From Static to Expanding Models of the Universe

Norbert Straumann, Uni Zürich


At the end of some historical remarks in the article on the 2011 Nobel Prize [1], it was announced that the author would indicate in a historical essay the interesting early history of cosmology. One of the reasons is that this is not even well-known among cosmologists, and is often distorted. In the words of the late Allen Sandage: "In 1929, Edwin Hubble published a paper that correlated redshifts of galaxies with distances he had estimated from calibration of their absolute magnitudes previously made in 1926. Writers of both popular accounts and technical textbooks have often described this as the discovery of the expanding universe. This is not so." [2]

Einstein's static model of the universe

On 8 February 1917, in the middle of the most terrible time during the First World War, Einstein gave a talk in the Preussian Academy of Sciences on an application of his general relativity on the universe as a whole. One week before the German military leadership had declared in the same city the unconstrained submarine war. Einstein’s first paper on cosmology [3] marks in many ways the beginning of modern cosmology.
Perhaps the main reason why Einstein turned so soon after the completion of general relativity to cosmology had much to do with Machian ideas on the origin of inertia, which played in those years an important role in Einstein’s thinking. His intention was to eliminate all vestiges of absolute space. He was, in particular, convinced that isolated masses cannot impose a structure on space at infinity. Einstein was actually thinking about the problem regarding the choice of boundary conditions at infinity already in spring 1916. In a letter to Michele Besso from 14 May 1916 he also mentions the possibility of the world being finite. A few months later he expanded on this in letters to Willem de Sitter. It is along these lines that he postulated a Universe that is spatially finite and closed, a Universe in which no boundary conditions are needed 1. He then believed that this was the only way to satisfy what he later [5] named Mach’s principle, in the sense that the metric field should be determined uniquely by the energy-momentum tensor.
In addition, Einstein assumed that the Universe was static. This was not unreasonable at the time, because the relative velocities of the stars as observed were small. (Recall that astronomers only learned later that spiral nebulae are independent star systems outside the Milky Way. This was definitely established when in 1924 Hubble found that there were Cepheid variables in Andromeda and also in other nearby galaxies. Einstein compares the observed small peculiar velocities of stars with the speed of light.)


These two assumptions were, however, not compatible with Einstein’s original field equations. For this reason, Einstein added the famous Λ-term, which is compatible with the principles of general relativity. The cosmological term is, in four dimensions, the only possible complication of the field equations if no higher than second order derivatives of the metric are allowed (Lovelock theorem). This remarkable uniqueness is one of the most attractive features of general relativity. (In higher dimensions additional terms satisfying this requirement are allowed.)
For the static Einstein universe the field equations with the cosmological term imply the two relations

where ρ is the mass density of the dust filled universe (zero pressure) and a is the radius of curvature. For Λ = 0 the density ρ would have to vanish. (We remark, in passing, that the Einstein universe is the only static dust solution; one does not have to assume isotropy or homogeneity.) Einstein was very pleased by this direct connection between the mass density and geometry, because he thought that this was in accord with Mach's philosophy.
Einstein concludes with the following sentences:

"In order to arrive at this consistent view, we admittedly had to introduce an extension of the field equations of gravitation which is not justified by our actual knowledge of gravitation. It has to be emphasized, however, that a positive curvature of space is given by our results, even if the supplementary term is not introduced. That term is necessary only for the purpose of making possible a quasi-static distribution of matter, as required by the fact of the small velocities of the stars."

To de Sitter he emphasized in a letter on 12 March 1917, that his cosmological model was intended primarily to settle the question "whether the basic idea of relativity can be followed through its completion, or whether it leads to contradictions". And he adds whether the model corresponds to reality was another matter.
Only later Einstein came to realize that Mach's philosophy is predicated on an antiquated ontology that seeks to reduce the metric field to an epiphenomenon of matter. It became increasingly clear to him that the metric field has an independent existence, and his enthusiasm for what he called Mach's principle later decreased. In a letter to F. Pirani he wrote in 1954: "As a matter of fact, one should no longer speak of Mach's principle at all." GR still preserves some remnant of Newton’s absolute space and time.

