Editorial note to "A Homogeneous Universe of Constant Mass and Increasing Radius accounting for the Radial Velocity of Extra--Galactic Nebulae" by Georges Lema\^itre (1927)

Abstract

This is an editorial note to accompany printing as a Golden Oldie in the
Journal of General Relativity and Gravitation of the fundamental article by
Georges Lema\^itre 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
Lema\^itre 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.

Full text

Gen Relativ Gravit (2013) 45:1619–1633
DOI 10.1007/s10714-013-1547-4
GOLDEN OLDIE EDITORIAL
Editorial note to:
Georges Lemaître,
A homogeneous universe of constant mass
and increasing radius accounting for the radial
velocity of extra-galactic nebulae
Jean-Pierre Luminet
Published online: 13 June 2013
© Springer Science+Business Media New York 2013
Keywords Expanding Universe ·Generalised Friedmann models ·
Georges Lemaître ·Golden Oldie
1 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.1Before 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 3-year fellowship from the Belgian government, enabling him to study abroad.
1A 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.
The republication of the original paper can be found in this issue following the editorial note and online
via doi:10.1007/s10714-013-1548-3.
J.-P. Luminet (B
)
Laboratoire Univers et Théories, Observatoire de Paris-CNRS UMR8102-Université Paris Diderot,
5 Place Jules Janssen, 92190 Meudon, France
e-mail: jean-pierre.luminet@obspm.fr
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1620 J.-P. Luminet
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
nebulae. 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 nebulae 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 nebulae 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 “steady-state” models [9,10], and still later in the 1980’s
to describe the hypothetical inflationary phase of the very early universe, see e.g. [11].
In the second article [12] he deduced that the relation between the relative speed of
test-particles 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 nebulae, he also realized that the model resulted in
an infinite Euclidean space, that he considered inadmissible: as a neo-Thomist 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 1926–27, 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 [13], 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.
2 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
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Editorial note to: A homogeneous universe of constant mass 1621
frequency (or wavelength) implied a radial speed of displacement of the source com-
pared to the observer. Radial speeds were thus indirectly measured by spectroscopy.
By 1917, Slipher (see [14] and references therein) had analyzed the spectra of 25
spiral nebulae, 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
nebulae 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 nebulae.
Slipher never published his final list,2but it was given in Arthur Eddington’s book
of 1923 [16], who noticed that “one of the most perplexing problems in cosmogony
is the great speed of spiral nebulae. Their radial velocities average about 600km.
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 space-time 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
nebulae.
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 nebulae 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 P3obtained
by identification of antipodal points of the simply-connected hypersphere S3;see[17]
for an explanation of such a choice) with the radius of curvature R(and consequently
the matter density ρ) being a function of time t, and a non-zero cosmological constant
λ. From Einstein’s field equations he obtains differential equations (Eqs. (2)–(3)) for
R(t)and ρ(t)almost identical to those previously obtained by Friedmann [18](atthe
time Lemaître was not aware of Friedmann’s work, see below). The difference is that
2For details see [15].
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1622 J.-P. Luminet
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 stress-energy 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 stress-energy 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 R=R =0,R=R0at
t=−∞and he adjusts the value of the cosmological constant such that λ=1/R0,
in the same way Einstein had adjusted the value of λin his static model with constant
radius.
As a consequence, the exact solution he obtains in Eq. (30) describes a monoto-
nous 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 def-
inite age—as well as the “monotonous solution of second species” found earlier by
Friedmann—will be later baptized the Eddington-Lemaître’s model (see below).
Lemaître does not provide a graph for R(t)but gives numerical values in a
table going from t=−∞to +∞. For the sake of clarity our Fig. 1depicts such
a graph.
Fig. 1 The 1927 Lemaître’s universe model, later named Eddington–Lemaître. The radius R0of the static
Einstein hypersphere is reached asymptotically for t=−∞. The origin of cosmic time is arbitrary, thus
the model does not pose any problem of age
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Editorial note to: A homogeneous universe of constant mass 1623
Fig. 2 Handwritten graph by Lemaître. This extraordinary diagram, plotted by Lemaître in 1927, but unpub-
lished until 1998 [21], depicts the time evolution of the radius of the universe with the cosmological constant
(denoted a), for a space of positive curvature. All the models start with a singularity in (x=0,t=0).Fora
sufficiently large cosmological constant a, the universe becomes open. The most recent cosmological data
are compatible with a Lemaître’s solution with positive curvature and accelerated expansion (top curve)
Lemaître conceived the static Einstein universe as a kind of pre-universe 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 evo-
lution 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 publi-
cations. 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 (Fig. 2).
As mentioned above, the 1927 article does not refer to the work of Friedmann,
published in Zeitschrift für Physik—although it was 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 [19], which had been largely discussed
in the scientific community. A plausible explanation is that Lemaître could not read the
German [20]. 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
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1624 J.-P. Luminet
[22], 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 appar-
ent 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 lim-
its 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 so-called 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,3and deduces
their distance from a recent empirical formula between the distance and the absolute
magnitude provided by Hubble [24], who himself took them from Hopmann [25].
Eventually, Lemaître is able to give the numerical figures for the initial and present-
day values of the radius of the universe, resp. R0=2,7×108pc and R=6×109pc.
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 R/120, whereas for a distance only greater than R/11,5 the whole
visible spectrum is displaced into the infrared—he could not imagine the space era
with infra-red and submillimeter telescopes placed on board of satellites.
We have seen above that Lemaître knew already all the solutions of Einstein’s equa-
tions 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.
3 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
