One hundred years ago this month, Albert Einstein presented the paper Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie (Cosmological Considerations in the General Theory of Relativity) to the Prussian Academy of Sciences in Berlin. This paper, published in the Proceedings of the Academy on February 15th 1917,1 laid the foundations for today’s Big Bang model of the universe.
Only a year before, Einstein had finally completed his great masterpiece, a new theory of gravity, space, and time known as the general theory of relativity. A fundamental tenet of the new theory was that space and time are not absolute, immutable quantities, but are malleable, interdependent and affected by the presence of matter; this behavior was encoded in the field equations of general relativity.2
Once completed, it was natural for Einstein to apply his new theory to all of space and time, i.e., to investigate whether general relativity could give a satisfactory model of the universe as a whole. As he remarked in a letter to the Dutch astronomer Willem de Sitter: “For me…it was a burning question whether the relativity concept can be followed through to the finish, or whether it leads to contradictions.”3 However, he found this no easy task assuming a universe that was static in time. The main difficulty was Einstein’s conviction that a satisfactory model of the universe should reflect both the Principle of Relativity (a principle that demanded that all frames of reference are equivalent) and an assumption he later named Mach’s Principle—that the inertia of a body is determined by the presence of other masses in the universe. (Inertia is a term physicists use to describe the reluctance of a body to move).
The Einstein World
In his seminal Cosmological Considerations paper of 1917, Einstein showed that general relativity could indeed give a consistent model of the static universe. However, two assumptions were necessary. First, the universe was of closed spatial geometry (like a three-dimensional sphere) and had no beginning or end in time. Second, a consistent solution necessitated the introduction of a new term to the field equations of relativity.
Some regarded the new term, known as the cosmological constant, as something of a fudge factor and claimed that it marred the symmetry and simplicity of the original field equations. However, general relativity certainly allowed the term; indeed it is a little-known fact that Einstein had noted the possibility of such an extension to the field equations in his original exposition of 1916. Now the cosmological constant found an important application, because it allowed a model of the universe that was consistent with Einstein’s views on the relativity of inertia.
Einstein’s Cosmological Considerations paper is a fascinating read as it contains a discussion of his views of the limitations of Newtonian cosmology and a description of his long path to a satisfactory relativistic model of the universe. We have recently completed a detailed review4 of the paper to mark the centenary its publication; we note here that Einstein’s analysis culminated in a simple relation between the radius of the cosmos R, the cosmological constant and the mean density of matter in the universe.
Einstein and alternate models of the universe
One puzzling aspect of Cosmological Considerations is Einstein’s failure to consider the stability of his static universe. His analysis predicted a direct relation between the size of the universe and the density of matter within it; but what happened if the density of matter varied from time to time, as one might expect? In fact, it was later shown that the Einstein World is highly unstable against perturbations in matter density;5 it is strange that Einstein did not notice this aspect of his model universe.
Only a few months after the publication of Cosmological Considerations, the Dutch astronomer Willem de Sitter noted that the addition of the cosmological constant term to the field equations allowed for an alternate cosmic solution, namely the case of a universe with no material content. In this paper, de Sitter replaced Einstein’s 3-dimensional matter-filled universe of closed spatial geometry with an empty 4-dimensional universe of closed spacetime geometry.6 Einstein was greatly perturbed by de Sitter’s model, as the existence of a vacuum solution for the universe was in direct conflict with his understanding of Mach’s Principle. A long debate between the two physicists ensued—however, we find no evidence in Einstein’s writings that he ever accepted de Sitter’s solution as a realistic model of the universe.
In 1922, the young Russian physicist Alexander Friedman suggested that non-static solutions of the Einstein field equations should be considered in models of the universe. Starting from the modified field equations, Friedman derived two differential equations linking the time evolution of the universe with the density of matter and the cosmological constant.7 Einstein did not welcome Friedman’s contribution. His first reaction was that the Russian had made a mathematical error. While this criticism was eventually retracted, an unpublished draft of Einstein’s retraction8 shows that he considered Friedman’s cosmology unrealistic.
In 1927, the Belgian physicist Georges Lemaître also derived time-varying equations for the radius of the universe from the modified field equations. Aware of astronomical observations of the recession of the spiral nebulae, and of emerging evidence that the nebulae constituted distinct galaxies far beyond the confines of the Milky Way, Lemaître linked the new astronomical observations with general relativity by suggesting that the data were evidence of a universe that was not static but expanding.9 Apparently, Einstein did not receive this work favourably either: in his memoirs,10 Lemaître recalled that Einstein described his model of an expanding universe as “abominable.”
