ds^2} &+2\operatorname{cotg}\chi\,\frac{d\vartheta}{ds}\frac{d\chi}{ds} -\sin\vartheta\cos\vartheta\left(\frac{d\varphi}{ds}\right)^2=0,\\ \frac{d^2\varphi}{ds^2} &+2\operatorname{cotg}\chi\,\frac{d\chi}{ds}\frac{d\varphi}{ds} +2\operatorname{cotg}\vartheta\,\frac{d\vartheta}{ds}\frac{d\varphi}{ds}=0,\\ \frac{d^2t}{ds^2}&=0; \end{aligned} \right\} \tag{29} \]

\(^1\) The question may arise why \(\lambda\) cannot be a negative number. The question is perfectly legitimate. However, if \(\lambda<0\) when \(R=\infty\), then \(\rho-\dfrac{3p}{c^2}=\dfrac{2\lambda}{\varkappa}<0\) (the density of ordinary matter is negative), which is why this solution must also be rejected.

In the case of the de Sitter world,

\[ \left. \begin{aligned} &\frac{d^{2}\chi}{ds^{2}} -\sin\chi\cos\chi\left[\left(\frac{d\vartheta}{ds}\right)^{2} +\sin^{2}\vartheta\left(\frac{d\varphi}{ds}\right)^{2}\right] -\frac{c^{2}}{R^{2}}\cos\chi\sin\chi\left(\frac{dt}{ds}\right)^{2}=0,\\ &\frac{d^{2}\vartheta}{ds^{2}} +2\cotg\chi\,\frac{d\vartheta}{ds}\frac{d\chi}{ds} -\sin\vartheta\cos\vartheta\left(\frac{d\varphi}{ds}\right)^{2}=0,\\ &\frac{d^{2}\varphi}{ds^{2}} +2\cotg\chi\,\frac{d\chi}{ds}\frac{d\varphi}{ds} +2\cotg\vartheta\,\frac{d\vartheta}{ds}\frac{d\varphi}{ds}=0,\\ &\frac{d^{2}t}{ds^{2}}-\tg\chi\,\frac{dt}{ds}\frac{d\chi}{ds}=0. \end{aligned} \right\} \tag{30} \]

Let us examine both cases in order.

Solution A. From formula (29) it follows that the time \(t\) is a linear function of \(s\). From the point of view of an observer who uses the coordinates \(\chi,\vartheta,\varphi\), and \(t\) to describe the world, the equation of motion of a body of small mass is obtained if, in the first three formulas (29), the letter \(s\) is replaced by the letter \(t\). It is easy to see that a body of small mass on which no forces act moves along the shortest line of three-dimensional space. The equations are also satisfied in the case where one sets \(\chi=\mathrm{const}\), \(\vartheta=\mathrm{const}\), \(\varphi=\mathrm{const}\). In other words, if a body of small mass is placed at any distance from the observer and is not given an initial velocity, then it will remain at rest with respect to the observer.

Solution B. If we try to set \(\chi=\mathrm{const}\), \(\vartheta=\mathrm{const}\), \(\varphi=\mathrm{const}\), then equations (30) are not satisfied (one must not forget that it must be

\[ c^{2}\cos^{2}\chi\left(\frac{dt}{ds}\right)^{2} -R^{2}\left\{\left(\frac{d\chi}{ds}\right)^{2} +\sin^{2}\chi\left[\left(\frac{d\vartheta}{ds}\right)^{2} +\sin^{2}\vartheta\left(\frac{d\varphi}{ds}\right)\right]\right\}=1). \]

A body of small mass on which no forces act cannot remain at rest with respect to the observer, but must necessarily move. Let us study this motion.

The last equation (30) has the form

\[ \frac{d}{ds}\left(\cos^{2}\chi\,\frac{dt}{ds}\right)=0. \]

We conclude from this that

\[ \frac{d}{ds}=\frac{\mathrm{const.}}{\cos^{2}\chi}\,\frac{d}{dt},\qquad \frac{d^{2}}{ds^{2}}= \left(\frac{\mathrm{const.}}{\cos^{2}\chi}\right)^{2} \left(\frac{d^{2}}{dt^{2}}+2\tg\chi\,\frac{d\chi}{dt}\frac{d}{dt}\right). \]

Let us draw the plane \(\varphi=\mathrm{const}\) in such a way that at the initial moment

\[ \frac{d\varphi}{dt}=0 \]

(we do not assume that at the initial moment the comet was necessarily at rest with respect to the observer). Then, in general, \(\varphi=\mathrm{const}\), i.e. the trajectory of a body moving by inertia will lie entirely in one plane. Further, we obtain

\[ \frac{d^{2}\chi}{dt^{2}} +2\operatorname{tg}^{2}\chi\left(\frac{d\chi}{dt}\right)^{2} -\sin\chi\cos\chi\left[\left(\frac{d\vartheta}{dt}\right)^{2}+\frac{c^{2}}{R^{2}}\right]=0, \]

\[ \frac{d^{2}\vartheta}{dt^{2}} +2\left(\operatorname{tg}\chi+\operatorname{cotg}\chi\right) \frac{d\chi}{dt}\frac{d\vartheta}{dt}=0. \]

Following de Sitter’s example, let us introduce, instead of \(\chi\), the variable

\[ P=R\operatorname{tg}\chi, \]

which, let us note in passing, at small distances from the observer practically does not differ from \(r=R\chi\), but on the horizon takes the value \(\infty\), and not \(\frac{\pi}{2}R\). The equations of motion of the comet are then simplified in a remarkable way, and we find

\[ \frac{d^{2}P}{dt^{2}}-P\left(\frac{d\vartheta}{dt}\right)^{2} =\frac{c^{2}}{R^{2}}P, \]

\[ \frac{d^{2}\vartheta}{dt} +\frac{2}{P}\frac{dP}{dt}\frac{d\vartheta}{dt}=0, \]

whence one immediately obtains the “area integral”

\[ P^{2}\frac{d\vartheta}{dt}=h \quad (h\text{—constant of integration}) \]

and the “integral of living forces”

\[ \left(\frac{dP}{dt}\right)^{2} +P^{2}\left(\frac{d\vartheta}{dc}\right)^{2} =\frac{c^{2}}{R^{2}}P^{2}+k \quad (k\text{—another constant}). \]

Eliminating the time \(t\), we obtain the differential equation of the trajectory

\[ \left(\frac{dP}{d\vartheta}\right)^{2}\frac{h^{2}}{P^{4}} +\frac{h^{2}}{P^{2}} =\frac{c^{2}}{R^{4}}P^{2}+k. \]

It is easy to integrate this equation; one need only introduce the new variable

\[ y=\frac{1}{2P^{2}}. \]

Then elementary calculations give

\[ \left(\frac{dy}{d\vartheta}\right)^2+4y^2=\frac{c^2}{h^2R^2}+\frac{2ky}{h^2}, \]

whence

\[ \vartheta=\pm \frac12 \int \frac{dy}{\sqrt{\left(\frac{c}{2hR}\right)^2+\frac{k}{2h^2}y-y^2}} \]

\[ =\pm \frac12 \arccos \frac{2y-\frac{k}{2h^2}} {\sqrt{\frac{c^2}{h^2R^2}+\frac{k^2}{4h^4}}} +\vartheta_0, \]

where \(\vartheta_0\) is a constant of integration. The final equation of the orbit is

\[ \mathrm{P}^2\left[1+\varepsilon\cos 2(\vartheta-\vartheta_0)\right] =\frac{2h^2}{k}, \tag{31} \]

where

\[ \varepsilon= \sqrt{1+\frac{4c^2h^2}{R^2k^2}}. \]

