The universe of Albert Einstein: an overview of Einstein’s static and dynamic models of the cosmos
Cormac O’Raifeartaigh,aBrendan McCann,aMichael O’Keeffe,a Werner Nahmb and Simon Mittonc
aSchool of Science and Computing, Waterford Institute of Technology, Waterford, Ireland
b School of Theoretical Physics, Dublin Institute for Advanced Studies, Dublin 2, Ireland
cDepartment of History and Philosophy of Science, University of Cambridge, Cambridge, UK
Author for correspondence:
We present an overview of Einstein’s relativistic models of the universe, from hisstatic model of 1917 to his evolving models of the 1930s, from an abandoned steady-state model to the Einstein-Rosen and Einstein-Straus models. Takingall of these workstogether, we findthatEinstein’s cosmologywasinformed by an ingenious mix of profound philosophy constrained byempirical observation. We argue that, contrary to conventional accounts, Einstein maintained a deep interest in cosmology throughout his career, althoughthis interestwas tempered by reservations concerning the validity of relativistic models of the cosmos at early epochs.
- Introduction
Einstein’s general theory of relativity has unquestionably played a central role in the development of modern cosmology. Yet, it is a surprising fact that the full range of Einstein’s own contributions to cosmology has been somewhat overlooked byphysicists, historians and biographers. Indeed, most discussions of Einstein’s cosmologyportray an image of abrilliant scientist whose cosmologywas limited by a narrowphilosophical prejudice for a static universe; and who proposed a single, rather minimalist model of the expanding universewhen the former model became untenable. For example, just two pages are afforded to Einstein’s cosmology in the definitive biography by Pais (Pais 1986 pp -); three pages in the biography by Clarke ( Clarke 1985 pp - ) and two in that by Isaccson (Isaccson pp- ). Such views can also be found in accounts of the history of modern cosmology such as that by North (North 1965 pp ) and by Kragh (Kragh 1996…). A similar outlook is found in more recent works. For example, one finds the statement ‘Einstein had little interest in applying relativity to a description of the universe’ in the recent anthology ‘Einstein: A Hundred Years of Relativity’ (Robinson 2005 p11), a book that contains contributions by notable physicists such as Freeman Dyson, Stephen Hawking and Steven Weinberg. Similarly, Einstein’s cosmology merited only two pages (in the appendix) in a recent Einstein encyclopedia (Calaprice, Kennefick and Schulmann 2015); meanwhile, remarkably few of the conferences celebrating the centenary of the general theory of relativity offered presentations on Einstein’s cosmology.[1]
On the basis of recent research into Einstein’s cosmological works, we find that his contributions to the fieldweremuch more substantial thanportrayed above. For example, we find that Einstein’s static model of 1917 was the most appropriate model at the time and that he had good reason to distrust the dynamic cosmologies of Alexander Friedman and Georges Lemaîtrewhen they were first mooted. We note that, withthe emergence of observational evidence for a cosmic expansion, Einstein explored a plethora of models of the expanding universe, from a‘steady-state’ model to evolving models of openand closed curvature: these models were tightly constrained by a consideration of empirical observations. We also find that Einstein proposed two further models of the cosmos during a period when few physicists took an active interest in the subject. We argue that his lack of interest in discussions concerning an origin for the universe did not arise from disinterest, but fromreservations concerning the application of the field equations to conditions of high matter density at early cosmic epochs.
2. Einstein’s cosmic model of 1917
Following his first full explication ofthegeneral theory of relativity (Einstein 1916), Einstein lost little time in applying his new theory of space, time and gravitation to the universe as a whole. As he remarked to Willem de Sitter soon afterwards, a key motivation was the clarification of the conceptual foundations of the general theory:“For me… it was a burning question whether the relativity concept can be followed through to the finish, or whether it leads to contradictions” (Einstein 1917a).The resulting paper ‘Cosmological Considerationsin the General Theory of Relativity’ (Einstein 1917b) can be regarded as the beginning of modern cosmology.
