Editorial note to “A Homogeneous Universe of Constant Mass and Increasing Radius accounting for the Radial Velocity of Extra–Galactic Nebulæ” by Georges Lemaître (1927)
Abstract
This is an editorial note to accompany printing as a Golden Oldie in the Journal of General Relativity and Gravitation^{1}^{1}1See http://www.mth.uct.ac.za/ cwh/goldies.html. of the fundamental article by Georges Lemaître first published in French in 1927, in which the author provided the first explanation of the observations on the recession velocities of galaxies as a natural consequence of dynamical cosmological solutions of Einstein’s field equations, and discovered the so–called Hubble law. We analyze in detail the scientific contents of this outstanding work, we describe how it remained unread or poorly appreciated until 1930, and we list and explain the differences between the 1927 and 1931 versions. Indeed the English translation published in 1931 in MNRAS was not perfectly faithful to the original text – it was updated. As it turned out very recently, the updates were done by Lemaître himself, but the discrepancies between the two texts caused a temporary stir among historians. Our new translation – given in the Appendix – follows the 1927 version exactly.
Introduction
As already pointed out in a previous Golden Oldie devoted to the Lemaître’s short note of 1931 which can be considered as the true “Charter” of the modern big bang theory [1], although the Belgian scientist was primarily a remarkable mathematician and a theoretical physicist, he stayed closely related to astronomy all his life and always felt the absolute need for confronting the observational data and the general relativity theory. This basic fact explains why as soon as 1927, while still a beginner in cosmology, he was the first one to be able to understand the recent observations on the recession velocities of galaxies as a natural consequence of dynamical cosmological solutions of Einstein’s field equations.^{2}^{2}2A number of other authors such as Hermann Weyl [2], Carl Wirtz [3], Ludwig Silberstein [4], Knut Lundmark [5] had looked for a relation that fit into the context of de Sitter’s static model which presented spurious radial velocities. Before examining in detail the contents of his outstanding article, let us summarize the road which, in the few preceding years, led the young Lemaître to the expanding universe (see e.g. [6]).
In 1923, the same year as he was ordained as a priest, Georges Lemaître obtained a 3year fellowship from the Belgian government, enabling him to study abroad. He spent the first year at the University of Cambridge, England, where he studied stellar astronomy, relativistic cosmology and numerical analysis under the direction of Arthur Eddington. He spent the second year at Harvard College Observatory in Cambridge, Massachusets, directed by Harlow Shapley who worked on the problem of nebulæ . Then he passed to the Massachusetts Institute of Technology (M.I.T.), where Edwin Hubble and Vesto Slipher were active. The first one measured the distances of nebulæ by observing variable stars of the Cepheid type, the second one estimated their radial velocities from their spectral shifts.
While following closely the experimental work of the American astronomers, who were going soon to found observational cosmology, Lemaître undertook a PhD thesis at M.I.T. with his compatriot Paul Heymans as advisor, on the gravitational field of fluids in general relativity – a theoretical subject suggested by Eddington. At the end of 1924, he attended a meeting in Washington which remained famous since the discovery of Cepheids in spiral nebulas was announced there by Edwin Hubble; this made it possible to prove the existence of galaxies external to ours, and Lemaître understood at once that this new design of “island universes” would have drastic consequences for the theories of relativistic cosmology.
On July 1925, his American stay ended and Lemaître had to go back to Belgium. In this same decisive year for observational cosmology, Lemaître obtained his first notable scientific results, concerning the cosmological solution found by De Sitter [7]. In the first article [8] he demonstrated how he could introduce new coordinates for the De Sitter universe which made the metric no more static, with a space of zero curvature and a scale factor depending exponentially on time. This metric would be used twenty years later by the keenest adversaries of the theory of the expanding universe in the framework of “steadystate” models [9], and still later in the 1980’s to describe the hypothetical inflationary phase of the very early universe, see e.g. [10]. In the second article [11] he deduced that the relation between the relative speed of testparticles and their mutual distances in the De Sitter universe was linear. It was the first time that the cosmological constant (when it is positive) was seen allotting the role of a “cosmic repulsion” forcing the worldlines of particles to recede with time. However, although he found this non–static feature to be promising because of its connection to the redshifts of nebulæ , he also realized that the model resulted in an infinite Euclidean space, that he considered inadmissible: as a neoThomist he did not accept the actual infinite and remained faithful to the finitude of space and matter throughout his career. Thus he had to seek for an alternative explanation, involving a truly non–static and spatially closed solution of Einstein’s equations.
In 192627, Lemaître went again to the United States, where he remained at M.I.T. during three quarters of the academic year. Back in Europe in June 1927, he was informed by letter that he got his PhD [12], having been exempted of oral defense. The same year, he was appointed professor at the University of Louvain and published his great article on the expanding universe.
Recession of galaxies and expanding universe
Since 1912, Vesto Slipher had undertaken a program of measurement of the radial velocities of spiral nebulae. Interpreted in terms of the Doppler effect, the shifts in frequency (or wavelength) implied a radial speed of displacement of the source compared to the observer. Radial speeds were thus indirectly measured by spectroscopy. By 1917, Slipher (see [13] and references therein) had analyzed the spectra of 25 spiral nebulæ, which he had observed at Lowell Observatory in Flagstaff, Arizona; 21 of them presented redshifts that could be interpreted as a systematic motion of recession (the exceptions were M81 and 3 galaxies from the Local Group). However, nobody suspected yet the repercussions that these preliminary data would have soon for the whole of cosmology, mainly due to the fact that the debate on whether spiral nebulæ were island universes went on. The evidence for the redshifts mounted mainly due to Slipher’s efforts, and by 1923 reached a score of 36 among 41 spiral nebulæ .
Slipher never published his final list,^{3}^{3}3For details see [14]. but it was given in Arthur Eddington’s book of 1923 [15], who noticed that “one of the most perplexing problems in cosmogony is the great speed of spiral nebulæ . Their radial velocities average about 600 km. per sec. and there is a great preponderance of velocities of recession from the solar system”. The influential British astronomer suggested that effects due to the curvature of spacetime should be looked for and referred to De Sitter’s model for a possible explanation.
Thanks to his various stays at Cambridge, England, and at M.I.T. (where he met Slipher personally), Lemaître was perfectly informed of these preliminary results, and he wanted to take account of the available data by using a new cosmological solution of Einstein’s equations.
As the title of his 1927 article clearly states, Lemaître was able to connect the expansion of space arising naturally from the non–static cosmological solutions of general relativity with the observations of the recession velocities of extragalactic nebulæ.
He begins to review the dilemma between the De Sitter and Einstein universe models. The De Sitter model ignored the existence of matter; however, it emphasized the recession velocities of spiral nebulæ as a simple consequence of the gravitational field. Einstein’s solution allowed for the presence of matter and led to a relation between matter density and the radius of the space – assumed to be a positively curved hypersphere; being strictly static due an adjustment of the cosmological constant, it could not, however, explain the recession of the galaxies. Lemaître thus looks for a new solution of the relativistic equations combining the advantages of the Einstein and De Sitter models without their inconveniences, i.e. having a material content and explaining at the same time the recession velocities.
For this, in the next section he assumes a positively curved space (as made precise in a footnote, with “elliptic topology”, namely that of the projective space obtained by identification of antipodal points of the simplyconnected hypersphere ; see [16] for an explanation of such a choice) with the radius of curvature (and consequently the matter density ) being a function of time , and a nonzero cosmological constant . From Einstein’s field equations he obtains differential equations (eq. (2)(3)) for and almost identical to those previously obtained by Friedmann [17] (at the time Lemaître was not aware of Friedmann’s work, see below). The difference is that Lemaître supposes the conservation of energy (eq. (4)) – this is the first introduction of thermodynamics in relativistic cosmology – and he includes the pressure of radiation as well as the term of matter density into the stressenergy tensor (he rightly considers the matter pressure to be negligible). Lemaître emphasizes the importance of radiation pressure in the first stages of the cosmic expansion. Now it is well known that, within the framework of big bang models, the approximation of zero pressure is valid only for times posterior to the big bang for approximately four hundred thousand years. Just like Einstein and De Sitter, Friedmann had made the assumption that the term of pressure in the stressenergy tensor was always zero. The equations derived by Lemaître are thus more general and realistic.
