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These comments on what people were thinking are gleaned from the literature and supplemented by private discussions and our own recollections. More is required for a satisfactory history of the subject, of course, but we hope we have captured the main themes of the discussion and the way the themes have evolved to the present appreciation of the situation.

A. Einstein's thoughts

Einstein disliked the idea of an island universe in asymptotically flat spacetime, because a particle could leave the island and move arbitrarily far from all the other matter in the universe, yet preserve all its inertial properties, which he considered a violation of Mach's idea of the relativity of inertia. Einstein's (1917) cosmological model accordingly assumes the universe is homogeneous and isotropic, on average, thus removing the possibility of arbitrarily isolated particles. Einstein had no empirical support for this assumption, yet it agrees with modern precision tests. There is no agreement on whether this is more than a lucky guess.

Motivated by the observed low velocities of the then known stars, Einstein assumed that the large-scale structure of the universe is static. He introduced the cosmological constant to reconcile this picture with his general relativity theory. In the notation of Eq. (12), one sees that a positive value of OmegaLambda0 can balance the positive values of OmegaM0 and OmegaR0 for consistency with addot = 0. The balance is unstable: a small perturbation to the mean mass density or the mass distribution causes expansion or contraction of the whole or parts of the universe. One sees this in Eq. (24): the mass distribution can be chosen so the two terms on the right hand side cancel, but the balance can be upset by redistributing the mass. (13)

Einstein did not consider the cosmological constant to be part of the stress-energy term: his form for the field equation (in the streamlined notation of Eq. [17]) is

Equation 34 (34)

The left hand side contains the metric tensor and its derivatives; a new constant of nature, Lambda, appears in the addition to Einstein's original field equation. One can equally well put Einstein's new term on the right hand side of the equation, as in Eq. (20), and count rhoLambda gµnu as part of the source term in the stress-energy tensor. The distinction becomes interesting when rhoLambda takes part in the dynamics, and the field equation is properly written with rhoLambda, or its generalization, as part of the stress-energy tensor. One would then say that the differential equation of gravity physics has not been changed from Einstein's original form; instead there is a new component in the content of the universe.

Having taken it that the universe is static, Einstein did not write down the differential equation for a(t), and so did not see the instability. Friedmann (1922, 1924) found the evolving homogeneous solution, but had the misfortune to do it before the astronomy became suggestive. Slipher's measurements of the spectra of the spiral nebulae - galaxies of stars - showed most are shifted toward the red, and Eddington (1924, pp. 161-2) remarked that that might be a manifestation of the second, repulsive, term in Eq. (24). Lemaître (1927) introduced the relation between Slipher's redshifts and a homogeneous matter-filled expanding relativistic world model. He may have been influenced by Hubble's work that was leading to the publication in 1929 of the linear redshift-distance relation (eq. [5]): as a graduate student at MIT he attended a lecture by Hubble.

In Lemaître's (1927) solution the expanding universe traces asymptotically back to Einstein's static case. Lemaître then turned to what he called the primeval atom, and we would term a Big Bang model. This solution expands from densities so large as to require some sort of quantum treatment, passes through a quasi-static approximation to Einstein's solution, and then continues expanding to de Sitter's (1917) empty space solution. To modern tastes this "loitering" model requires incredibly special initial conditions, as will be discussed. Lemaître liked it because the loitering epoch allows the expansion time to be acceptably long for Hubble's (1929) estimate of H0, which is an order of magnitude high.

The record shows Einstein never liked the Lambda term. His view of how general relativity might fit Mach's principle was disturbed by de Sitter's (1917) solution to Eq. (34) for empty space (Tµnu = 0) with Lambda > 0. (14),(15) Pais (1982, p. 288) points out that Einstein in a letter to Weyl in 1923 comments on the effect of Lambda in Eq. (24): "According to De Sitter two material points that are sufficiently far apart, continue to be accelerated and move apart. If there is no quasistatic world, then away with the cosmological term." We do not know whether at this time Einstein was influenced by Slipher's redshifts or Friedmann's expanding world model.

The earliest published comments we have found on Einstein's opinion of Lambda within the evolving world model (Einstein, 1931; Einstein and de Sitter, 1932) make the point that, since not all the terms in the expansion rate Eq. (11) are logically required, and the matter term surely is present and likely dominates over radiation at low redshift, a reasonable working model drops OmegaK0 and OmegaLambda0 and ignores OmegaR0. This simplifies the expansion rate equation to what has come to be called the Einstein-de Sitter model,

Equation 35 (35)

where rhoM is the mass density in non-relativistic matter; here OmegaM = 8pi G rhoM / (3H2) is unity. The left side is a measure of the kinetic energy of expansion per unit mass, and the right-hand side a measure of the negative of the gravitational potential energy. In effect, this model universe is expanding with escape velocity.

Einstein and de Sitter point out that Hubble's estimate of H0 and de Sitter's estimate of the mean mass density in galaxies are not inconsistent with Eq. (35) (and since both quantities scale with distance in the same way, this result is not affected by the error in the distance scale that affected Hubble's initial measurement of H0). But the evidence now is that the mass density is about one quarter of what is predicted by this equation, as we will discuss.