3Strömberg [23] relied himself on redshifts measured by Slipher and included some globular clusters in
addition to spiral nebulae.
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Editorial note to: A homogeneous universe of constant mass 1625
and completely inaccessible journal”, as is sometimes claimed [26], instead of one of
the prestigious astronomical journals of the time.4As rightly pointed out by Lambert
[27], 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 nebulae
and I had the impression that Einstein was hardly aware of the astronomical facts.
At the university, everyone began to speak in German” [28]. Einstein’s response to
Lemaître shows the same unwillingness to change his position that characterized his
former response to Friedmann (see e.g. [29]): he accepted the mathematics, but not a
physically expanding universe!
In 1928 H. P. Robertson published an article [30] 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
v=cd/Rwhere dis the distance of the nebula and Rthe 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 Lemaître5and that would be taken by Hub-
ble one year later. From this he calculated R=2×1027 cm, and a proportional-
ity constant of 464km/s/Mpc (that he did not calculate, the figure can be found in
[31]). The main interest of Robertson’s work (see also [32]) 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 [33] used the experimental data on the Doppler redshifts mostly
given by Slipher (who was not quoted) and found a linear velocity-distance rela-
tion v=Hr with H=465 ±50 km/s/Mpc for 24 objects and 513 ±60 km/s/Mpc
4The paper was reprinted later in 1927 in vol. 4 of Publications du Laboratoired’Astronomie et de Géodésie
de l’Université de Louvain, still less suited for widespread dissemination.
5He did not know the Lemaître’s articles of 1925 and 1927.
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1626 J.-P. Luminet
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 uni-
verse models. He stated “The outstanding feature, however, is the possibility that the
velocity-distance 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 expan-
sion. 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 [34], he tackled the theoretical interpretation of the obser-
vations 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 biogra-
pher 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’ ”
[35]. 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.
[36]) and Hubble himself, who was fiercely territorial; in a letter to De Sitter dated
21 August 1930, he wrote “I consider the velocity-distance relation, its formulation,
testing and confirmation, as a Mount Wilson contribution and I am deeply concerned
in its recognition as such” (quoted in [37]).
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
[38], 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 [39] 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?”
[40], 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 Edding-
ton 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 inter-
mediary solutions between those of Einstein and De Sitter. I made these investigations
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Editorial note to: A homogeneous universe of constant mass 1627
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 nebulae is
always a receding one from time minus infinity to plus infinity.”6Lemaître 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 [41]).
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 [41]).
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 [42]).
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 [43]). In late May, De Sitter published a discussion about the expansion of the
universe [44], where he wrote “A dynamical solution of the equations (4) with the
line-element (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 [45]. Then he published an important article [46] 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 1,200million
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
withamassMgreater or smaller than the mass MEof the Einstein model, but he
6From a copy kept at the Archives Lemaître of Louvain-la-Neuve, quoted in [27].
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1628 J.-P. Luminet
rejected the two solutions, arguing that, for M>ME, “it seems to require a sudden
and peculiar beginning of things”, whereas for M<ME, “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 [47].
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 phys-
ical 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 [48].
4 The English translation and discrepancies
A great deal has been written on the topic of who really discovered the expanding
universe [49–56]. The French astronomer Paul Couderc [57] was probably the first
one to rightly underline the priority of Lemaître over Hubble, but since Lemaître
himself never claimed any priority (see [58] for more details), the case was not much
discussed.
An intriguing discrepancy between the original French article and its English trans-
lation had already been quoted by various authors (e.g. [41–43]): the important para-
graph discussing the observational data and Eq. (24) where Lemaître gave the rela-
tion 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
R/R=0,68 ×10−27cm−1”. It was found curious that the crucial paragraphs assess-
ing the Hubble law were dropped so that, either due to Eddington’s blunder7or some
other mysterious reason, Lemaître was never recognized as the discoverer of the expan-
sion of the universe. De facto Lemaître was eclipsed and multitudes of textbooks pro-
claim Hubble as the discoverer of the expanding universe, although Hubble himself
never believed in such an explanation [59].
Suddenly, in 2011, a burst of accusations has flared up against Hubble, from the sus-
picion that a censorship was exerted either on Lemaître by the editor of the M.N.R.A.S.
[60] or on the editor by Hubble himself [37]—suspicion based on the “complex per-
sonality” 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
[61], with the help of the Archives Lemaître at Louvain and the Archives of the Royal
Astronomical Society (see also [27] 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
7Until very recently the identity of the translator was not assessed, generally assumed to be that of Eddington
himself.
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Editorial note to: A homogeneous universe of constant mass 1629
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. [62].
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 [18,63] distinguished the spherical space S3and the
projective space P3, that he called the elliptical space. As recalled by Lemaître in
the first paragraph, P3has a (comoving) volume π2R3instead of 2π2R3
0for S3,
and the longest closed straight line is πRinstead of 2πR. The main cosmological
difference is due to the presence, in S3, of an antipodal point associated to any
point, and in particular to the observer, at a distance of πRprecisely. This was
considered as an undesirable fact, so that cosmological models with P3seemed
preferable than those with S3.
Eddington [16] also referred to elliptical space as an alternative more attractive than
S3, and Lemaître also adopted this point of view.8We 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 [62], 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 explain-
ing the fact that extragalactic nebulae 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 nebulae [...]” 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 up-to-date.
– Section 1, sixth paragraph
8Note that elliptical space is not simply connected but multiply connected, see e.g. [17].
123