De Sitter model

Surprisingly to Einstein, de Sitter discovered in the same year, 1917, a completely different static cosmological model which also incorporated the cosmological constant, but was anti-Machian, because it contained no matter [6]. For this reason, Einstein tried to discard it on various grounds (more on this below). The original form of the metric was:

Here, the spatial part is the standard metric of a three-sphere of radius R, with R = (3/Λ)1/2. The model had one very interesting property: For light sources moving along static world lines there is a gravitational redshift, which became known as the de Sitter effect. This was thought to have some bearing on the redshift results obtained by Slipher. Because the fundamental (static) worldlines in this model are not geodesic, a freely-falling object released by any static observer will be seen by him to accelerate away, generating also local velocity (Doppler) redshifts corresponding to peculiar velocities. In the second edition of his book [7], published in 1924, Eddington writes about this:
"de Sitter's theory gives a double explanation for this motion of recession; first there is a general tendency to scatter (...); second there is a general displacement of spectral lines to the red in distant objects owing to the slowing down of atomic vibrations (...), which would erroneously be interpreted as a motion of recession."
I do not want to enter into all the confusion over the de Sitter universe. One source of this was the apparent singularity at r = R = (3/Λ)1/2. This was at first thoroughly misunderstood even by Einstein and Weyl. ("The Einstein-de Sitter-Weyl-Klein Debate" is now published in Vol. 8 of the Collected Papers [4].) At the end, Einstein had to acknowledge that de Sitter's solution is fully regular and matter-free and thus indeed a counter example to Mach's principle. But he still discarded the solution as physically irrelevant because it is not globally static. This is clearly expressed in a letter from Weyl to Klein, after he had discussed the issue during a visit of Einstein in Zürich [8]. An important discussion of the redshift of galaxies in de Sitter's model by H. Weyl in 1923 should be mentioned. Weyl introduced an expanding version 2 of the de Sitter model [9]. For small distances his result reduced to what later became known as the Hubble law. Independently of Weyl, Cornelius Lanczos introduced in 1922 also a non-stationary interpretation of de Sitter's solution in the form of a Friedmann spacetime with a positive spatial curvature [10]. In a second paper he also derived the redshift for the non-stationary interpretation [11].


From static to expanding world models

Until about 1930 almost everybody believed that the Universe was static, in spite of the two fundamental papers by Friedmann [12] in 1922 and 1924 and Lemaître's independent work [13] in 1927. These path breaking papers were in fact largely ignored. The history of this early period has - as is often the case - been distorted by some widely read documents. Einstein too accepted the idea of an expanding Universe only much later. After the first paper of Friedmann, he published a brief note claiming an error in Friedmann's work; when it was pointed out to him that it was his error, Einstein published a retraction of his comment, with a sentence that luckily was deleted before publication: "[Friedmann's paper] while mathematically correct is of no physical significance". In comments to Lemaître during the Solvay meeting in 1927, Einstein again rejected the expanding universe solutions as physically unacceptable. According to Lemaître, Einstein was telling him: "Vos calculs sont corrects, mais votre physique est abominable". It appears astonishing that Einstein - after having studied carefully Friedmann's papers - did not realize that his static model is unstable, and hence that the Universe has to be expanding or contracting. On the other hand, I found in the archive of the ETH many years ago a postcard of Einstein to Weyl from 1923, related to Weyl's reinterpretation of de Sitter's solution, with the following interesting sentence: "If there is no quasi-static world, then away with the cosmological term".
It also is not well-known that Hubble interpreted in 1929 the redshifts of radiation emitted by distant 'nebulae' in the framework of the de Sitter model, as had been suggested by Eddington.