All this changed in 1929, when the American astronomer Edwin Hubble published the first definitive evidence of a linear relation between the recession of the spiral nebulae and their radial distance.11 Many physicists accepted Hubble’s results as evidence of a universe that was expanding on the largest scales, and a variety of relativistic time-varying models of the cosmos were proposed. Einstein himself lost little time in abandoning his static cosmology at this point. In the early 1930s, he published two distinct models of the expanding universe, one of closed spatial curvature12 and one of Euclidean geometry.13 In each case, he also abandoned the cosmological constant, stating that the term was both unsatisfactory (it gave an unstable solution in static models) and redundant (relativity could describe the expanding universe without the term).
In 1931, Lemaître made a second bold advance. Noting that an expanding universe was undoubtedly smaller, denser and hotter in the past, he suggested that our universe may have originated as a small primeval fireball, expanding and cooling over billions of years.14 This idea, later dubbed the 'big bang' model of the cosmos, had few supporters at first, but won favour with the discovery of several pieces of supporting evidence in the 1960s. The most important of these discoveries, a faint universal radiation left over from the infant universe known as the cosmic microwave background, remains a field of intense research for today’s cosmologists.
“My biggest blunder”
In his memoirs,15 the émigré Russian scientist George Gamow reported that Einstein once described the cosmological constant as “my biggest blunder.” Some doubt16 has been cast on Gamow’s claim in recent years; however, we have learnt that other physicists made similar reports.17 Certainly, it is intriguing to think that Einstein might have predicted the expansion of the universe many years before Hubble’s observations had he not introduced the cosmological constant to the field equations. On the other hand, it must be remembered that Einstein’s task in 1917 was to investigate whether relativity could describe the known universe, i.e., a universe that was assumed to be static. If Einstein did make the “greatest blunder” comment, it is likely that he was referring to his failure to notice that the cosmological constant did not provide a stable, static model of the universe.
Today, the cosmological constant has made a dramatic return to relativity due to the discovery of an acceleration in the expansion of the cosmos (a phenomenon known as dark energy). It might therefore be argued that Einstein’s real blunder was to abandon the term in the 1930s. However, such a view is once again somewhat ahistorical, as evidence of an accelerated expansion was not known to him at the time.
The Einstein World makes a comeback
The reader will be surprised to learn that Einstein’s static model of the universe has once again become a topic of interest for some cosmologists today. In attempts to avoid the well-known problem of a big bang singularity, theorists have become interested in the possibility of a universe that inflates from a static Einstein World. Whether this scenario, known as the emergent universe,18 can offer a plausible, consistent description of the early universe is not yet known, but it is intriguing to think that, like the cosmological constant, the Einstein World might yet make a dramatic comeback.
A longer version of this article can be found at Physics Today.
1. Einstein, A. 1917. Sitz. König. Preuss. Akad. 142-152. Eng. transl. CPAE 6 (Doc. 43).
2. Einstein, A. 1916. Ann. Physik. 49: 769-822. Eng. transl. CPAE 6 (Doc.30)
3. Letter from Albert Einstein to Willem de Sitter, March 12th, 1917. Eng. transl. CPAE 8 (Doc. 311).
4. O’Raifeartaigh, C., O’Keeffe M., Nahm W. and S. Mitton. 2017. https://arxiv.org/abs/1701.07261
5. Eddington, A.S. 1930. Month. Not. Roy. Astron. Soc. 90, 668-678
6. De Sitter, W. 1917. Month. Not. Roy. Astron. Soc. 78: 3-28
7. Friedman, A. 1922. Zeit. Physik. 10: 377-386.
8. Nussbaumer, H. and L. Bieri 2009. Discovering the Expanding Universe (CUP) p92.
9. Lemaȋtre, G. 1927. Annal. Soc. Sci. Brux. A. 47: 49-59
10. Lemaȋtre, G. 1958. Rev. Quest. Sci. 129: 129-132
11. Hubble, E. 1929. Proc. Nat. Acad. Sci. 15: 168-173
12. Einstein, A. 1931. Sitz. König. Preuss. Akad. 235-237.
13. Einstein, A. and W. de Sitter. 1932. Proc. Nat. Acad. Sci. 18 (3): 213-214
14. Lemaȋtre, G. 1931. Nature 127: 706.
15. Gamow, G. 1970. My Worldline (Viking Press) p44.
16. Livio, M. 2013. Brilliant Blunders (Simon&Schuster) p233
17. Taylor E.F. and J.A. Wheeler. 2000. Exploring Black Holes: Introduction to General Relativity (Addison Wesley) p11
18. Ellis, G.F.R. and R. Maartens 2004. Class. Quant. Grav. 21: 223—239