One hundred years ago this month, as bitter war raged throughout Europe, 37-year-old Albert Einstein presented the paper “Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie” (“Cosmological considerations in the general theory of relativity”) at the weekly meeting of the Prussian Academy of Sciences in Berlin. The corresponding paper, published a week later in the proceedings of the academy, laid the foundations of modern theories of the universe.
Applying general relativity to the cosmos
Only a year before, Einstein had finally completed his great masterwork, a new theory of gravity, space, and time known as the general theory of relativity. From a scientific point of view, it is hardly surprising that Einstein quickly turned his attention to cosmology. A fundamental tenet of the general theory was that the geometric structure of a region of spacetime is not an independent, self-determined entity but is determined by mass–energy. In modern notation, that idea is expressed as the field equations
where Gμν is a four-dimensional tensor that describes the geometry of a region of spacetime and Tμν is a four-dimensional tensor that describes the flux of mass–energy within that region (the quantity κ is a constant known as the Einstein constant). Once Einstein had completed the theory, it was natural for him to ask if general relativity could deliver a consistent model of all of spacetime—a plausible model of the universe as a whole. As he remarked in a letter to the Dutch astronomer Willem de Sitter, “For me, though, it was a burning question whether the relativity concept can be followed through to the finish or whether it leads to contradictions.”
Einstein soon found that, assuming a universe with a static distribution of matter (evidence to the contrary did not emerge until 1929), it was no easy task to obtain a satisfactory solution to the field equations for the case of the universe as a whole. The main difficulty was his insistence that a model of the cosmos should reflect both the principle of relativity, which demanded that all frames of reference be equivalent, and an assumption he later named Mach’s principle—that the inertia of a body is determined entirely by the presence of other masses in the universe.
The Einstein universe
The outcome of those deliberations was Einstein’s “Cosmological considerations” paper of 1917. His ingenious breakthrough was to postulate that we inhabit a universe of closed spatial geometry. Relativity could deliver a satisfactory model of the known universe if it was assumed that the cosmos had the geometry of a three-dimensional sphere—unbounded spatially, yet finite in content.
However, the Einstein universe came at a price. In his analysis, Einstein found that a nonzero solution to the field equations could be obtained only if a new term was introduced to the equations according to:
To some, the new term λgμν, known as the cosmological constant term, marred the symmetry and simplicity of the original field equations. However, general relativity certainly permitted the term; indeed 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.
“Cosmological considerations” is a fascinating read, as it contains a detailed discussion of the limitations of Newtonian cosmology and a description of Einstein’s “long and winding path” to a consistent relativistic model of the universe. Einstein’s analysis culminated in a simple relation between the cosmological constant λ, the mean density of matter ρ, and the radius of the cosmos R according to
Puzzling aspects of the 1917 paper
One puzzling aspect of Einstein’s “Cosmological considerations” paper is that he made no attempt to estimate the size of his model universe from equation 3. After all, even a rough approximation of the mean density of matter in the universe could have given some estimate of the cosmic radius R. Instead he merely declared at the end of the paper that the model was logically consistent: “At any rate, this view is logically consistent, and from the standpoint of the general theory of relativity lies nearest at hand; whether, from the standpoint of present astronomical knowledge, it is tenable, will not here be discussed.”
An estimate of the size of the Einstein universe can be found in Einstein’s correspondence around that time. Taking a value of ρ = 10-22 g/cm3 from astronomers for the mean density of matter in the Milky Way, he obtained from equation 3 an estimate of 107 light-years for the radius of his model universe. However, he appears to have distrusted that result on the basis of its being much larger than contemporaneous estimates of the distance to the farthest stars (104 light-years). Indeed, in a well-known lecture in 1921, he remarked that one should not equate the mean density of matter in our galaxy with that of the universe in such calculations.
A second puzzle associated with the “Cosmological considerations” paper is Einstein’s failure to consider the stability of his model universe. After all, the quantity ρ in equation 3 represented a mean value for the density of matter in the universe; one could expect a variation in that parameter from time to time, which raises the question of the stability of the model against such perturbations. Indeed, it was later shown that the Einstein universe is highly unstable against perturbations in matter density (a slight increase in density would trigger an inexorable contraction, while a slight decrease would result in a runaway expansion). It is strange that Einstein did not notice that aspect of his model.