The trajectory of the comet, as is seen from equation (31), is a hyperbola (in polar coordinates \(\mathrm{P}\) and \(\vartheta\), whose pole coincides with the center of the hyperbola). Indeed, if we introduce

\[ x=\mathrm{P}\cos(\vartheta-\vartheta_0) \]

\[ y=\mathrm{P}\sin(\vartheta-\vartheta_0) \]

and, moreover,

\[ a=h\sqrt{\frac{2}{k(\varepsilon+1)}},\qquad b=h\sqrt{\frac{2}{k(\varepsilon-1)}}, \]

then we obtain

\[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1. \]

The physical meaning of the semiaxes \(a\) and \(b\) of the hyperbola is very simple: let us note that, when the comet under consideration moves along this hyperbola, its “velocity” \(v\), as is seen from the integral of vis viva, increases with the “distance” \(\mathrm{P}\) according to the law

\[ v^2=\frac{c^2}{R^2}\mathrm{P}^2+k. \]

When the comet is closest to the observer \((\vartheta=\vartheta_0)\), the distance and the velocity are respectively equal to

\[ \mathrm{P}_0=h\sqrt{\frac{2}{k(\varepsilon+1)}},\qquad v_0=\sqrt{\frac{2c^2h^2}{(\varepsilon+1)kR^2}} = k\,\frac{ch}{R}\sqrt{\frac{2}{k(\varepsilon-1)}}, \]

whence

\[ a=\mathrm{P}_0,\quad b=\frac{R}{c}v_0. \]

If \(v_0=0\), then \(b=0\), and the hyperbola degenerates into a straight line. The constants \(\mathrm{P}_0\) and \(v_0\) may be substituted for \(k\) and \(h\). We have \(h=\mathrm{P}_0v_0\) (for at the distance \(\mathrm{P}_0\) the velocity must be perpendicular to the radius vector; therefore the areal constant is equal to the product of \(\mathrm{P}_0\) and \(v_0\)); the expressions given for \(\mathrm{P}_0\) and \(v_0\) yield

\[ k\frac{(\varepsilon+1)}{2}=\frac{h^2}{\mathrm{P}_0^4}=v_0^2,\qquad \frac{k(\varepsilon-1)}{2}=\frac{c^2h^2}{R^2v_0^2}=\frac{c^2}{R^2}\mathrm{P}_0^2. \]

Eliminating \(\varepsilon\), we obtain

\[ k=v_0^2-\frac{c^2}{R^2}\mathrm{P}_0^2, \]

whence, by means of the vis-viva integral, we find

\[ v^2=\frac{c^2}{R^2}\mathrm{P}^2+v_0^2-\frac{c^2}{R^2}\mathrm{P}_0^2. \]

If we recall that \(v^2=\left(\dfrac{d\mathrm{P}}{dt}\right)^2+\mathrm{P}^2\left(\dfrac{d\vartheta}{dt}\right)^2\), where \(\dfrac{d\mathrm{P}}{dt}=v_\rho\) is the component of the velocity along the line of sight, and \(\mathrm{P}\dfrac{d\vartheta}{dt}=\dfrac{h}{\mathrm{P}}\) is the transverse component, then, subtracting \(\dfrac{h^2}{\mathrm{P}^2}=\dfrac{\mathrm{P}_0^2v_0^2}{\mathrm{P}^2}\) from \(v^2\), we obtain for the radial component the following formula of de Sitter:¹

\[ \frac{v_\rho}{c}=\pm\frac{\mathrm{P}}{R}\sqrt{\left(1-\frac{\mathrm{P}_0^2}{\mathrm{P}^2}\right)\left(1+\frac{v_0^2}{c^2}\frac{R^2}{\mathrm{P}^2}\right)}. \tag{32} \]

¹ However, in de Sitter’s American paper of 1930 [6] this formula, owing to a misprint, is given in an incorrect form (namely, in the last parenthesis \(1+\dfrac{v_0^2}{c^2\mathrm{P}^2}\) is printed instead of \(1+\dfrac{v_0^2R^2}{c^2\mathrm{P}^2}\)), as a result of which he erroneously concludes that if \(\mathrm{P}\gg \mathrm{P}_0\), then \(\dfrac{v_\rho}{c}\) coincides with \(\pm\dfrac{\mathrm{P}}{R}\), independently of the magnitude of \(v_0\). In the Dutch article [5] this formula is written correctly.

If in a body possessing small mass and moving by inertia along the hyperbola (31) there are atoms emitting a definite line spectrum, then an observer located at the origin of coordinates may perhaps wish to measure the radial velocity of the moving body by means of the Doppler effect. This effect is very easy to calculate. If (from the point of view of the observer situated at the origin of coordinates) a light signal from the moving comet was sent at the instant of time \(t\), then the observer will receive it at the instant of time

\[ t+\frac{R}{c}\int_{0}^{\chi}\frac{d\chi}{\cos\chi}. \]

The next light signal was sent at the instant of time \(t+dt\), when the coordinates of the moving body were no longer \(\chi\) and \(\vartheta\), but \(\chi+d\chi\) and \(\vartheta+d\vartheta\). Our observer will receive it at the instant of time

\[ t+dt+\frac{R}{c}\int_{0}^{\chi+d\chi}\frac{d\chi}{\cos\chi}. \]

The interval of time between the reception of the two signals will be

\[ dt+\frac{R}{c}\frac{d\chi}{\cos\chi}. \]

Since \(P=-R\operatorname{tg}\chi\), it follows from formula (32), which gives \(\frac{1}{c}\frac{dP}{dt}\), that

\[ \frac{R}{c}\,d\chi=\pm \sin\chi\cos\chi \sqrt{\left(1-\frac{P_0^2}{R^2\operatorname{tg}^2\chi}\right) \left(1+\frac{v_0^2}{c^2\operatorname{tg}^2\chi}\right)}\,dt. \]

In order to compare the interval of time between the reception of the signals, measured by the observer’s clock, with the interval of time between the sending of the signals, measured by the clock on the comet (and this is what the Doppler effect consists in), let us note that it must be

\[ \frac{1}{c}\,ds= \sqrt{ \cos^2\chi -\frac{R^2}{c}\left(\frac{d\chi}{dt}\right)^2 -\frac{R^2}{c^2}\sin^2\chi\left(\frac{d\vartheta}{dt}\right)^2 }\,dt = \]

\[ =\sqrt{ \cos^2\chi -\sin^2\chi\cos^2\chi \left(1-\frac{P_0^2}{R^2\operatorname{tg}^2\chi}\right) \left(1+\frac{v_0^2}{c^2\operatorname{tg}^2\chi}\right) -\frac{R^2}{c^2}\sin^2\chi \left(\frac{P_0v_0}{R^2\operatorname{tg}^2\chi}\right)^2 }\,dt. \]

*

From these formulas it follows that, if the wavelength of the spectral line sent from the comet was \(\lambda\), then the observer at the origin of coordinates will receive radiation with wavelength \(\lambda+d\lambda\), where

\[ \frac{\lambda+d\lambda}{\lambda} = \frac{dt+\dfrac{R}{c}\dfrac{d\chi}{\cos\chi}}{\dfrac{1}{c}\,ds} = \]

\[ = \frac{ 1\pm \sin\chi\sqrt{\left(1-\dfrac{P_0^2}{R^2\operatorname{tg}^2\chi}\right) \left(1+\dfrac{v_0^2}{c^2\operatorname{tg}^2\chi}\right)} }{ \sqrt{ \cos^2\chi-\sin^2\chi\cos^2\chi \left(1-\dfrac{P_0^2}{R^2\operatorname{tg}^2\chi}\right) \left(1+\dfrac{v_0^2}{c^2\operatorname{tg}^2\chi}\right) -\dfrac{P_0^2v_0^2\cos^2\chi}{R^2c^2\operatorname{tg}^2\chi} } } \tag{33} \]