Lemaître shows how the Einstein and De Sitter models are particular solutions of the general equations. Next he chooses as initial conditions at and he adjusts the value of the cosmological constant such that , in the same way Einstein had adjusted the value of in his static model with constant radius.
As a consequence, the exact solution he obtains in eq. (30) describes a monotonous expanding universe, which, when one indefinitely goes back in time, approaches in an asymptotic way the Einstein static solution, while in the future it approaches asymptotically an exponentially expanding De Sitter universe.
This model, deprived of initial singularity and, consequently, not possessing a definite age – as well as the “monotonous solution of second species” found earlier by Friedmann – will be later baptized the EddingtonLemaître’s model (see below).
Lemaître does not provide a graph for but gives numerical values in a table going from to . For the sake of clarity our figure 1 depicts such a graph.
Lemaître conceived the static Einstein universe as a kind of preuniverse out of which the expansion had grown as a result of an instability. As a physical cause for the expansion he suggested the radiation pressure itself, due to its infinite accumulation in a closed static universe, but he did not develop this (erroneous) idea.
While giving preference to this particular model in his article, Lemaître nevertheless calculated separately the whole of dynamical homogeneous cosmological solutions, since he had the general formula (eq. (11)) making it possible to calculate the time evolution of all the homogeneous isotropic models with positive curvature. The Lemaître archives at the University of Louvain keep a red pad with the inscription “1927”, which contains the galley proofs of his article, some notes in handwriting connected with the paper, and two diagrams which (unfortunately) do not appear in any of his publications. These diagrams depict the time evolution of the space scale factor depending on the value of the cosmological constant for all homogeneous and isotropic solutions of Einstein’s equations with positive curvature of space.
As mentioned above, the 1927 article does not refer to the work of Friedmann, published in Zeitschrift für Physik – although one of the best known journals in theoretical physics at that time. This absence seems strange if one remembers the two notes by Einstein published in the same review [18], which had been largely discussed in the scientific community. A plausible explanation is that Lemaître could not read the German [19]. Friedmann’s articles were pointed out to Lemaître by Einstein himself, during their meeting at the 1927 Solvay Conference. The reference to Friedmann thus appears for the first ime in a text of 1929 written in French, La grandeur de l’espace [21], in which Lemaître thanks “Mr. Einstein for the kindness that he showed by announcing to me the important work of Friedmann which includes several of the results contained in my note on a homogeneous universe”. The reference will also appear in the 1931 English translation of Lemaître’s article, see below.
The exceptional interest of Lemaître’s work is to provide the first interpretation of cosmological redshifts as a natural effect of the expansion of the universe within the framework of general relativity, instead of a real motion of galaxies: as it is written down in eq.(23), space is constantly expanding and consequently increases the apparent separations between galaxies. This idea will prove to be one of the most profound discoveries of our time.
The relation of proportionality (23) between the recession velocity and the distance is an approximation valid at not too large distances which can be used “within the limits of the visible spectrum”. Then, using the available astronomical data, Lemaître provides the explicit relation of proportionality in eq. (24), with a factor 625 or 575 km/s/Mpc, depending on his choice of observations which presented an enormous scatter. This is the first determination of the socalled Hubble law and the Hubble constant, that should as well have been named Lemaître’s law.
For this the Belgian scientist uses a list of 42 radial velocities compiled by Gustav Strömberg, a Swedish astronomer at the Mount Wilson Observatory,^{4}^{4}4Strömberg [22] relied himself on redshifts measured by Slipher and included some globular clusters in addition to spiral nebulæ . and deduces their distance from a recent empirical formula between the distance and the absolute magnitude provided by Hubble [23], who himself took them from Hopmann [24]. Eventually, Lemaître is able to give the numerical figures for the initial and presentday values of the radius of the universe, resp. and . At the very end he points out that the largest part of the universe will be forever out of reach of the visible spectrum, since the maximum distance reached by the Mt Wilson telescope is only , whereas for a distance only greater than the whole visible spectrum is displaced into the infrared – he could not imagine the space era with infrared and submillimeter telescopes placed on board of satellites.
We have seen above that Lemaître knew already all the solutions of Einstein’s equations for homogeneous and isotropic universes. The reason why he privileged a very particular model, adjusting the cosmological constant in order to have no beginning of time, is due to his overestimate of the Hubble constant: as is well known, the latter gives an order of magnitude of the duration of the expansion phase; with the estimate of about 600 km/s/Mpc found by Lemaître, this period is about one billion years only, a number less than the age of the Earth estimated by the geologists of the time. Thus the model with exponential expansion and no beginning allowed to reconcile the theory with both astronomical and geological data.
First reactions
The significance of Lemaître’s work has remained mostly unnoticed for three years, not exclusively (but partly) due to the fact that it was published in French in an “obscure and completely inaccessible journal”, as is sometimes claimed [25], instead of one of the prestigious astronomical journals of the time.^{5}^{5}5The paper was reprinted later in 1927 in vol. 4 of Publications du Laboratoire d’Astronomie et de Géodésie de l’Université de Louvain, still less suited for widespread dissemination. As rightly pointed out by Lambert [26], the Annales de la Société Scientifique de Bruxelles published some articles in English, had an excellent scientific level and therefore were displayed in a large number of academic libraries and observatories all around the world; also French could be read by a much larger scientific audience than today. Indeed, the main obstacle to a larger diffusion of Lemaître’s article was that most of the physicists of the time, such as Einstein and Hubble, could not accept the idea of a non–static universe. This was not the case with Eddington; unfortunately, his former mentor, to whom Lemaître had sent a copy, either forgot to read it in time, or he had not understood its importance.
From 24 to 29 October 1927 the Fifth Solvay Conference in Physics took place in Brussels, one of the great meetings of world science. The Solvay Conference was devoted to the new discipline of quantum mechanics, whose problems disturbed many physicists. Among them was Einstein. For Lemaître, it was the opportunity to discuss with the father of general relativity. He later reported himself on this meeting: “While walking in the alleys of the Parc Léopold, [Einstein] spoke to me about an article, little noticed, which I had written the previous year on the expansion of the universe and which a friend had made him read. After some favorable technical remarks, he concluded by saying that from the physical point of view that appeared completely abominable to him. As I sought to prolong the conversation, Auguste Piccard, who accompanied him, invited me to go up by taxi with Einstein, who was to visit his laboratory at the University of Brussels. In the taxi, I spoke about the speeds of nebulas and I had the impression that Einstein was hardly aware of the astronomical facts. At the university, everyone began to speak in German” [27]. Einstein’s response to Lemaître shows the same unwillingness to change his position that characterized his former response to Friedmann (see e.g. [28]): he accepted the mathematics, but not a physically expanding universe!
In 1928 H. P. Robertson published an article [29] in which he wanted to replace De Sitter’s metric by a “mathematically equivalent in which many of the apparent paradoxes inherent in [De Sitter’s solution] were eliminated”. He got the formula where is the distance of the nebula and the radius of curvature of the universe, but in the framework of a static solution. Robertson used the same set of observations as had been taken by Lemaître^{6}^{6}6He did not know the Lemaître’s articles of 1925 and 1927. and that would be taken by Hubble one year later. From this he calculated cm, and a proportionality constant of 464 km/s/Mpc (that he did not calculate, the figure can be found in [30]). The main interest of Robertson’s work (see also [31]) is that he was the first to search in detail for all the mathematical models satisfying a spatially homogeneous and isotropic universe – which imply strong symmetries in the solutions of Einstein’s equations.
In 1929, Hubble [32] used the experimental data on the Doppler redshifts mostly given by Slipher (who was not quoted) and found a linear velocitydistance relation with km/s/Mpc for 24 objects and km/s/Mpc for 9 groups. The law was strictly identical to Lemaître’s Eq.(24), with almost the same proportionality factor, but Hubble did not make the link with expanding universe models. He stated “The outstanding feature, however, is the possibility that the velocitydistance relation may represent the De Sitter effect”. In fact Hubble never read Lemaître’s paper; he interpreted the galaxy redshifts as a pure Doppler effect (due to a proper radial velocity of galaxies) instead of as an effect of space expansion. And throughout his life he would stay skeptical about the general relativistic interpretation of his observations. For instance, in the 202 pages of his book of 1936 The Realm of the Nebulae [33], he tackled the theoretical interpretation of the observations only in a short ultimate paragraph on page 198, in which he quoted Einstein, De Sitter, Friedmann, Robertson, Tolman and Milne. As pointed out by his biographer G. Christianson, Hubble was chary of “all theories of cosmic expansion long after most astronomers and physicists had been won over. When queried about the matter as late as 1937, he sounded like an incredulous schoolboy: ‘Well, perhaps the nebulae are all receding in this peculiar manner. But the notion is rather startling’ ” [34]. Indeed the fact that the expansion of the universe was discovered by Hubble is a myth that was first propagated by his collaborator Humason as soon as 1931 (see e.g. [35]) and Hubble himself, who was fiercely territorial; in a letter to De Sitter dated 21 August 1930, he wrote “I consider the velocitydistance relation, its formulation, testing and confirmation, as a Mount Wilson contribution and I am deeply concerned in its recognition as such” (quoted in [36]).