Einstein’s reaction to alternate cosmological models
Some fascinating insights into Einstein’s cosmology can be gleaned from his reaction to alternate models of the universe. Only a few months after the publication of “Cosmological considerations,” de Sitter noted that the modified field equations, shown in equation 2, allowed for an alternate cosmic solution, namely a universe with no material content. In his paper “On Einstein’s theory of gravitation, and its astronomical consequences,” de Sitter replaced Einstein’s three-dimensional matter-filled universe of closed spatial geometry with an empty four-dimensional universe of closed spacetime geometry.
Einstein was greatly perturbed by de Sitter’s model universe. Quite apart from the fact that the model bore little relation to the real world, 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. In compiling research for a review of “Cosmological considerations,” my colleagues and I found 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 Friedmann suggested that nonstatic solutions of the Einstein field equations should be considered in relativistic models of the universe. Starting from equation 2 and assuming a positive spatial curvature for the cosmos, Friedmann derived two differential equations linking the time evolution of the universe with the density of matter and the cosmological constant. However, Einstein did not welcome Friedmann’s contribution. Einstein’s first reaction was that the Russian had made a mathematical error. Although that criticism was eventually retracted, an unpublished draft of Einstein’s retraction shows that he considered Friedmann’s cosmology unrealistic.
In 1927 the Belgian physicist Georges Lemaître independently derived differential equations for the radius of the universe that were almost identical to the Friedmann equations. Aware of astronomical observations of the recession of the spiral nebulae, and of emerging evidence of the enormous distances to the nebulae, Lemaître saw the observations as evidence for an expanding universe. Einstein did not view this work favorably either; in fact, Lemaître later reported that Einstein described his expanding model as “abominable.”
All that changed in 1929, when American astronomer Edwin Hubble published the first evidence of a linear relation between the redshifts of the spiral nebulae and their radial distance. Many theorists viewed Hubble’s results as evidence of a nonstatic universe and proposed a variety of relativistic time-varying models of the cosmos. Einstein himself lost little time in abandoning his static cosmology at that point. In the early 1930s, he published two distinct models of the expanding universe, one of positive spatial curvature and one of Euclidean geometry. In each case, he also abandoned the cosmological constant, stating that the term was both unsatisfactory (it gave an unstable solution) and redundant (relativity could describe expanding models of the universe without the term). In 2014 my colleagues and I discovered that Einstein also attempted a steady-state model of the expanding universe in those years; however, he soon abandoned the idea.
Some years later, the Russian scientist George Gamow reported in his memoirs that Einstein once described the cosmological constant as his “biggest blunder.” Although some doubt has recently been cast on Gamow’s claim, our research team has learned that at least two other physicists made similar reports. 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. However, it must be remembered that Einstein’s task in 1917 was to investigate whether relativity could describe the known universe, that is, a universe that was assumed to be static. If Einstein did make the “biggest blunder” comment, he may have been referring to his failure to notice the instability of his model.
Today the term cosmological constant has made a dramatic return to the field equations due to the observation of an acceleration in the expansion of the cosmos. 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 retrospective, because evidence of an accelerated expansion was not known to him.
In recent years, the Einstein universe has once more become a topic of interest in theoretical cosmology. In attempts to avoid the well-known problem of a big bang singularity, some theorists have become interested in the possibility of a universe that inflates from a static Einstein universe, a scenario known as the emergent universe. Whether the emergent universe will 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 universe might yet make a dramatic comeback.
Cormac O’Raifeartaigh lectures at the Waterford Institute of Technology in Ireland and is a Fellow of the Institute of Physics. He and collaborators Michael O’Keeffe, Werner Nahm, and Simon Mitton recently submitted a review of Einstein’s 1917 model to the European Physical Journal H; a preprint can be found at https://arxiv.org/abs/1701.07261.