Such is the exact formula\(^1\) for the Doppler effect. If the observer knows the distance to the comet (i.e. its coordinate \(P\)), and if he may assume that \(R\operatorname{tg}\chi \gg P_0\) and \(v_0 \ll c\), then by means of the formula

\[ \frac{\lambda+d\lambda}{\lambda} = \frac{1\pm \sin\chi}{\sqrt{\cos^2\chi-\sin^2\chi\cos^2\chi}} = \frac{1\pm \sin\chi}{\cos^2\chi} \]

or

\[ \frac{d\lambda}{\lambda} = \operatorname{tg}^2\chi \pm \frac{\operatorname{tg}^2\chi}{\cos^2\chi} = \frac{\sin^2\chi\pm \sin\chi}{1-\sin^2\chi} = \pm\frac{P}{R} \sqrt{1+\frac{P^2}{R^2}+\frac{P^3}{R^4}} \]

he will be able to determine the radius of the world. If \(\sin\chi\) is a small quantity and one may neglect its cube and fourth power, then the approximate formula is obtained

\[ \frac{d\lambda}{\lambda} = \pm \sin\chi+\sin^2\chi . \tag{34} \]

We see that the Doppler effect, which can be measured by the observer, arises from two causes: the first part of the effect, which may have either the sign \(+\) or the sign \(-\),

\(^1\) Weyl [3] derived the formula \(\dfrac{d\lambda}{\lambda}=\operatorname{tg}\chi\) (without the sign \(\pm\)). Silberstein [4] contrasts with this formula his own, namely \(\dfrac{d\lambda}{\lambda}=\pm \sin\chi\), and sees Weyl’s error not only in the fact that, instead of \(\pm\), he puts only the sign \(+\), but also in the fact that instead of \(\sin\chi\) Weyl has \(\operatorname{tg}\chi\) (in fact, at this degree of approximation it is all the same whether it is \(\sin\chi\) or \(\operatorname{tg}\chi\)).

occurs as a consequence of the motion of the comet along its hyperbolic orbit, caused by inertial forces (this motion may occur both in one direction and in the other); the second part of the effect, quadratic rather than linear with respect to \(\sin\chi\), is caused by the presence of the factor \(\cos\chi\) in \(c\,dt\), i.e. by the “slowing of time” at great distances from the observer. The first part of the effect may be called the real Doppler effect, while the second, as having nothing in common with the motion of the light source, may be called the apparent Doppler effect. Both effects increase as the light source recedes from the observer.

In the Einstein world, determined by solution \(A\), nothing of the kind can occur. The speed with which a body of small mass moves in this world is in no way connected with its distance from the observer; nor does the second (apparent) part of the Doppler effect occur in the Einstein world. The same applies also to solution \(C\).

Already de Sitter saw in the presence of the indicated Doppler effect a possibility for an empirical test of solution \(B\) and for comparing this solution with solution \(A\). In his principal paper [2] in Monthly Notices he points out that the velocities of objects remote from us, measured spectroscopically, to which first of all the spiral nebulae belong, are unusually large in comparison with the velocities of stars near us. In 1917 the available empirical material was very small; de Sitter could point to only three nebulae whose velocities had been measured sufficiently well; these were

Object	Radial velocity
N. G. C. 224	\(-311\) km/sec
N. G. C. 1068	\(+925\) km/sec
N. G. C. 4594	\(+1185\) km/sec

[Radial velocities are measured from the displacement of spectral lines according to the formula

\[ \frac{v}{c}=\frac{\delta\lambda}{\lambda}; \]

velocities directed away from the observer (displacement toward the red end) are regarded as positive, and velocities directed toward him (displacement ...]

toward the violet side), negative]. The distances to the spiral nebulae had not yet been measured at that time, and de Sitter took, quite at random, as a probable distance to these objects the number \(10^5\) parsecs \(=\) \(3 \times 10^{23}\) cm (we saw in § 1 that the distance to N. G. C. 224 is actually equal to \(2.8 \times 10^5\) parsecs, so that the order of magnitude had been guessed correctly by de Sitter).

In doing so de Sitter made the following argument: the displacement toward the red end, due to the quadratic effect, has a constant sign, whereas the displacements caused by the linear effect have different signs, and therefore, if one operates with mean quantities, only the displacement due to the quadratic effect will remain. In fact, for the three nebulae mentioned the mean is \(+600\) km/sec (a displacement toward the red end). For this displacement de Sitter applies the formula

\[ \frac{\delta\lambda}{\lambda}=\frac{1}{2}\sin^2\chi \]

(the factor \(1/2\) was evidently introduced by de Sitter by mistake: he proceeded from the assumption that, in the case of a nebula at rest with respect to the observer, one should have \(\frac{1}{c}\,ds=\cos\chi\,dt\), and that the whole Doppler effect is due to the difference between the observer’s time \(dt\) and the “proper time” of the nebula \(\frac{1}{c}\,ds\); hence

\[ \frac{\lambda+d\lambda}{\lambda} = \frac{dt}{\frac{1}{c}\,ds} = \frac{1}{\cos\chi} \]

and

\[ \frac{d\lambda}{\lambda} = \frac{1-\cos\chi}{\cos\chi}\sim \frac{1}{2}\sin^2\chi \]
).

If one assumes that the “distance” which plays a role in astronomy in estimating the brightness of objects is precisely \(R\sin\chi\), and not \(R\chi\) or \(R\tg\chi\) (for the surface of a sphere described about the origin of coordinates is proportional to \(R^2\sin^2\chi\), and therefore the intensity of light must decrease inversely proportionally to \(R^2\sin^2\chi\)), then from the numbers adopted by de Sitter it followed that

\[ \frac{\delta\lambda}{\lambda} = \frac{600}{300\,000} = \frac{1}{2}\sin^2\chi = \frac{1}{2}\left(\frac{3\times10^{23}}{R}\right)^2, \]

whence

\[ R = 3 \times 10^{23}\sqrt{250} = 4.8 \times 10^{24}\ \text{cm} = 4.8A, \]

a figure quite absurd from the present-day point of view (de Sitter supplied it with the remark: “of course this result, derived from only three nebulae, has no practical value”).