One month only after Hubble’s article, Tolman joined the game of searching for an explanation of recession velocities, but still in the framework of a static solution [37], as he said “the correlation between distance and apparent radial velocity of the extra–galactic nebulae obtained by Hubble, and the recent measurement of the Doppler effect for a very distant nebula made by Humason at the Mount Wilson Observatory, make it desirable to consider once more the theoretical relations between distance and Doppler effect which could be expected from the form of line element for the universe proposed by De Sitter”. One year later, Tolman published another article [38] where he suggested that the expansion was due to the conversion of matter into radiation, an idea already proposed by Lemaître in his 1927 article, who again was not quoted.
A new opportunity for the recognition of Lemaître’s model arose early in 1930. In January, in London, a discussion between Eddington and De Sitter took place at a meeting of the Royal Astronomical Society. They did not know how to interpret the data on the recession velocities of galaxies. Eddington suggested that the problem could be due to the fact that only static models of the universe were hitherto considered; he nicely formulated the situation as follows: “Shall we put a little motion into Einstein’s world of inert matter, or shall we put a little matter into de Sitter’s Primum Mobile?” [39], and called for new searches in order to explain the recession velocities in terms of dynamical space models.
Having read the report of the meeting of London, Lemaître understood that Eddington and De Sitter posed a problem which he had solved three years earlier. He thus wrote to Eddington to remind him about his communication of 1927 and requested him to transmit a copy to de Sitter: “Dear Professor Eddington, I have just read the February n of the Observatory and your suggestion of investigating non statical intermediary solutions between those of Einstein and De Sitter. I made these investigations two years ago. I consider a universe of curvature constant in space but increasing with time. And I emphasize the existence of solution in which the motion of the nebulæ is always a receding one from time minus infinity to plus infinity.”^{7}^{7}7From a copy kept at the Archives Lemaître of LouvainlaNeuve, quoted in [26]. Lemaître precised: “I had occasion to speak of the matter with Einstein two years ago. He told me that the theory was right and is all which needs to be done, that it was not new but had been considered by Friedmann, he made critics against which he was obliged to withdraw, but that from the physical point of view it was ‘tout à fait abominable’ ” (quoted in [40]).
The British astrophysicist was one of the most prominent figures of science at the time, and was in the best possible position to play a key role in the recognition of the Lemaître’s results. This time he paid attention to Lemaître’s contribution, dispatched a copy to De Sitter in Holland and H. Shapley in the United States. Eddington was somewhat embarrassed. According to George McVittie, at the time a research student of Eddington working with him on the stability of the Einstein’s static model, “[I remember] the day when Eddington, rather shamefacedly, showed me a letter from Lemaître which reminded Eddington of the solution to the problem which Lemaître had already given. Eddington confessed that although he had seen Lemaître’s paper in 1927 he had forgotten completely about it until that moment” (quoted in [40]).
On March 19th, Eddington accompanied his invoice of Lemaître’s paper to De Sitter in Leiden by the following comment: “It was the report of your remarks and mine at the [Royal Astronomical Society] which caused Lemaître to write to me about it. At this time, one of my research students, McVittie, and I had been worrying at the problem and made considerable progress; so it was a blow to us to find it done much more completely by Lemaître (a blow attenuated, as far as I am concerned, by the fact that Lemaître was a student of mine)” (reported in [41]).
De Sitter answered Lemaître very favorably in a letter dated March 25th, 1930, and the Belgian physicist replied to him on April 5th (these letters are fully displayed in [42]). In late May, De Sitter published a discussion about the expansion of the universe [43], where he wrote “A dynamical solution of the equations (4) with the lineelement (5) (7) and the material energy tensor (6) is given by Dr. G. Lemaître in a paper published in 1927, which had unfortunately escaped my notice until my attention was called to it by Professor Eddington a few weeks ago.”
On his side, Eddington reworked his communication to the following meeting of the Royal Astronomical Society in May, to bring Lemaître’s work to the attention of the world [44]. Then he published an important article [45] in which he reexamined the Einstein static model and discovered that, like a pen balanced on its point, it was unstable: any slight disturbance in the equilibrium would start the increase of the radius of the hypersphere; thus he adopted Lemaître’s model of the expanding universe – which will be henceforward referred to as the Eddington–Lemaître model – and calculated that the original size of the Einstein universe was about 1200 million light years, of the same order of magnitude as that estimated by Lemaître in 1927. Interestingly enough, Eddington also considered the possibility of an initial universe with a mass greater or smaller than the mass of the Einstein model, but he rejected the two solutions, arguing that, for , “it seems to require a sudden and peculiar beginning of things”, whereas for , “the date of the beginning of the universe is uncomfortably recent”.
Eventually, Eddington sponsored the English translation of the 1927 Lemaître’s article for publication in the Monthly Notices of the Royal Astronomical Society [46].
Then, with the support of Eddington and De Sitter, Lemaître suddenly rose to become a celebrated innovator of science. He was invited to London in order to take part in a meeting of the British Association on the relation between the physical universe and spirituality. But in the meantime he had considerably progressed in his investigations of relativistic cosmologies, and instead of promoting his model of 1927, he dared to propose that the Universe expanded from an initial point which he called the “Primeval Atom”. Then cosmology experienced a second paradigmatic shift [47].
The English translation and discrepancies
A great deal has been written on the topic of who really discovered the expanding universe [48]. The French astronomer Paul Couderc [49] was probably the first one to rightly underline the priority of Lemaître over Hubble, but since Lemaître himself never claimed any priority (see [50] for more details), the case was not much discussed.
An intriguing discrepancy between the original French article and its English translation had already been quoted by various authors (e.g. [40–42]): the important paragraph discussing the observational data and eq. (24) where Lemaître gave the relation of proportionality between the recession velocity and the distance (in which the determination of the constant that later became known as Hubble’s constant appears) was replaced by a single sentence: “From a discussion of available data, we adopt ”. It was found curious that the crucial paragraphs assessing the Hubble law were dropped so that, either due to Eddington’s blunder^{8}^{8}8Until very recently the identity of the translator was not assessed, generally assumed to be that of Eddington himself. or some other mysterious reason, Lemaître was never recognized as the discoverer of the expansion of the universe. De facto Lemaître was eclipsed and multitudes of textbooks proclaim Hubble as the discoverer of the expanding universe, although Hubble himself never believed in such an explanation [51].
Suddenly, in 2011, a burst of accusations has flared up against Hubble, from the suspicion that a censorship was exerted either on Lemaître by the editor of the M.N.R.A.S. [52] or on the editor by Hubble himself [36] – suspicion based on the “complex personality” of Hubble, who strongly desired to be credited with determining the Hubble constant.
The controversy was ended by Mario Livio, from the Space Telescope Institute [53], with the help of the Archives Lemaître at Louvain and the Archives of the Royal Astronomical Society (see also [26] for additional details). It is not the scope of the present note to enter into the explanations that solve the conundrum, it is sufficient to say that it is now certain that Lemaître himself translated his article, and that he chose to delete several paragraphs and notes without any external pressure. On the contrary, he was encouraged to add comments on the subject; but the Belgian scientist, who had indeed new ideas, preferred to publish them in a separate article, published in the same issue of M.N.R.A.S. [54].
For the present purpose it is much more interesting to list in detail all the discrepancies – as far as we know a little work that has never been done – in order to better understand how the preoccupations of Lemaître had changed since 1927, and how the question which he had in mind in 1931 was less the expansion of space than the deep cause of it, how it started and how the first galaxies could form.