In the work, Einstein set himself the task of obtaining solutions to the field equations
for the case of the universe at large.[2]Modelling the cosmos as a pressure-free fluidwhose four-velocity at each point represented the average motion of matter at that point, he constructed the matter–energy tensor as , wheredenotesthe scalar density of matter and denotes the contravariant velocity four-vector.In his analysis, Einstein employed two philosophicalconstraints. The first was the assumption that a consistent model of the cosmos should reflecthis understanding of Mach’s principle:“In a consistent theory of relativity, there can be no inertia relative to “space”, but onlyan inertia of masses relative to one another”(Einstein 1917b). This condition implied that space could not exist independent of matter and thatthe spatial components of the metric tensor of the field equations should therefore vanish at infinity, a boundary condition that Einstein found difficult to satisfy. His solution was to banish the boundaries by assuming a cosmos of closed spatial geometry: “I have not succeeding in formulating boundary conditions for spatial infinity. Nevertheless, there is still a possible way out…for if it were possible to regard the universe as a continuum, which is finite (closed) with respect to is spatial dimensions, we should have no need at all of any such boundary conditions” (Einstein 1917b).
A second constraint was the assumption of aquasi-static distribution of matter, and therefore a static spacetimemetric: “The most important fact that we draw from experience as to the distribution of matter is that the relative velocities of the stars are very small as compared with the velocity of light. So I think for the present we may base our reasoning on the following approximate assumption. There is a system of reference relative to which matter may be looked upon as being permanently at rest” (Einstein 1917b).
Adopting the above two philosophical constraints, and assuming for simplicity a universe that was homogenous and isotropicon the largest scales, Einsteinfound that a consistent solutionto the field equations could not be obtained.[3]Hissolution wasto propose amodification of the field equations allowed by relativity: “However, the system of equations ..allows a readily suggested extension which is compatible with the relativity postulate... On the left hand side of the field equation…we may add the fundamental tensor , multiplied by a universal constant,, at present unknown, without destroying the general covariance.”(Einstein 1917b).[4]Thus Einstein was led to a famous modification of the field equations, according to:
He immediately noted that the modified field equations were compatible with observation for small values of the cosmological constant: “This field equation, with sufficiently small, is in any case compatible with the facts of experience derived from the solar system”(Einstein 1917b).
Employing the modified field equations, Einstein then attaineda solution for the case of the cosmos, a solution in whichthe mean density of matter and the radius of the cosmos were determined by the new cosmological constant according to:
“Thus the newly introduced universal constant λ defines both the mean density of distribution ρwhich can remain in equilibrium and also the radius R and the volumeof spherical space”(Einstein 1917b).[5]
Summarizing the paper, Einstein noted that, while untested empirically, his static, spherical model of the cosmos wasthe simplest consistent model that could be extracted from the general theory, although the prediction of a static, non-zero mean density of matter had necessitated a modification of the field equations: “Thus the theoretical view of the actual universe…is the following. The curvature of space is variable in time and place, according to the distribution of matter, but we may roughly approximate to it by means of a spherical space. 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. 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. …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” (Einstein 1917b).
2.1 Einstein’s static model in retrospect
A detailed discussion of Einstein’s static model can be found in many historical accounts of modern cosmology such as (North 1965 p …; Rindler 1972 p…; Straumann 1985 p…..; Nussbaumer 2006 p… ). We note herethat Einstein’sassumption of a static cosmoswas entirely reasonable at the time. Astronomical observations of the redshift of light from the spiral nebulae (Slipher 1915; Slipher 1917) were not known to Einstein in these years, nor were the unimaginable distances to the nebulae (Hubble 1925). Indeed, many years were to elapse before the discovery of a linear relation between the redshift and distance for the nebulae (Hubble 1929), the first evidence for a cosmic expansion. Thus, the introduction of the cosmological constant to the field equations was a logical step at the time, given that the general theory allowed it:as discussed by Ellis, the frequent description of the cosmological constant as a ‘blunder’ is ahistorical (Ellis 1986).