As empirical material accumulated, astronomers’ views on the significance of the large velocities of objects distant from us gradually changed. In his book The Mathematical Theory of Relativity (Cambridge, 1923), Eddington published a table compiled by Slipher (Lowell Observatory in Arizona), presenting a summary of what was known in February 1922 about the radial velocities of spiral nebulae. The table contains data on 41 nebulae; of these, 36 have a positive radial velocity (are moving away from us), while the remaining 5 have a negative one (including N.G.C. 221 and the Andromeda nebula N.G.C. 224, which are moving toward us at a velocity of 300 km/sec). The largest radial velocity among those given in Slipher’s table belongs to the nebula N.G.C. 584 and amounts to \(+1800\) km/sec.^1 From the table it was evident that, among objects distant from us, the overwhelming majority possess positive radial velocities. At the present time (1930) this should be regarded as a fully established empirical fact: de Sitter [5] processed a very extensive body of observational material from the Mount Wilson Observatory, setting himself the goal of establishing a correlation between a nebula’s radial velocity and its distance; distances had been measured by reliable methods only for a few nebulae, but de Sitter expanded this material, estimating the distances to many other nebulae from their brightness and apparent diameter. It turned out that between distance and radial

^1 At present, as de Sitter states, referring to unpublished materials of the Mount Wilson Observatory, the greatest measured radial velocity found is \(+12000\) km/sec.

with the velocity measured from the displacement of the spectral lines, there exists the correlation

\[ \frac{v}{c}=\frac{r}{2000}, \]

where \(v\) is the radial velocity, \(c\) the velocity of light, and \(r\) the distance (i.e., evidently \(R\sin\chi\)), measured in units \(A\). But even in 1923, when only Slipher’s material, first published by Eddington, was known, it was clear that the explanation proposed by de Sitter, based on the quadratic, i.e. apparent, Doppler effect, could not be correct. If it were indeed correct, then a linear effect of variable sign would be superposed on the quadratic effect of constant sign, and only the mean values would show a red shift.

From de Sitter’s solution there in no way follows the necessity of an overwhelmingly predominant number of positive radial velocities over negative ones; to be sure, such a predominance does not contradict de Sitter’s solution, but this is very little consolation. Eddington and Weyl tried, by various truths and untruths, to explain this predominance as allegedly inherent in matter in the de Sitter world through a tendency to scattering; if matter really possessed such a property, then, taking of the two signs \(+\) and \(-\) only the first, for not too large distances we would obtain the formula

\[ \frac{\delta\lambda}{\lambda}=\sin\chi \]

[since in formula (34) one could neglect the quadratic effect in comparison with the linear one]. Comparing with de Sitter’s empirical formula, we obtain for the radius of the de Sitter universe the value

\[ R=2000\ A=2\times 10^{27}\ \text{cm}, \]

i.e., a number of approximately the same order of magnitude as the radius of the Einstein universe obtained in § 3,

Ludwik Silberstein [4] vigorously protested against the “tendency toward scattering” discovered by Weyl and Eddington, believing that such a tendency in no way follows from de Sitter cosmology; in this he was entirely right, since the de Sitter world is empty: no universal properties of matter can follow from de Sitter cosmology, into which matter in general can be brought only as contraband, in the form of “comets,” i.e., “des riens visibles” (bodies possessing no mass). The absence of a tendency toward scattering, of course, also does not follow from de Sitter’s solution. Thus the dispute that flared up over the “tendency toward scattering,” despite the abundant use of ingenuity and irony on the part of the disputing parties, was a completely sterile dispute. Unwilling to agree with the compelling arguments of Weyl and Eddington concerning the necessity of retaining in the formula for the Doppler effect only the sign \(+\), Silberstein fell into the completely opposite extreme: he closed his eyes to the spiral nebulae and took up the distribution of velocities among nearer objects, such as globular star clusters and even the stars of our Milky Way. In Silberstein’s opinion, there exists a correlative relation between the distance of these objects and their measured radial velocity; moreover, in the latter only the absolute magnitude is of interest, not the sign. In this opinion Silberstein, so far as we know, was entirely alone; but we should not treat his views with irony, as Eddington would have us do, since in any case they contain no more, and rather less, confusion than Eddington’s and Weyl’s views on the notorious “tendency toward scattering.” Silberstein’s conclusions should be rejected not from the standpoint of the theory of relativity, but from the astronomical standpoint. The study of the distribution of velocities among stars led Silberstein to the necessity of upholding much smaller values for the radius of the de Sitter world than all other authors did. Thus, for example, in one of his latest publications [6] he states that the radius of the world is \(1.55 \times 10^6\) parsecs,

i.e., about \(5 \times 10^{24}\) cm, or \(5\ A\). Although he says that in such a universe several million stellar systems similar to our Milky Way can freely fit, the reader who has looked through our § 1 will agree that the figure \(5 \times 10^{24}\) cm is absurdly small for an astronomer.1 According to Eddington (who is apparently quite right on this question), the correlation between the velocities of stars and their distances, discovered by Silberstein, is essentially reducible to the rotation of the Milky Way about its axis: if, from the velocities on which Silberstein based his calculations, one subtracts the correction due to this rotation, studied in Oort’s works, then almost nothing remains of the correlation.

From this review it is clear that until very recently the position of relativistic cosmology was extremely unfavorable. Solution \(A\), i.e. Einstein’s space-time, had to be rejected as in no way explaining the regularity that undoubtedly exists in the distribution of velocities of distant objects; along with it, solution \(C\), its special case, also had to retire modestly. There remained only solution \(B\), about which its zealous supporters stubbornly asserted that it was precisely this solution that agreed with the astronomical facts. But we have just seen that such “healthy optimism” was hardly appropriate in this case. Solution \(B\) explained only the increase in the absolute values of velocities with distance from the observer, but could not explain the constant sign of these velocities. And indeed, how could anything at all be favorable in such a cosmology, which declared the world completely empty and permitted the intro-

to leave only a vanishingly small test body as an instrument for measuring the field of inertia! As such a “test body,” such a “comet,” the supporters of de Sitter’s solution proposed considering a spiral nebula, a galactic system consisting of tens of billions of suns, and, moreover, surrounded on all sides by similar galactic systems at average distances of \(10^{24}\) cm from one another. Quite apart from the fact that a zero mean density of matter in the universe is extremely improbable (as the founder of the theory of relativity asserts in his Princeton lectures, and it seems that one must agree with him on this), the very magnitude of the radius of the world, which is obtained by comparing the formula \(\frac{\delta\lambda}{\lambda}=\sin\chi\) with the facts, is a strong argument against the possibility of de Sitter cosmology. At least for de Sitter himself, in that work [5] in which he renounced his cosmology and burned what he had worshiped, this was the most decisive argument. A radius of \(2000\,A\), which approximately coincides with the radius of \(2300\,A\) computed from the density \(\rho=2\times10^{-28}\ \mathrm{g/cm^3}\) by the formulas of Einsteinian cosmology, is in itself extraordinarily suspicious. Our telescopes penetrate into the depths of space to \(150\,A\); if the radius of the universe is \(2000\,A\), then the mass of the universe (which, after all, is probably still greater than the part of it enclosed within a sphere of radius \(150\,A\)) is too large for the ultimate mean density to be regarded as zero and for the universe to deviate only slightly from the condition of homogeneity imposed by cosmological theories. All this leads to the conclusion that solution \(B\), so celebrated in all courses on the theory of relativity, must also be rejected. In 1930 this was rather unexpectedly acknowledged by the most zealous paladins of solution \(B\)—de Sitter and Eddington.

In the following paragraph there will be set forth those views which they now propagate.