Section 1, first paragraph
The footnote “We consider simply connected elliptic space, i.e. without antipodes” is suppressed from the 1931 translation.
As soon as 1917, De Sitter [17][55] distinguished the spherical space and the projective space , that he called the elliptical space. As recalled by Lemaître in the first paragraph, has a (comoving) volume instead of for , and the longest closed straight line is instead of . The main cosmological difference is due to the presence, in , of an antipodal point associated to any point, and in particular to the observer, at a distance of precisely. This was considered as an undesirable fact, so that cosmological models with seemed preferable than those with .
Eddington [15] also referred to elliptical space as an alternative more attractive than , and Lemaître also adopted this point of view.^{9}^{9}9Note that elliptical space is not simply connected but multiply connected, see e.g. [16]. We can infer that he suppressed his footnote because in any case, topology has no influence on the dynamics, which was the very purpose of his work, and because in the meantime he had published an extended discussion on the subject [54], which he merely points out in reference 4 of the 1931 version.

Section 1, second paragraph
In the 1931 translation, the original sentence “[…] it is of great interest as explaining the fact that extragalactic nebulæ seem to recede from us with a huge velocity […]” is replaced by “[…] it is of extreme interest as explaining quite naturally the observed receding velocities of extragalactic nebulæ […]” to acknowledge the fact that, due to the post–1927 observational work of Hubble and Humason, the receding velocities had acquired a firm observational status.