A technical pointworth notingis that Einstein’s introduction of the cosmological constant arose from a consideration of the problem of gravitational collapse in the Newtonian universe. In his 1917 paper, Einstein noted that the Newtonian paradox could be removed by modifying Poisson’s equation from to, and proposed a similar modification for the case of the field equations of relativity.[6] Although the analogy was not exact (Heckmann 1942 p15; Harvey and Schucking 1999), it is interesting thatthe famous cosmological constant of the field equations originated from a consideration of Newtonian gravity. Einstein’scontemporaneousview of the term can be seen in a rather prescientletter to de Sitter: “In any case, one thing stands. The general theory of relativity allows the addition of the term in the field equations. One day, our actual knowledge of the composition of the fixed-star sky, the apparent motions of fixed stars, and the position of spectral lines as afunction of distance, will probably have come far enough for us to be ableto decide empirically the question of whether or not vanishes. Conviction is a good mainspring, but a bad judge!” (Einstein 1917c).
Not long after the publication of Einstein’s static model, Erwin Schrödinger noted that the new cosmological constant term could be placed on the right hand side of the field equations, as a negative energy density term in the matter-energy tensor (Schrödinger 1918). Einstein’s response was that this proposal was equivalent to his original formulation (Einstein 1918a). Schrödinger also suggested that the term might not be time invariant (Schrödinger 1918).[7] However, this latter suggestion was too speculative for Einstein: “That means, one not only has to start out from the hypothesis of a non-observable negative density in the interstellar spaces, but also has to postulate a hypothetical law about the spacetime distribution of this mass density. The course taken byHerr Schrödinger does not appear passable to me because it leads too deeply into the thicket of hypotheses” (Einstein 1918a).
A great deal has been written about Einstein’s view of the cosmological constant in later years. For example, the emigré Russian physicist George Gamow claimed that Einstein once declared the term “my greatest blunder” (Gamow 1956; Gamow 1970 p44), while others have cast doubt on this claim (Straumann 2002; Earman; Livio 2013 pp 233-241). We will not enter this debate here. Instead, we note that Einstein’s willingness to introduce the term to the field equations in 1917 in order to achieve a consistent static model of the cosmos indicates that he attached great importance to a successful description of the universe by the general theory. However, it is quite curious that Einstein failed to notice that his static model suffered from a fatal flaw: it was unstable against the slightest perturbation, as later pointed out implicitly by Georges Lemaître (Lemaître1927) and explicitly by Arthur Eddington (Eddington 1930).
2.2The de Sitter model
Within a few months of the publication of Einstein’s first model of the cosmos, the Dutch astronomer and theorist Willem de Sitter noted that the modified field equations allowed for an alternate cosmological solution, namely the case of a universe empty of matter (de Sitter 1917a: de Sitter 1917b; de Sitter 1917c). Einstein was greatly perturbed byde Sitter’s solution as the concept of a universe empty of matter was in direct conflict with his understanding of Mach’s principle. He made his reservations public in a paper titled ‘Critical Comment on a Solution of the Gravitational Field Equations Given by Mr De Sitter’: “In my opinion, the general theory of relativity is a satisfying system only if it shows that the physical qualities of space are completely determined by matter alone. Therefore no-fieldmust exist (that is no space-time continuum is possible) without matter that generates it” (Einstein 1918b). Einstein also had a technical objection to the de Sitter model, namely that itappeared to containa spacetimesingularity:“However, g vanishes also for r = , and it seems that no choice of co-ordinates can remove this discontinuity…Until the opposite is proven, we have to assume that the de Sitter solution has a genuine singularity on the surface r = in the finite domain; i.e., it does not satisfy the field equations…for any choice ofco-ordinates” (Einstein 1918b). Thus, Einstein suggested that the de Sitter universe was not truly empty, but that its matter was contained at the horizon: “the de Sitter system does not look like a world free of matter, but rather like a world whose matter is concentrated entirely on the surface r = ” (Einstein 1918b).