§ 5. Nonstatic Solutions of the Cosmological Problem

The Einstein solution and the de Sitter solution are static solutions of equations (15). This means that both in the Einstein world and in the de Sitter world one can choose a coordinate system such that the coefficients \(g_{ik}\) in the expression \(ds^2=\sum_{i,k=1}^{4} g_{ik}\,dx_i\,dx_k\) are not functions of time, but depend only on the remaining coordinates, spatial in the strict sense of the word. In fact, both line elements (13) and (27) already satisfy these conditions. However, there is no need to confine oneself to such static solutions. As early as 1922 the late Russian mathematician A. A. Friedman [7] considered a four-dimensional world with the line element

\[ ds^2=-R^2\,[d\chi^2+\sin^2\chi\,(d\vartheta^2+\sin^2\vartheta\,d\varphi^2)]+c^2\,dt^2, \]

in which, unlike the Einstein world, \(R\) is regarded not as a constant quantity, but as a function of time. However, Friedman restricted his solution by the condition \(p=0\), i.e. he required that the world with such a line element contain no radiant energy; this restriction is not physically necessary. Friedman’s work was half forgotten,¹ and only in 1927 did Lemaître [8] again consider the same line element, applying it to a world filled, in addition to matter, also with radiant energy. If we ta—

¹ It is possible that this was helped by an objection made to Friedman by Einstein [7]. This objection, however, is itself based on a gross error: incorrectly applying the formula of covariant differentiation, Einstein asserts that the fourth of the equations expressing the well-known theorem on the vanishing of the divergence of the tensor \(T_{ik}\) (the equation of conservation of energy) reads \(\dfrac{d\rho}{dt}=0\). In reality, however, under the condition \(p=0\) it reads

\[ \frac{1}{R^3}\frac{d}{dt}\left(R^3\rho\right)=0, \]

being a special case of equation (37), derived in the text.

to the numerical values of the quantities \(T_{ik}\) determined by formula (21), it will turn out that

\[ T_{11}=\frac{\lambda R^{2}-1}{\varkappa} -\frac{1}{\varkappa c^{2}}\left[\left(\frac{dR}{dt}\right)^{2} +2R\frac{d^{2}R}{dt^{2}}\right], \]

\[ T_{22}=\sin^{2}\chi\,T_{11},\qquad T_{33}=\sin^{2}\vartheta\,T_{22}, \]

\[ T_{44}=\frac{c^{2}}{\varkappa}\left(\frac{3}{R^{2}}-\lambda\right) +\frac{3}{\varkappa R^{2}}\left(\frac{dR}{dt}\right)^{2},\qquad T_{ik}=0\ \text{for } i\ne k . \]

Comparing this with equations (15), we obtain

\[ \left. \begin{gathered} \frac{d\varkappa}{ds}=\frac{d\vartheta}{ds}=\frac{d\varphi}{ds}=0,\qquad \left(\frac{dt}{ds}\right)^{2}=\frac{1}{c^{2}},\\[4pt] p=\frac{c^{2}}{\varkappa}\left(\lambda-\frac{1}{R^{2}}\right) -\frac{1}{\varkappa}\left[\frac{1}{R^{2}}\left(\frac{dR}{dt}\right)^{2} +\frac{2}{R}\frac{d^{2}R}{dt^{2}}\right],\\[4pt] \rho=\frac{1}{\varkappa}\left[\frac{3}{R^{2}}-\lambda +\frac{3}{c^{2}R^{2}}\left(\frac{dR}{dt}\right)^{2}\right], \end{gathered} \right\} \tag{35} \]

whence

\[ \rho_{0}=\rho-\frac{3p}{c^{2}} =\frac{6}{\varkappa R^{2}}\left[\frac{1}{c^{2}}R\frac{d^{2}R}{dt^{2}} +\frac{1}{c^{2}}\left(\frac{dR}{dt}\right)^{2}+1\right] -\frac{\lambda}{\varkappa}. \tag{36} \]

It is not difficult to verify the validity of the equation

\[ \frac{1}{R^{3}}\frac{d}{dt}(R^{3}\rho) +\frac{3}{R}\frac{dR}{dt}\frac{p}{c^{2}}=0, \tag{37} \]

to which one can at once give a curious physical interpretation. For this purpose let us introduce the volume of the world \(V\). Since at any given moment of time \(t\) the geometry of three-dimensional space is the same as in the case of Einstein’s solution, we can use formula (20) and write \(V=2\pi^{2}R^{3}\). There exists, however, an opinion (especially attractive in the case of de Sitter’s solution with its “horizon”) that to each point in our physical space there correspond not one but two diametrically opposite points of the spherical space with line element \(R\sqrt{d\chi^{2}+\sin^{2}\chi\,(d\vartheta^{2}+\sin^{2}\vartheta\,d\varphi^{2})}\); in that case one must use the formula \(V=\pi^{2}R^{3}\). In any event, \(V\) coincides with \(R^{3}\) up to a constant factor. Therefore from equation (37) it follows that

\[ d(V\rho c^{2})+p\,dV=0. \]

Since the mass of the world, equal to \(V\rho\), corresponds, according to the well-known principle of the equivalence of energy and mass,

energy is \(E=V\rho c^2\), then the energy of the universe \(E=V\rho c^2\) and its volume \(V\) are connected by the equation

\[ dE+p\,dV=0, \tag{33} \]

entirely analogous to the equation of adiabatic expansion or compression of a gas. The masses of the universe (material and radiative) change with time; with them the dimensions of the universe also change. With the expansion of the universe (and we shall see that it is precisely this that must occur in order for the observed velocities of spiral nebulae to be explained), its mass and its energy (provided only that \(\rho\ne 0\)) decrease. Thus, 200 years after Newton’s death, his old supposition that the energy of the universe continuously decreases has unexpectedly been revived. There is no contradiction of the law of conservation of energy in equation (38); on the contrary, it is a consequence of the differential form of the law of conservation of energy [readers familiar with the general theory of relativity will readily convince themselves that equation (87) is one of four equations whose totality represents the relativistic law of conservation of energy and momentum: the “divergence of the tensor of matter and energy \(T_{ik}\) is equal to zero”]. If the energy of the universe decreases, this is because it is spent (an unfamiliar phrase to the physicist’s ear!) on the work of adiabatic expansion of the universe.

In order to consider the question more closely, it is necessary to find the dependence of the radius of the universe \(R\) on time. Strictly speaking, equations (36) alone do not allow one to answer this important question. Any dependence of \(R\) on time will, when substituted into these equations, give the corresponding dependences for \(p\) and for \(\rho\). Therefore some new physical hypothesis is necessary. If we recall that \(\rho=\rho_0+\dfrac{3p}{c^2}\) and substitute in (38)

\[ E=V\rho c^2=V\rho_0c^2+3pV=M_0c^2+3pV, \]

where \(M_0\) is the material mass of the universe, then instead of (38) we obtain

\[ c^2\,dM_0+4p\,dV+3V\,dp=0. \tag{81} \]

Lemaître makes the hypothesis that the material mass of the universe

constant ($M_0=\mathrm{const}$), whence $4p\,dV+3V\,dp=0$ and finally,

\[ pV^{\frac{4}{3}}=\mathrm{const}. \]

The world expands like an adiabatically expanding gas with a ratio of specific heats of 1.33. We shall arrive at the same conclusion $M_0=\mathrm{const}$ if, following Friedmann, we make the hypothesis $p=0$. This hypothesis is also made by Eddington [9], justifying it by the simplification of the calculations. However, an assumption of this kind cannot be physically justified, since, according to astrophysical data, the material mass of the world is continuously melting away, turning into radiation. This must be taken into account when discussing equation (38). Tolman [10] insisted on this; apparently he began to develop this circle of ideas quite independently of the works of Friedmann and Lemaître. The same question is investigated by de Sitter [5], who, however, comes to the conclusion that the influence of the annihilation of matter *** must be very small.^1