Einstein and de Sitter (1932) remark that the curvature term in Eq. (11) is "essentially determinable, and an increase in the precision of the data derived from observations will enable us in the future to fix its sign and determine its value." This is happening, 70 years later. The cosmological constant term is measurable in principle, too, and may now have been detected. But Einstein and de Sitter say only that the theory of an expanding universe with finite mean mass density "can be reached without the introduction of" Lambda.

Further to this point, in the appendix of the second edition of his book, The Meaning of Relativity, Einstein (1945, p. 127) states that the "introduction of the `cosmologic member'" - Einstein's terminology for Lambda - "into the equations of gravity, though possible from the point of view of relativity, is to be rejected from the point of view of logical economy", and that if "Hubble's expansion had been discovered at the time of the creation of the general theory of relativity, the cosmologic member would never have been introduced. It seems now so much less justified to introduce such a member into the field equations, since its introduction loses its sole original justification, - that of leading to a natural solution of the cosmologic problem." Einstein knew that without the cosmological constant the expansion time derived from Hubble's estimate of H0 is uncomfortably short compared to estimates of the ages of the stars, and opined that that might be a problem with the star ages. The big error, the value of H0, was corrected by 1960 (Sandage, 1958, 1962).

Gamow (1970, p. 44) recalls that "when I was discussing cosmological problems with Einstein, he remarked that the introduction of the cosmological term was the biggest blunder he ever made in his life." This certainly is consistent with all of Einstein's written comments we have seen on the cosmological constant per se; we do not know whether Einstein was also referring to the missed chance to predict the evolution of the universe.

13 To help motivate the introduction of Lambda, Einstein (1917) mentioned a modification of Newtonian gravity physics that might make the theory well defined when the mass distribution is homogeneous. In Einstein's example, similar to what was considered by Seeliger and Neumann in the mid 1890s, the modified field equation for the gravitational potential varphi is nabla2 varphi - lambda varphi = 4pi G rhoM. This allows the nonsingular homogeneous static solution varphi = -4pi G rhoM / lambda. In this example the potential for an isolated point mass is the Yukawa form, varphi propto e-lambda1/2r / r. Trautman (1965) points out that this is not the nonrelativistic limit of general relativity with the cosmological term. Rather, Eq. (24) follows from nabla2 varphi = 4pi G(rhoM - 2rhoLambda), where the active gravitational mass density of the Lambda term is rhoLambda + 3pLambda = - 2rhoLambda. Norton (1999) reviews the history of ideas of this Seeliger-Neumann Yukawa-type potential in gravity physics. Back.

14 North (1965) reviews the confused early history of ideas on the possible astronomical significance of de Sitter's solution for an empty universe with Lambda > 0; we add a few comments on the physics that contributed to the discovery of the expanding world model. Suppose an observer in de Sitter's spacetime holds a string tied to a source of light, so the source stays at fixed physical distance r << HLambda-1. The source is much less massive than the observer, the gravitational frequency shift due to the observer's mass may be neglected, and the observer is moving freely. Then the observer receives light from the source shifted to the red by deltalambda / lambda = - (HLambdar)2 / 2. The observed redshifts of particles moving on geodesics depend on the initial conditions. Stars in the outskirts of our galaxy are held at fixed mean distances from us by their transverse motions. The mean shifts of the spectra of light from these stars include this quadratic de Sitter term as well as the much larger Doppler and ordinary gravitational shifts. The prescription for initial conditions that reproduces the linear redshift-distance relation for distant galaxies follows Weyl's (1923) principle: the world particle geodesics trace back to a near common position in the remote past, in the limiting case of the Friedmann-Lemaître model at OmegaM0 rightarrow 0. This spatially homogeneous coordinate labeling of de Sitter's spacetime, with space sections with negative curvature, already appears in de Sitter (1917, Eq. [15]), and is repeated in Lanczos (1922). This line element is the second expression in our Eq. (15) with a propto cosh HLambdat. Lemaître (1925) and Robertson (1928) present the coordinate labeling for the spatially-flat case, where the line element is ds2 = dt2 - e2HLambdat(dx2 + dy2 + dz2) (in the choice of symbols and signature in Eqs. [15] and [27]). Lemaître (1925) and Robertson (1928) note that particles at rest in this coordinate system present a linear redshift-distance relation, v = HLambdar, at small v. Robertson (1928) estimated HLambda, and Lemaître (1927) its analog for the Friedmann-Lemaître model, from published redshifts and Hubble's galaxy distances. Their estimates are not far off Hubble's (1929) published value. Back.

15 To the present way of thinking the lengthy debate about the singularity in de Sitter's static solution, chronicled in North (1965), seems surprising, because de Sitter (1917) and Klein (1918) had presented de Sitter's solution as a sphere embedded in 4 plus 1 dimensional flat space, with no physical singularity. Back.

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