Section 1, third paragraph
The sentence “This relation forecasted the existence of masses enormously greater than any known when the theory was for the first time compared with the facts” is replaced by “This relation forecasted the existence of masses enormously greater than any known at the time”.

Section 1, third paragraph
The footnote giving reference to Hubble’s article of 1926 is suppressed because it is no more uptodate.

Section 1, sixth paragraph
The two footnotes are suppressed. They both give geometrical details and subtleties about the De Sitter solution that Lemaître probably judged not appropriate for a journal such as M.N.R.A.S, more devoted to astronomy than to geometry. These details came mainly from an article by K. Lanczos and the 1925 articles of Lemaître himself. In the 1931 version, he added at the end of the article the bibliographic references to Lanczos and himself without development, and added references to H. Weyl and P. du Val.

Section 2
Between eqs. (4) and (5) the sentence “It is suitable for an interesting interpretation” has disappeared for the sake of economy.

Section 4
The paragraphs from “Radial velocities of 43 extragalactic nebulæ […]” up to “This relation enables us to calculate ”, as well as the three footnotes, are suppressed and replaced by “From a discussion of available data, we adopt ”. This is precisely the part of the 1927 article where Lemaître discusses the astronomical data on the redshifts, the errors in the distance estimates, where he gives the relation of proportionality between the velocity and distance, and in footnotes, the references to Strömberg and Lundmark, as well as his calculation of two possible values of the constant of proportionality of 575 and 670, depending on how the data are grouped. The original eq. (24) is truncated to a pure numerical one, whereas the original gives precisely what is called the Hubble’s law.
In a letter dated 9 March 1931 addressed to William H. Smart, the editor of M.N.R.A.S., Lemaître writes: “I send you a translation of the paper. I did not find advisable to reprint the provisional discussion of radial velocities which is clearly of no actual interest, and also the geometrical one, which could be replaced by a small bibliography of ancient and new papers on the subject” (quoted in [53]). The choice of Lemaître is quite comprehensible because the data he used in 1927 gave only very imperfectly the linear relation , whereas in 1931 the new data from Hubble allowed to validate this relationship in a much more precise manner, see figure 3 for comparative plots. Also because, as he explained himself in 1950, in 1927 he had not at his disposal data concerning clusters of galaxies, and he added that Hubble’s law could not be proved without the knowledge of the clusters of galaxies” [56]. Here we find again one of the characteristic features of Lemaître’s personality already mentioned, namely the crucial importance he always gave to experimental data.