^1 It should be noted that de Sitter [5], Eddington [9], and others give to equations (15) not the interpretation that was given above in our text, but a somewhat different one; by $\dfrac{dv_i}{ds}$ they understand not the velocity of individual parts of matter (for example, galaxies), but some average velocity (for example, the velocity of streams of galaxies); correspondingly, by $p$ is meant the pressure not of radiant energy alone, but also the pressure of streams of galaxies, calculated as if, in the place of galaxies, there were separate molecules (according to the formulas of kinetic theory such a pressure is equal to two thirds of the kinetic energy per unit volume). Such an interpretation, like ours, is in accord with the principle of relativity; it can be rigorously shown that the two interpretations are equivalent to one another; however, there is a difference between our requirement that individual galaxies be at rest relative to one another, and Eddington’s and de Sitter’s requirement that there be no resultant velocities in the disordered “thermal” motion of galaxies. On strict examination, de Sitter’s and Eddington’s interpretation leads to more complicated formulas, since the gas pressure enters into the expression for the energy density with coefficient $\dfrac{3}{2}$, and not with coefficient 3, as does the pressure of radiant energy. Therefore, if $\rho$ is the total density of mass (material and radiant), and $p$ is the total pressure, then one must intro-

Let us introduce, following de Sitter, instead of \(\rho\) and \(p\) the variables

\[ \alpha=\chi\left(\rho-\frac{3p}{c^2}\right)R^3=\chi\rho_0R^3,\qquad \beta=\frac{\chi}{c^2}pR^4. \]

Then instead of equation (39) one obtains

\[ R\frac{d\alpha}{dt}+3\frac{d\beta}{dt}=0. \tag{40} \]

Since the material mass of the world decreases, \(\frac{d\alpha}{dt}\) is negative, and de Sitter makes the hypothesis

\[ \frac{1}{\alpha}\frac{d\alpha}{dt}=-\frac{\gamma}{R}\frac{dR}{dt}, \]

where \(\gamma\) is a positive constant. If this hypothesis is made, then it follows that

\[ \alpha=\alpha_0R^{-\gamma}, \]

where \(\alpha_0\) is a constant of integration. Equation (40) then gives

\[ \beta=\beta_0+\frac{\gamma\alpha_0}{3(1-\gamma)}R^{1-\gamma}, \]

whence

\[ p=\frac{c^2}{\chi}\frac{\beta_0}{R^4} +\frac{\gamma c^2\alpha_0}{3(1-\gamma)\chi}\frac{1}{R^{3+\gamma}}. \]

If one introduces the volume of the world \(V=2\pi^2R^3\), then the amount of radiant energy in the universe proves to be equal to

\[ 2\pi^2R^3\cdot 3p =\frac{6\pi^2c^2}{\chi}\left(\frac{\beta_0}{R} +\frac{\gamma\alpha_0}{3(1-\gamma)}\frac{1}{R^\gamma}\right). \]

This quantity, as we see, decreases as the world expands. Thus, as de Sitter says,

the quantity \(\rho_0=\rho-3\frac{p}{c^2}\) is not the “rest mass” of matter, as de Sitter and Eddington seem to assert. Practically, however, the “pressure” of the streams of galaxies is quite negligible, and as the world expands it decreases sooner than the pressure of the radiant energy filling space. Therefore both Eddington and de Sitter ultimately come to the necessity of neglecting the kinematic pressure (the damping of individual motions, according to Eddington), and the difference between the two interpretations disappears. The pressure of radiant energy, however, is also negligible and becomes ever smaller, since despite the continuous “melting away” of matter, the universe expands so rapidly that the density of the radiant energy existing in space must constantly decrease (see also note 2 to p. 136).

“theory gives an answer to the old question of what happens to the energy continuously radiated by the stars into space. This energy is all expended on the adiabatic expansion of the world, and even proves insufficient for it.”

A remarkable property of the Friedmann–Lemaître cosmology consists in the fact that it gives a completely clear and simple explanation of the fact, hitherto puzzling, that all spiral nebulae have one and the same sign of radial velocity. If we write the equations of motion of a body moving by inertia in the Friedmann–Lemaître world, it turns out that

\[ \frac{d^2\chi}{ds^2} +\frac{2}{R}\frac{dR}{dt}\frac{dt}{ds}\frac{d\chi}{ds} -\sin\chi\cos\chi\left(\frac{d\vartheta}{ds}\right)^2 -\sin\chi\cos\chi\,\sin^2\vartheta\left(\frac{d\varphi}{ds}\right)^2 =0, \]

\[ \frac{d^2\vartheta}{ds^2} +\frac{2}{R}\frac{dR}{dt}\frac{dt}{ds}\frac{d\vartheta}{ds} +2\operatorname{cotg}\chi\,\frac{d\chi}{ds}\frac{d\vartheta}{ds} -\sin\vartheta\cos\vartheta\left(\frac{d\varphi}{ds}\right)^2 =0, \]

\[ \frac{d^2\varphi}{ds^2} +\frac{2}{R}\frac{dR}{dt}\frac{dt}{ds}\frac{d\varphi}{ds} +2\operatorname{cotg}\chi\,\frac{d\chi}{ds}\frac{d\varphi}{ds} +2\operatorname{cotg}\vartheta\,\frac{d\vartheta}{ds}\frac{d\varphi}{ds} =0, \]

\[ \frac{d^2 t}{ds^2} +\frac{R}{c^2}\frac{dR}{dt}\left(\frac{d\chi}{ds}\right)^2 +\frac{R}{c^2}\frac{dR}{dt}\sin^2\chi\left(\frac{d\vartheta}{ds}\right)^2 +\frac{R}{c^2}\frac{dR}{dt}\sin^2\chi\,\sin^2\vartheta\left(\frac{d\varphi}{ds}\right)^2 =0. \]

These equations are satisfied if we set \(\chi=\mathrm{const}\), \(\vartheta=\mathrm{const}\), \(\varphi=\mathrm{const}\), and \(t\) is a linear function of \(s\). This means that a body which was at rest with respect to the observer will remain in this state, i.e. will not change its coordinates \(\chi,\vartheta,\varphi\). In such rest with respect to the observer is found the overwhelming mass of the material universe surrounding him. But the distances to the immobile objects of this universe, measured not in fractions of the radius of the world but in centimeters, are changing all the time, since the scale in which the distances prove constant is a scale changing in time. It is easy to calculate the Doppler effect resulting from this. If a light signal is sent at the moment of time \(t\) from the point with coordinates \(\chi,\vartheta,\varphi\), then the time \(t+\tau\) arr—

of this signal by an observer situated at the origin of coordinates will be determined by the equality

\[ \chi=\int_t^{t+\tau}\frac{c}{R}\,dt, \]

and since the path of the light waves is the straight line \(\varphi=\mathrm{const}\), \(\vartheta=\mathrm{const}\), and the time of their propagation is determined by the fact that \(ds^2=c^2dt^2-R^2d\chi^2=0\). The next signal is sent from a point with the same coordinates \(\chi,\vartheta\), and \(\varphi\) at the instant of time \(t+dt\); the moment of reception of the signal will no longer be \(t+\tau\), but \(t+\tau+d\tau\), where

\[ \chi=\int_{t+dt}^{t+\tau+d\tau}\frac{c}{R}\,dt =\int_t^{t+\tau}\frac{c}{R}\,dt +\left(\frac{c}{R}\right)_{t+\tau}d\tau -\left(\frac{c}{R}\right)_t dt, \]

whence

\[ d\tau:dt=\left(\frac{c}{R}\right)_t:\left(\frac{c}{R}\right)_{t+\tau} =(R)_{t+\tau}:(R)_t . \]