A long debate ensued concerning the relative merits of the cosmic models of Einstein and de Sitter (dubbed ‘solution A’ and ‘solution B’ respectively by de Sitter). We shall not revisit this debate here,[8]but notethat the latter model attracted some attention during the 1920s because of a prediction thatradiation in the de Sitter universe would be red-shifted,[9]a prediction that chimed with emerging astronomical observations of the spiral nebulae(Slipher 1915, 1917). Despite a great deal of work by theorists such as Hermann Weyl, Cornelius Lanczos and -, thede Sitter model constituteda source ofsomeconfusion amongst theorists for many years; it waseventually shownthat its static character was amisapprehension (Weyl 1923;Lemaître 1925).
- Towards the dynamic universe
In 1922, the young Russian physicist Alexander Friedman suggested that non-stationarysolutions to the Einstein field equations shouldbe considered in relativistic models of the cosmos. In his first paper on the subject, Friedman explored time-varying models of positive spatial curvature for various values of the cosmological constant (Friedman 1922) and demonstrated that the Einstein and de Sitter models were special cases of this more general class of solutions.[10]Soon afterwards, Friedman noted that time-varying cosmologies allowed the possibility of negative spatial curvature, and published an analysis of time-varying models of hyperbolic curvature in 1924 (Friedman 1924). Thus, Friedman had explored almost all the main theoretical possibilities for the evolution of the cosmos and its geometry by 1924.[11]
Einstein did not view Friedman’s time-varying cosmology with enthusiasm. His first reaction was that Friedman had made a mathematical error (Einstein 1922). When it transpired that the error lay in Einstein’s correction, he duly retracted it (Einstein 1923a); however, an unpublished draft of Einstein’s retraction makes clear his view ofFriedman’s cosmology:“to this a physical significance can hardly be ascribed’” (Einstein 1923b).[12]
Unaware of Friedman’s time-varying analysis, the Belgian physicist Georges Lemaître proposed an expanding model of the cosmos in 1927. A theoretician with significant training in astronomy, Lemaître was aware of V.M. Slipher’s observations of the redshifts of the spiral nebulae (Slipher 1915, 1917), and of preliminary estimates of the vast distances to the nebulae by Edwin Hubble (Kragh 1996 p29; Farrell 2009 p78, p90; Lambert 2005 ....). Interpreting the redshifts of the nebulae as evidence of a relativistic expansion of space, Lemaîtrederived a universe of expanding radius from Einstein’s field equations, and estimated a rate of cosmic expansion from average values of the velocities and distances of the nebulae from Slipher and Hubble respectively.[13] This work received very little attention at first, probably because it was published in French in a little-known Belgian journal (Lemaître 1927). However, Lemaître discussed the model directly with Einstein at the 1927 Solvay conference, only to have it dismissed with the forthright comment:“Voscalculssont corrects, maisvotre physique est abominable” (Lemaître 1958).[14]
In 1929, Edwin Hubble published the first empirical evidence of a linear relation between the redshifts of the spiral nebulae (now known to be extra-galactic) and their radial distance (Hubble 1929).[15] By this stage, it had also been established that the static models of Einstein and de Sitter presented problems of a theoretical nature: Einstein’s universe was not stable (Lemaître 1927; Eddington 1930) while de Sitter’s universe was not truly static (Weyl 1923; Lemaître 1925). In consequence, theorists began to take the notion of a cosmic expansion seriously,[16],[17] and a variety of relativistic time-varying models of the cosmos of the Friedman-Lemaître typewere soon proposed (Eddington 1930, 1931: de Sitter 1930a, 1930b; Tolman 1930a, 1930b, 1931, 1932; Heckmann 1931, 1932; Robertson 1932, 1933).