1630 J.-P. Luminet
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 interpreta-
tion” has disappeared for the sake of economy.
– Section 4
The paragraphs from “Radial velocities of 43 extra-galactic nebulae [...]”up
to “This relation enables us to calculate R0”, as well as the three footnotes,
are suppressed and replaced by “From a discussion of available data, we adopt
R/R=0,68 ×10−27cm−1”. 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 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 [61]). The
choice of Lemaître is quite comprehensible because the data he used in 1927 gave
only very imperfectly the linear relation v=Hd, whereas in 1931 the new data
from Hubble allowed to validate this relationship in a much more precise manner,
see Fig. 3for 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” [64]. 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 100-inch Mount Wilson telescope estimated by Hubble
from R/120 to R/200.
– 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,
123

Editorial note to: A homogeneous universe of constant mass 1631
Fig. 3 Comparison between the data used by Lemaître in 1927 (left) to yield the first empirical value of the
rate of expansion of the Universe as 575km/s/Mpc (reconstructed in [31]), and the radial velocity–distance
diagram published by Hubble in 1929, with a best slope of 530km/s/Mpc (right)
De Sitter and himself in 1930, and eventually two popular expositions given by
him in 1929 (in French) and by De Sitter in 1931.
Comment by the Golden Oldies editor: A brief biography of Georges Lemaître was
printed together with another Golden Oldie by him, in Gen. Relativ. Gravit. 29, 639
(1997), doi:10.1023/A:1018803604510.
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