The duration of the interval between the sending of the two signals, measured by clocks located on the spiral nebula, will still be equal to \(dt\), since \(d\chi=d\varphi=d\vartheta=0\), and consequently \(\frac{1}{c}ds=dt\). Therefore for the Doppler effect we shall have

\[ \frac{\lambda+d\lambda}{\lambda}=\frac{d\tau}{dt},\quad \text{whence}\quad \frac{\delta\lambda}{\lambda} =\frac{(R)_{t+\tau}-(R)_t}{(R)_t} \]

or

\[ \frac{\delta\lambda}{\lambda} =\frac{\tau}{(R)_t}\left(\frac{dR}{dt}\right)_{t+\eta\tau}, \]

where \(0\leq \eta\leq 1\). If the time interval \(\tau\), during which the ray of light reaches us from the nebula, is not so large that the radius of the world has time to change appreciably during this interval, then one may approximately put

\[ \frac{\delta\lambda}{\lambda}=\frac{\tau}{R}\frac{dR}{dt}; \]

and since, to the same order of approximation, \(\tau=\frac{r}{c}\), where \(r\) is the distance to the nebula measured by the formula \(r=R\chi\) (and not \(r=\sin\chi\), as, strictly speaking, one must put in order to

to remain in agreement with the astronomical determination of distances, then

\[ \frac{\delta \lambda}{\lambda}=\frac{r}{R}\frac{dR}{c\,dt}. \tag{41} \]

From this formula it is clear that the Doppler effect must have one and the same sign for all remote objects and be proportional to the distance. If only we assume that \(\frac{dR}{dt}>0\), i.e. that the radius of the universe is not decreasing, but increasing, then formula (41) will be in complete agreement with all observations (more consistent with the astronomical determination of distance would be the formula \(\frac{\delta\lambda}{\lambda}=\sin\left(\frac{r}{R}\right)\cdot\frac{dR}{c\,dt}\), but the difference between the two formulas at present apparently lies beyond the limits of reliability of the available empirical material). If we compare formula (41) with de Sitter’s empirical formula, it turns out that

\[ \frac{1}{R}\frac{dR}{dt}=\frac{3}{2}\times 10^{-17}\ \mathrm{sec}^{-1}, \]

whence follows the astonishing result that, at the present rate of expansion of the universe, its radius must double in \(\log 2\cdot \frac{2}{3}\,10^{17}\ \mathrm{sec}=4.62\times 10^{16}\ \mathrm{sec}=1.5\times 10^9\) years, which represents, as Eddington indicates, a time interval of merely “geological” order.

The value found for \(\frac{1}{R}\frac{dR}{dt}\) makes it possible to estimate the quantity \(\gamma\) (de Sitter’s hypothetical constant). Comparing the absolute luminosities of spiral nebulae with their presumed masses, de Sitter finds that, on average, each galaxy loses per second \(3\times 10^{-24}\) of its mass through radiation. Extending this result to the entire universe, one may put approximately \(\frac{1}{a}\frac{da}{dt}=-3\times 10^{-24}\), whence it follows that \(\gamma=2\times 10^{-7}\).

Therefore, as de Sitter concludes, in the first approximation one may quite freely put \(\gamma=0\), i.e. regard the material mass of the world as a constant quantity.

If this supposition is made, it turns out that \(\alpha=\mathrm{const}\), \(\beta=\mathrm{const}\), and the second of equations (35) gives

\[ \frac{1}{\varkappa}\left(\frac{3}{R^{2}}-\lambda\right) +\frac{3}{c^{2}\varkappa R^{2}}\left(\frac{dR}{dt}\right)^{2} = \frac{\alpha}{\varkappa R^{3}} +\frac{3\beta}{\varkappa R^{4}}. \]

This differential equation is integrated by means of elliptic functions. The computations of de Sitter [5] relating to this point, which represent generalizations of Friedmann’s solution (for whom \(\beta=0\)), are not, however, of great physical interest, since numerical quantities from these computations can be obtained only with the aid of special hypotheses. One of them (which Eddington and de Sitter regard as the most plausible) consists in the following: before the universe began to expand, its radius was constant for an infinite (or at any rate very long) time. If this radius is denoted by \(R_{0}\), then one must have

\[ \frac{1}{\varkappa}\left(\frac{3}{R_{0}^{2}}-\lambda\right) = \frac{\alpha}{\varkappa R_{0}^{3}} + \frac{3\beta}{\varkappa R_{0}^{4}}. \]

The constancy of the radius \(R\) means that the line element \(ds^{2}\) did not differ from that which occurs in Einstein’s solution. If it is assumed that space contained the greatest mass which it is in general capable of containing, then it turns out (see 26) that \(\alpha=2R_{0}\), \(\beta=0\), \(R_{0}=\dfrac{1}{\sqrt{\lambda}}\).

Therefore the differential equation by which \(R\) is determined takes the form

\[ \frac{3}{R^{2}}-\lambda + \frac{3}{c^{2}R^{2}}\left(\frac{dR}{dt}\right)^{2} = \frac{2}{\sqrt{\lambda}\,R^{3}}. \tag{42} \]

Since observations show that at the present time

\[ \frac{1}{R}\frac{dR}{dt}=\frac{3}{2}\times10^{-17}, \]

the present radius of the universe satisfies the equation

\[ \frac{3}{R^{2}}-\lambda+\frac{3}{4}\times10^{-54} = \frac{2}{R^{3}\sqrt{\lambda}}. \tag{43} \]

On the other hand, if it is assumed that the present density of matter in the universe is \(2\times10^{-28}\ \mathrm{g/cm^{3}}\), then this same radius also satisfies the equation

\[ 2\times10^{-28}R^{3} = \frac{2\lambda}{\varkappa}\left(\frac{1}{\sqrt{\lambda}}\right)^{3}; \]

since \(\dfrac{2\lambda}{\varkappa}\) was the density of matter in the universe before it began to expand, and \(\dfrac{1}{\sqrt{\lambda}}\) its radius. Since

\[ \varkappa=1.87\times 10^{-27}, \]

we obtain

\[ \frac{1}{\sqrt{\lambda}}=1.87\times 10^{-55} R^{3}. \]

Combining this equation with (43), we find

\[ 3-\lambda R^{2}+\frac{0.75\times 10^{-54}}{1.87\times 10^{-55} R\sqrt{\lambda}} =\frac{2}{R\sqrt{\lambda}}, \]

whence, approximately,

\[ R\sqrt{\lambda}=\frac{R}{R_{0}}=2. \]

The present radius of the universe is twice as large as its initial radius. Moreover,

\[ \lambda=1.5\times 10^{-54}\ \mathrm{cm}^{-2}, \]

\[ R=1.6\times 10^{27}\ \mathrm{cm}=1600\ A. \]

These figures, as de Sitter himself says, cannot lay any great claim to reliability. Nevertheless, let us add to them some other numerical data that follow from them. The mass of the universe, which according to (26) must be equal to \(\dfrac{4\pi^{2}}{\varkappa}R_{0}\), turns out to be \(1.7\times 10^{55}\ \mathrm{g}\), or \(0.8\times 10^{22}\ \odot\). The number of protons in the universe is \(10^{79}\).