Section 6
The item 4 of the 1927’s conclusions, giving the radius of the universe as 1/5th the radius of Einstein’s hypersphere, is suppressed, and in the next sentence, Lemaître changes the range of the 100inch Mount Wilson telescope estimated by Hubble from to .

Added references
Whereas the 1931 translation does not contain footnotes, it provides at the end new references that could not be given in the 1927 article: to Friedmann’s article of 1922 and Einstein’s comments on it, the article of Tolman about models of variable radius of 1923, the developments of his own model given by Eddington, De Sitter and himself in 1930, and eventually two popular expositions given by him in 1929 (in French) and by De Sitter in 1931.
References
[1] J.P. Luminet, Editorial note to The beginning of the world from the point of view of quantum theory, Gen. Relativ. Gravit. 43, n10, 29112928 (2011).
[2] H. Weyl, Zur allgemeinen relativitätstheorie, Physikalische Zeitschrift, 24, 230–232 (1923). English translation: H. Weyl, On the general relativity theory, Gen. Relativ. Gravit., 35, 1661 Ð 1666 (2009).
[3] C. Wirtz, De Sitters Kosmologie und die Radialbewegungen der Spiralnebel, Astronomische Nachtrichten, 222, 21 (1924).
[4] L. Silberstein, The Curvature of de Sitter’s SpaceTime Derived from Globular Clusters, M.N.R.A.S., 4 363 (1924).
[5] K. Lundmark, The Motions and the Distances of Spiral Nebulae, M.N.R.A.S., 85, 865 (1925).
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APPENDIX
A HOMOGENEOUS UNIVERSE OF CONSTANT MASS AND INCREASING RADIUS, ACCOUNTING FOR THE RADIAL VELOCITY OF EXTRAGALACTIC NEBULAE
Note by Abbé G. Lemaître
(Translation from the French original by J.P. Luminet)
1. Generalities.
According to the theory of relativity, a homogeneous universe may exist not only when the distribution of matter is uniform, but also when all positions in space are completely equivalent, there is no center of gravity. The radius of space is constant, space is elliptic with uniform positive curvature , the lines starting from a same point come back to their starting point after having travelled a path equal to , the total volume of space is finite and equal to , straight lines are closed lines going through the whole space without encountering any boundary().^{1}^{1}footnotetext: We consider simply connected elliptic space, i.e. without antipodes.
Two solutions have been proposed. That of de Sitter ignores the existence of matter and supposes its density equal to zero. It leads to special difficulties of interpretation which we will be referred to later, but it is of great interest as explaining the fact that extragalactic nebulæ seem to recede from us with a huge velocity, as a simple consequence of the properties of the gravitational field, without having to suppose that we are at a point of the universe distinguished by special properties.
The other solution is that of Einstein. It pays attention to the obvious fact that the density of matter is not zero and it leads to a relation between this density and the radius of the universe. This relation forecasted the existence of masses enormously greater than any known when the theory was for the first time compared with the facts. These masses have since been discovered, the distances and dimensions of extragalactic nebulæ having become established. From Einstein’s formula and recent observational data, the radius of the universe is found to be some hundred times greater than the most distant objects which can be photographed by our telescopes ().^{2}^{2}footnotetext: Cf. Hubble E. ExtraGalactic Nebulæ, ApJ., vol. 64, p. 321, 1926. M Wilson Contr. N 324.
Each theory has its own advantage. One is in agreement with the observed radial velocities of nebulæ, the other with the existence of matter, giving a satisfactory relation between the radius and the mass of the universe. It seems desirable to find an intermediate solution which could combine the advantages of both.
At first sight, such an intermediate solution does not appear to exist. A static gravitational field with spherical symmetry has only two solutions, that of Einstein and that of de Sitter, if the matter is uniformly distributed without pressure or internal stress. De Sitter’s universe is empty, that of Einstein has been described as containing as much matter as it can contain. It is remarkable that the theory can provide no mean between these two extremes.
The solution of the paradox is that the de Sitter’s solution does not really meet all the requirements of the problem ().^{3}^{3}footnotetext: Cf. K. Lanczos, Bemerkung zur de Sitterschen Welt, Phys. Zeitschr. vol. 23, 1922, p. 539, and H. Weyl, Zur allgemeinen Relativitätstheorie, Id., vol. 24, 1923, p. 230, 1923. We follow the point of view of Lanczos here. The worldlines of nebulæ form a bunch with ideal center and real axial hyperplane; space orthogonal to these worldlines is formed by the hyperspheres equidistant from the axial plane. This space is elliptic, its variable radius being minimum at the moment corresponding to the axial plane. Following the assumption of Weyl, the worldlines are parallel in the past; the normal hypersurfaces representing space are horospheres, the geometry of space is thus Euclidean. The spatial distance between nebulæ increases as the parallel geodesics which they follow recede one from the other proportionally to , where is the proper time and the radius of the universe. The Doppler effect is equal to , where is the distance from the source at the moment of observation. Cf. G. Lemaître, Note on de Sitter’s universe, Journal of mathematics and physics, vol. 4, n3, May 1925, or Publications du Laboratoire d’Astronomie et de Géodesie de l’Université de Louvain, vol. 2, p. 37, 1925. For the discussion of the de Sitter’s partition, see P. Du Val, Geometrical note on de Sitter’s world, Phil. Mag. (6), vol. 47, p. 930, 1924. Space is constituted by hyperplanes orthogonal to a time line described by the introduced center, the trajectories of nebulæ are the trajectories orthogonal to these planes, in general they are no more geodesics and they tend to becoming lines of null length when one approaches the horizon of the center, i.e. the polar hyperplane of the central axis with respect to the absolute one. Space is homogeneous with constant positive curvature; spacetime is also homogeneous, for all events are perfectly equivalent. But the partition of spacetime into space and time disturbs the homogeneity. The selected coordinates introduce a center to which nothing corresponds in reality; a particle at rest somewhere else than at the center does not describe a geodesic. The coordinates chosen destroy the homogeneity that exists in the data for the problem and produce the paradoxical results which appear at the socalled ”¡horizon”¿ of the center. When we use coordinates and a corresponding partition of space and time of such a kind as to preserve the homogeneity of the universe, the field is found to be no longer static ; the universe becomes of the same form as that of Einstein, with a radius of space no longer constant but varying with the time according to a particular law ().^{4}^{4}footnotetext: If we restrict the problem to two dimensions, one of space and one of time, the partition of space and time used by Sitter can be represented on a sphere: the lines of space are provided by a system of great circles which intersect on a same diameter, and the lines of time are the parallels cutting orthogonally the lines of space. One of these parallels is a great circle and thus a geodesic, it corresponds to the center of space, the pole of this great circle is a singular point corresponding to the horizon of the center. Of course the representation must be extended to four dimensions and the time coordinate must be assumed imaginary, but the defect of homogeneity resulting from the choice of the coordinates remains. The coordinates respecting the homogeneity require taking a system of meridian lines as lines of time and the corresponding parallels for lines of space, whereas the radius of space varies with time.
In order to find a solution combining the advantages of those of Einstein and de Sitter, we are led to consider an Einstein universe where the radius of space (or of the universe) is allowed to vary in an arbitrary way.