Lemaître discovers the expanding universe

We repeat what we said in [1] about Lemaître's key role in the founding period of cosmology. He was the first person who seriously proposed an expanding universe as a model of the real universe. He derived in his crucial paper of 1927 the general redshift formula, and showed that it leads for small distances to a linear relation, known as Hubble's law. He also estimated the Hubble constant H0 based on Slipher's redshift data for about 40 nebulae, and Hubble's 1925 distance determination to Andromeda, as well as the the magnitudes of nebulae published by him in 1926. Two years before Hubble he found a value only somewhat higher than the one Hubble obtained in 1929. (Actually, Lemaître gave two values for H0.) But this seminal work was almost completely ignored. The general attitude is well illustrated by the following remark of Eddington at a Royal Society meeting in January, 1930: "One puzzling question is why there should be only two solutions. I suppose the trouble is that people look for static solutions."
Lemaître, who had been for a short time a post-doctoral student of Eddington, read this remark in a report on the meeting published in Observatory, and wrote to Eddington pointing out his 1927 paper. Eddington had seen that paper, but had completely forgotten about it. But now he was greatly impressed and recommended Lemaître's work in a letter to Nature. He also arranged for a translation which appeared in MNRAS [13]. Eddington also "pointed out that it was immediately deducible from his [Lemaître's] formulae that Einstein's world is unstable, so that an expanding or a contracting universe is an inevitable result of Einstein's law of gravitation."
Lemaître's successful explanation of Hubble's improved data, carefully analysed by de Sitter in a series of papers, finally changed the viewpoint of the majority of workers in the field. At this point, after a stay with Eddington, and a visit to the Mount Wilson Observatory, Einstein rejected the cosmological term as superfluous and no longer justified [14]. At the end of the paper in which he published his new view, Einstein adds some remarks about the age problem which was quite severe without the Λ-term, since Hubble's value of the Hubble parameter was then about seven times too large. Einstein is, however, not very worried and suggests two ways out. First he says that the matter distribution is in reality inhomogeneous and that the approximate treatment may be illusionary. Then he adds that in astronomy one should be cautious with large extrapolations in time.
After the Λ-force was rejected by its inventor, other cosmologists, such as Eddington and Lemaître, retained it. One major reason was that it solved the problem of the age of the Universe when the Hubble time scale was thought to be only 2 billion years (corresponding to the value H0 ~ 500 km s-1 Mpc-1 of the Hubble constant). This was even shorter than the age of the Earth. In addition, Eddington and others overestimated the age of stars and stellar systems.
For this reason, the Λ-term was employed again and a model was revived which Lemaître had singled out from the many solutions of the Friedmann-Lemaître equations 3. This so-called Lemaître hesitation universe is closed and has a repulsive Λ-force (Λ > 0), which is slightly greater than the value chosen by Einstein. It begins with a big bang and has the following two stages of expansion. In the first the Λ-force is not important, the expansion is decelerated due to gravity and slowly approaches the radius of the Einstein universe. At about the same time, the repulsion becomes stronger than gravity and a second stage of expansion begins which eventually inflates. In this way a positive Λ was employed to reconcile the expansion of the Universe with the age of stars.
Lemaître was also the first who associated in 1933 the cosmological constant with vacuum energy. Actually, Pauli made before the advent of the new quantum mechanics some simple, but profound remarks on this issue [15].
To a minority of cosmologists who had read the French original of Lemaître's 1927 paper, it was known that a few paragraphs were deleted in the translation, notably the one in which Lemaître assessed the evidence for linearity of the distance-velocity relation and estimated the expansion rate. Fortunately, the origin of this curious fact has very recently been completely cleared up [16]. It was Lemaître himself who translated his original paper. The correspondence of him with the editor of MNRAS, quoted in [16], shows that Lemaître was not very interested in establishing priority. He saw no point in repeating in 1931 his findings four years earlier, since the quality of the observational data had in the meantime been improved. This is one of the reasons that Hubble was elevated to the discoverer of the expanding universe.
For much more on all this I refer again to the recent excellent book [17] of our Swiss colleagues Harry Nussbaumer and Lydia Bieri.


1 The spatial geometry in Einstein's model is that of a three-sphere, i.e., the surface of a sphere in four-dimensional Euclidean space. This is a prototype of a highly symmetric compact manifold without boundary.
2 The de Sitter model has many different interpretations, depending on the choice of the velocity field for the subdominant matter flow.
3 I recall that Friedmann included the Λ-term in his basic equations. I find it remarkable that for the negatively curved solutions he pointed out that these may be open or compact (but not simply connected).



[1] N. Straumann, The 2011 Nobel Prize in Physics, SPG Mitteilungen, Nr. 36, Januar 2012.
[2] A. Sandage, Preface to [17].
[3] A. Einstein, Sitzungsber. Preuss. Akad. Wiss. phys.-math. Klasse VI, 142 (1917). See also: [4], Vol. 6, p. 540, Doc. 43.
[4] A. Einstein, The Collected Papers of Albert Einstein, Vols. 1-12, Princeton University Press, 1987–. See also: [http://www.].
[5] A. Einstein, On the Foundations of the General Theory of Relativity. Ref. [4], Vol. 7, Doc. 4.
[6] W. de Sitter, Proc. Acad. Sci., 19, 1217 (1917); and 20, 229 (1917).
[7] A. S. Eddington, The Mathematical Theory of Relativity. Chelsea Publishing Company (1924). Third (unaltered) Edition (1975). See especially Sect. 70.
[8] Letter from Hermann Weyl to Felix Klein, 7 February 1919; see also Ref. [5], Vol. 8, Part B, Doc. 567.
[9] H. Weyl, Phys. Zeits. 24, 230, (1923); Phil. Mag. 9, 923 (1930).
[10] C. Lanczos, Phys. Zeits. 23, 539 (1922).
[11] C. Lanczos, Zeits. f. Physik 17, 168 (1923).
[12] A. Friedmann, Z.Phys. 10, 377 (1922); 21, 326 (1924).
[13] G. Lemaître, L’univers en expansion. Ann. Soc. Sci. de Bruxelles 47, 49 (1927). Translated in MNRAS 91, 483 (1931).
[14] A. Einstein, S. B. Preuss. Akad. Wiss. (1931), 235.
[15] N. Straumann, Wolfgang Pauli and Modern Physics, Space Science Reviews 148, 25 (2009).
[16] M. Livio, Nature 479, 208-211 (2011).
[17] H. Nussbaumer and L. Bieri, Discovering the Expanding Universe, Cambridge University Press (2009).


[Released: May 2012]