The differential equation (42), satisfied by the radius, can be integrated in finite form without elliptic functions. If, for convenience of notation, we introduce the variable \(z=R\sqrt{\lambda}\), then by elementary manipulations one can obtain the relation

\[ 2\log\left(\sqrt{z+2}+\sqrt{z}\right) +\frac{1}{\sqrt{3}}\log\frac{\sqrt{3z}-\sqrt{z+2}}{\sqrt{3z}+\sqrt{z+2}} = \]

\[ = c\sqrt{\frac{\lambda}{3}}\,t+\mathrm{const}. \]

From this relation it is easy to see that for very large negative \(t\) the variable \(z\) must be close to \(1\)

(introduced by Einstein with radius \(\frac{1}{\sqrt{\lambda}}\)), and for very large positive \(t\) the variable \(z\) is proportional to \(\cosh\left(c\sqrt{\frac{\lambda}{3}}\,t\right)\). If we substitute \(R=R_0\cosh\left(c\sqrt{\frac{\lambda}{3}}\,t\right)\) into equation (42), then for large \(t\) the right-hand side may freely be set equal to zero, and it turns out that the equation is satisfied in that case if \(R_0=\sqrt{\frac{3}{\lambda}}\). Therefore, for very large positive \(t\), \(ds^2\) is expressed by the formula

\[ ds^2=-\frac{3}{\lambda}\cosh^2\left(c\sqrt{\frac{\lambda}{3}}\,t\right)\left[d\chi^2+\sin^2\chi\left(d\vartheta^2+\sin^2\vartheta\,d\varphi^2\right)\right]+c^2\,dt^2. \]

It is not difficult to verify from formulas (35) that such a world-measure corresponds to a world in which the density of matter and of radiant energy continuously decreases, tending to zero. The complete dispersal of matter, at least the complete dispersal of galaxies—this is the fate toward which our world is tending. The starting point of its development was Einstein’s universe with radius \(0.8\times10^{27}\) cm. Such a universe, according to Eddington, is unstable: indeed, if in equations (35) we put \(p=0\), then we obtain

\[ -\frac{6}{\varkappa}\frac{d^2R}{dt^2}=Rc^2\left(\rho-\frac{2\lambda}{\varkappa}\right), \]

whence it follows that for equilibrium the equality \(\rho=\frac{2\lambda}{\varkappa}\) is necessary. If for some accidental reason the mean density increases \(\left(\rho>\frac{2\lambda}{\varkappa}\right)\), then the radius of the world decreases, and this leads to a further increase of the density. If, however, the density becomes smaller than its equilibrium value, then the radius of the world grows and the density decreases still more strongly. Thus the universe with constant radius is unstable; the equilibrium must at some time be disturbed, and the static solution must be replaced by a dynamic one.

It should not be thought, however, that the new cosmological theory, which found in Eddington and in de Sit-

so many ardent supporters, evokes no objections and raises no fundamental doubts. The strangest point of this theory is the extraordinary rapidity with which the radius of the universe increases. It is difficult to reconcile these new conceptions with those long intervals which, in the opinion of cosmogonists, have elapsed since the birth of the stars of our galactic system (thus, for example, in Jeans’s opinion, the age of our galaxy is approximately equal to \(10^{13}\) years). “The rapid expansion of the universe now taking place,” says Eddington, “is decidedly incompatible with ideas of billions of years. It is unlikely that more than \(10^{10}\) years have now passed since the moment when the radius of the universe was one and a half times greater than its initial value. If the sun had really existed for \(5 \times 10^{12}\) years, then it is strange that it did not decide for so long to surround itself with a system of planets and did so only when the universe began to enter a state of expansion. At the present time the radius of the universe doubles every one and a half billion years. In \(10^{10}\) years the spiral nebulae will be 10 stellar magnitudes fainter than now. If the assumptions about billions of years were correct, astronomers would have to count it as a special good fortune that they appeared just in time to observe the spiral nebulae—these interesting, but rapidly vanishing, celestial objects.”

Cosmological theory will undoubtedly undergo many further changes. Above all, it will have to extend its time scales, which are nevertheless extremely constraining for cosmogonists. ****

Editorial Notes

* It must be emphasized that knowledge of the four coordinates determines only the space-time background of the phenomenon, but not its qualitative content, which is not exhausted, of course, by the numerical values of the coordinates.

** The word “matter” here, as also below, is used in accordance with the modern terminology established in physics, and not in

in the general philosophical sense. From the point of view of dialectical materialism, radiant energy, too, is, of course, matter.

*** By “annihilation of matter,” modern physics understands the transformation of an electron and a proton, upon collision, into radiant energy. From the point of view of dialectical materialism, no “annihilation of matter” occurs here, of course, since radiant energy is also matter.

**** A much weaker point of the theory is the very treatment of the problem of irreversibility, which requires fundamental reworking from the methodological point of view.

LITERATURE

1. A. Einstein. Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie, Berl. Sitzungsber. 142. 1917.

2. W. de Sitter. On Einstein’s Theory of Gravitation and its Astronomical Consequences. Third Paper. Mont. Not. Roy. Astr. Soc. 78, 3 (1917). See also Proc. Akad. Amsterdam. 19, 1217 (1917) and 20, 229 (1917). See also A. Einstein, Berl. Sitzungsber. 270 (1918), and W. de Sitter, Proc. Akad. Amsterdam. 20, 1309 (1918).

3. H. Weyl. Zur allgemeinen Relativitätstheorie. Phys. ZS. 24, 230 (1923). See also Raum—Zeit—Materie. 5. Aufl. 322—323 (1923).

4. L. Silberstein. The Theory of Relativity. 2-nd edition, p. 519. (1924).

5. W. de Sitter. On the distances and radial velocities of extragalactic nebulae etc. Proc. Nat. Acad. U. S. A. 16, 474 (1930); see also De snelheden der extragalactische nevels en hunne verklaring door de relativiteitstheorie. Proc. Akad. Amsterdam, 39. 82 (1930). Astronomical material is given by de Sitter especially in Bulletin Astr. Inst., Netherlands 185. (1930).

6. L. Silberstein. The Radius of Space. Nature. 123. 618 (1929) (cablegram from New York); New Determination of the Curvature Radius of Space-time. Ibid. 124, 179 (1929). See also his book The Size of the Universe (1930) (on it A. S. Eddington. Space and its Properties. Nature, 125, 849 (1930).

7. A. Friedman. Über die Krümmung des Raumes, ZS. f. Phys. 10, 377 (1922); On the curvature of space. Zh. R. F. Kh. O., phys. 56, 596 (1924); A. Einstein. Bemerkung zu der Arbeit von A. Friedman, ZS. f. Phys. 11, 326 (1922); see also A. Friedman, Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes. ZS. f. Phys. 21, 326 (1924).

8. G. Lemaître. Un univers homogène de masse constante et de rayon croissant, rendant compte de la vitesse radiale des nébuleuses extra-galactiques. Ann. de la Soc. Scientif. de Bruxelles, 47, 49 (1927).

9. A. S. Eddington. On the Instability of Einstein’s Spherical World. Mont. Not. Roy. Astr. Soc. 90, 668 (1930).

10. R. C. Tolman. The Effect of the Annihilation of Matter on the Wave-Length of Light from the Nebulae. Proc. Nat. Acad. U. S. A. 16, 320 (1930); More Complete Discussion of the Time-Dependence of the Nonstatic Line Element for the Universe. Ibid. 16, 409 (1930). On the Estimation of Distances in a Curved Universe with a Non-Static Line Element. Ibid. 16, 511 (1930).

1. From Silberstein’s point of view, the distances to spiral nebulae must be many times smaller than those measured by Hubble. Since the apparent brightness of a nebula must be inversely proportional to \(R^2\sin^2\chi\) (and not, for example, to \(R^2\tan^2\chi\)), the distance measured by the Cepheid method must be \(R\sin\chi\), i.e. must be smaller than the radius of the world, if only the relation used by Hubble between the period of the Cepheids and their absolute brightness is indeed valid. It is difficult for the contemporary astronomer to regard this “if” as unobjectionable. ↩↩