2. Einstein universe of variable radius. Field equations. Conservation of energy.
As in Einstein’s solution, we liken the universe to a rarefied gas whose molecules are the extragalactic nebulæ . We suppose them so numerous that a volume small in comparison with the universe as a whole contains enough nebulæ to allow us to speak of the density of matter. We ignore the possible influence of local condensations. Furthermore, we suppose that the nebulæ are uniformly distributed so that the density does not depend on position.
When the radius of the universe varies in an arbitrary way, the density, uniform in space, varies with time. Furthermore, there are generally internal stresses which, in order to preserve the homogeneity, must reduce to a simple pressure, uniform in space and variable with time. The pressure, being twothirds of the kinetic energy of the molecules, is negligible with respect to the energy associated with matter; the same can be said of interior stresses in nebulæ or in stars belonging to them. We are thus led to put . Nevertheless it might be necessary to take into account the radiationpressure of electromagnetic energy travelling through space; this energy is weak but it is evenly distributed through the whole of space and might provide a notable contribution to the mean energy. We shall keep the pressure in the general equations as the average radiationpressure of light, but we shall write when we discuss the application to astronomy.
We denote the density of total energy by , the density of radiation energy by , and the density of the energy condensed in matter by .
We identify and with the components and of the material energy tensor, and with . Working out the contracted Riemann tensor for a universe with a lineelement given by
where is the elementary distance in a space of radius unity, and the radius of space is a function of time, we find that the field equations can be written
and
Accents denote derivatives with respect to ; is the cosmological constant whose value is unknown, and is the Einstein constant whose value is in C.G.S. units ( in natural units).
The four identities expressing the conservation of momentum and of energy reduce to
which is the conservation of energy equation. This equation can replace (3). It is suitable for an interesting interpretation. Introducing the volume of space , it can be written
showing that the variation of total energy plus the work done by radiationpressure is equal to zero.
3. Case of a universe of constant total mass.
Let us seek a solution for which the total mass remains constant. We can write
where is a constant. Taking account of the relation
existing between the various kinds of energy, the principle of conservation of energy becomes
whose integration is immediate; and, being a constant of integration,
and therefore
By substitution in (2) we have to integrate
or
When and vanish, we obtain the de Sitter solution ()^{5}^{5}footnotetext: Cf. Lanczos, l.c.
The Einstein solution is found by making and constant. Writing in (2) and (3) we find
or
and from (6)
The Einstein solution does not result from (14) alone, it also supposes that the initial value of is zero. Indeed, if, in order to simplify the notation, we write
and put in (11) and , it follows that
For this solution the two equations (13) are of course no longer valid. Writing
we have from (14) and (15)
The value of , the radius of the universe computed from the average density by Einstein’s equations (17), has been found by Hubble to be
We shall see later that the value of can be computed from the radial velocities of the nebulæ; can then be found from (18). Finally, we shall show that a solution introducing a relation substantially different from (14) would lead to consequences not easily acceptable.
4. Doppler effect due to the variation of the radius of the universe
From (1) giving the line element of the universe, the equation for a light ray is
where and relate to spatial coordinates. We suppose that the light is emitted at the point and observed at . A ray of light emitted slightly later starts from at time and reaches at time . We have therefore
where and are the values of the radius at the time of emission and at the time of observation . is the proper time; if is the period of the emitted light, is the period of the observed light. Moreover, can also be considered as the period of the light emitted under the same conditions in the neighbourhood of the observer, because the period of the light emitted under the same physical conditions has the same value everywhere when reckoned in proper time. Therefore
measures the apparent Doppler effect due to the variation of the radius of the universe. It equals the ratio of the radii of the universe at the instants of observation and emission, diminished by unity. is that velocity of the observer which would produce the same effect. When the source is near enough, we can write approximately
where is the distance of the source. We have therefore
Radial velocities of 43 extragalactic nebulæ are given by Strömberg ().^{6}^{6}footnotetext: Analysis of radial velocities of globular clusters and non galactic nebulæ. Ap.J. vol. 61, p. 353, 1925. M Wilson Contr., N 292.
The apparent magnitude of these nebulæ can be found in the work of Hubble. It is possible to deduce their distance from it, because Hubble has shown that extragalactic nebulæ have approximately equal absolute magnitudes (magnitude at 10 parsecs, with individual variations ), the distance expressed in parsecs is then given by the formula .
One finds a distance of about parsecs, varying from a few tenths to 3,3 megaparsecs. The probable error resulting from the dispersion of absolute magnitudes is considerable. For a difference in absolute magnitude of , the distance exceeds from 0,4 to 2,5 times the calculated distance. Moreover, the error is proportional to the distance. One can admit that, for a distance of one megaparsec, the error resulting from the dispersion of magnitudes is of the same order as that resulting from the dispersion of velocities. Indeed, a difference of magnitude of value unity corresponds to a proper velocity of 300 Km/s, equal to the proper velocity of the sun compared to nebulæ . One can hope to avoid a systematic error by giving to the observations a weight proportional to , where is the distance in megaparsecs.
Using the 42 nebulæ appearing in the lists of Hubble and Strömberg (),^{7}^{7}footnotetext: Account is not taken of N.G.C. 5194 which is associated with N.G.C. 5195. The introduction of the Magellanic clouds would be without influence on the result. and taking account of the proper velocity of the Sun (300 Km/s in the direction , ), one finds a mean distance of 0,95 megaparsecs and a radial velocity of 600 Km/sec, i.e. 625 Km/sec at parsecs ().^{8}^{8}footnotetext: By not giving a weight to the observations, one would find 670 Km/sec at parsecs, 575 Km/sec at parsecs. Some authors sought to highlight the relation between and and obtained only a very weak correlation between these two terms. The error in the determination of the individual distances is of the same order of magnitude as the interval covered by the observations and the proper velocity of nebulæ (in any direction) is large (300 Km/sec according to Strömberg), it thus seems that these negative results are neither for nor against the relativistic interpretation of the Doppler effect. The inaccuracy of the observations makes only possible to assume proportional to and to try to avoid a systematic error in the determination of the ratio . Cf. Lundmark, The determination of the curvature of space time in de Sitter’s world, M.N., vol. 84, p. 747, 1924, and Strömberg, l.c.
We will thus adopt
This relation enables us to calculate . We have indeed by (16)
where we have set
On the other hand, from (18) and (26)
and therefore
With the adopted numerical data (24) for and (19) for , we have
We have therefore
Integral (16) can easily be computed. Writing
it can be written
If is the fraction of the radius of the universe travelled by light during time , we have also from (20)
The following table gives values of and for different values of .





1 

19  
2 

9  
3 


4 

4  
5 

3  
10 

1  
15 


20 

0  
25 



The constants of integration are adjusted to make and vanish for in place of 21,5. The last column gives the Doppler effect computed from (22). The approximate formula (23) would make proportional to and thus to . The error committed by adopting this equation is only 0.005 for . The approximate formula may therefore be used within the limits of the visible spectrum.
5. The meaning of equation (14).
The relation (14) between the two constants and has been adopted following Einstein’s solution. It is the necessary condition that quartic under the radical in (11) may have a double root giving on integration a logarithmic term. For simple roots, integration would give a square root, corresponding to a minimum of as in de Sitter’s solution (12). This minimum would generally occur at time of the order of , say years, i.e. quite recently for stellar evolution. It thus seems that the relation existing between the constants and must be close to (14) for which this minimum is removed to the epoch at minus infinity (). ^{9}^{9}footnotetext: If the positive roots were to become imaginary, the radius would vary from zero upwards, the variation slowing down in the neighbourhood of the modulus of the imaginary roots. For a relation substantially different from (14), this slowing down becomes weak and the time of evolution after leaving becomes again of the order of .
6. Conclusion.
We have found a solution such that:
1. The mass of the universe is a constant related to the cosmological constant by Einstein’s relation
2. The radius of the universe increases without limits from an asymptotic value for .
3. The recession velocities of extragalactic nebulæ are a cosmical effect of the expansion of the universe. The initial radius can be computed by formulæ (24) and (25) or by the approximate formula .
4. The radius of the universe is of the same order of magnitude as the radius deduced from density according to Einstein’s formula
This solution combines the advantages of the Einstein and de Sitter solutions.
Note that the largest part of the universe is forever out of our reach. The range of the 100inch Mount Wilson telescope is estimated by Hubble to be parsecs, or about . The corresponding Doppler effect is 3000 Km/sec. For a distance of it is equal to unity, and the whole visible spectrum is displaced into the infrared. It is impossible that ghost images of nebulæ or suns would form, as even if there were no absorption these images would be displaced by several octaves into the infrared and would not be observed.
It remains to find the cause of the expansion of the universe. We have seen that the pressure of radiation does work during the expansion. This seems to suggest that the expansion has been set up by the radiation itself. In a static universe, light emitted by matter travels round space, comes back to its starting point and accumulates indefinitely. It seems that this may be the origin of the velocity of expansion which Einstein assumed to be zero and which in our interpretation is observed as the radial velocity of extragalactic nebulæ.