7.1 Is the Universe Closed? 
The unboundedness of space has a greater empirical certainty than any experience of the external world, but its infinitude does not in any way follow from this; quite the contrary. Space would necessarily be finite if one assumed independence of bodies from position, and thus ascribed to it a constant curvature, as long as this curvature had ever so small a positive value. 
B. Riemann, 1854 
Very soon after arriving at the final form of the field equations, Einstein began to consider their implications with regard to the overall structure of the universe. His 1917 paper presented a simple model of a closed spherical universe which "from the standpoint of the general theory of relativity lies nearest at hand". In order to arrive at a quasistatic distribution of matter he found it necessary to introduce the "cosmological term" to the field equations (as discussed in Section 5.8), so he based his analysis on the equations 
where l is the cosmological constant. Before invoking the field equations we can consider the general form of a metric that is suitable for representing the largescale structure of the universe. First, we ordinarily assume that the universe would appear to be more or less the same when viewed from the rest frame of any galaxy, anywhere in the universe (at the present epoch). This is sometimes called the Cosmological Principle. 
Then, since the universe on a large scale appears (to us) highly homogenous and isotropic, we infer that these symmetries apply to every region of space. This greatly restricts the class of possible metrics. In addition, we can choose, for each region of space, to make the time coordinate coincide with the proper time of the typical galaxy in that region. Also, according to the Cosmological Principle, the coefficients of the spatial terms of the (diagonalized) metric should be independent of location, and any dependence on the time coordinate must apply symmetrically to all the space coordinates. From this we can infer a metric of the form 

where s(t) is some (still to be determined) dimensionless scale factor dependent on time, and ds is the total space differential with no time dependence. In other words, the “shape” of the spatial metric is fixed, but the scale factor is a function of time. Recall that for a perfectly flat Euclidean space the differential line element is 

where r^{2} = x^{2} + y^{2} + z^{2}. On the other hand, if we want to allow our space (at a given coordinate time t) to have curvature, the Cosmological Principle suggests that the (large scale) curvature should be the same everywhere and in every direction. Thus the Gaussian curvature of every twodimensional tangent subspace has the same value at every point. Now suppose we embed a Euclidean threedimensional space (x,y,z) in a fourdimensional space (w,x,y,z) whose metric is 

where k is either +1 or −1. If k = +1 the fourdimensional space is Euclidean, whereas if k = −1 it is pseudoEuclidean (like the Minkowski metric). In either case the fourdimensional space is "flat", i.e., has zero Riemannian curvature. We then consider a threedimensional subspace comprising a sphere (or pseudosphere), i.e., the locus of points satisfying the condition 

where A is a constant characteristic length. From this we have w^{2} = A^{2}  kr^{2}, and therefore 

Substituting this into the fourdimensional line element above gives the metric for the threedimensional sphere (or pseudosphere) 

Taking this as the spatial part of our overall spacetime metric (2), replacing r with Ar (so r is now dimensionless), and defining R(t) = As(t), we arrive at 

This metric, with k = +1 and R(t) = constant, was the basis of Einstein's 1917 paper, and it was subsequently studied by Alexander Friedmann in 1922 with both possible signs of k and with variable R(t). The general form was rediscovered by Robertson and Walker (independently) in 1935, so it is now often referred to as the RobertsonWalker metric. Notice that with k = +1 this metric essentially corresponds to polar coordinates on the "surface" of a sphere projected onto the "equatorial plane", so each value of r corresponds to two points, one in the Northern and one in the Southern hemisphere. We could remedy this by making the change of variable r ® r/(1 + kr^{2}/4), which (in the case k = +1) amounts to stereographic projection from the North pole to a tangent plane at the South pole. In terms of this transformed radial variable the RobertsonWalker metric has the form 

As noted above, Einstein originally assumed R(t) = constant, i.e., he envisioned a static unchanging universe. He also assumed the matter in the universe was roughly "stationary" at each point with respect to these cosmological coordinates, so the only nonzero component of the stressenergy tensor in these coordinates is T_{tt} = r where r is the density of matter (assumed to be uniform, in accord with the Cosmological Principle). On this basis, the field equations imply 

Here the symbol R denotes the assumed constant value of R(t) (not to be confused with the Ricci curvature scalar). This explains why Einstein was originally led to introduce a nonzero cosmological constant l, because if we assume a static universe and the Cosmological Principle, the field equations of general relativity can only be satisfied if the density r is proportional to the cosmological constant. However, it was soon pointed out that this static model is unstable, so it is apriori unlikely to correspond to the physical universe. Moreover, astronomical observations subsequently indicated that the universe (on the largest observable scale) is actually expanding, so we shouldn't restrict ourselves to models with R(t) = constant. If we allow R(t) to be variable, then the original field equations, without the cosmological term (i.e., with l = 0), do have solutions. In view of this, Einstein decided the cosmological term was unnecessary and should be excluded. 
Interestingly, George Gamow was working with Friedmann in Russia in the early 1920's, and later recalled that "Friedmann noticed that Einstein had made a mistake in his alleged proof that the universe must necessarily be stable". Specifically, Einstein had divided through an equation by a certain quantity, even though that quantity was zero under a certain set of conditions. As Gamow notes, "it is well known to students of high school algebra" that division by zero is not valid. Friedmann realized that this error invalidated Einstein's argument against the possibility of a dynamic universe, and indeed under the condition that the quantity in question vanishes, it is possible to satisfy the field equations with a dynamic model, i.e., with a model of the form given by the RobertsonWalker metric with R(t) variable. It's worth noting that Einstein's 1917 paper did not actually contain any alleged proof that the universe must be static, but it did suggest that a nonzero cosmological constant required a nonzero density of matter. Shortly after Einstein's paper appeared, de Sitter gave a counterexample (see Section 7.6), i.e., he described a model universe that had a nonzero l but zero matter density. However, unlike Einstein's model, it was not static. Einstein objected strenuously to de Sitter's model, because it showed that the field equations allowed inertia to exist in an empty universe, which Einstein viewed as "inertia relative to space", and he still harbored hopes that general relativity would fulfill Mach's idea that inertia should only be possible in relation to other masses. It was during the course of this debate that (presumably) Einstein advanced his "alleged proof" of the impossibility of dynamic models (with the errant division by zero?). However, before long Einstein withdrew his objection, realizing that his argument was flawed. Years later he recalled the sequence of events in a discussion with Gamow, and made the famous remark that it had been the biggest blunder of his life. This is usually interpreted to mean that he regretted ever considering a cosmological term (which seems to have been the case), but it could also be referring to his erroneous argument against the idea of a dynamic universe, resulting from his unfortunate "division by zero". 
In any case, the Friedmann universes (with and without cosmological constant) became the "standard model" for cosmologies. If k = +1 the manifold represented by the RobertsonWalker metric is a finite spherical space, so it is called "closed". If k = 0 or 1 the metric is typically interpreted as representing an infinite space, so it is called "open". However, it's worth noting that this need not be the case, because the metric gives only local attributes of the manifold; it does not tell us the overall global topology. For example, we discuss in Section 7.4 a manifold that is everywhere locally flat, but that is closed cylindrically. This shows that when we identify "open" (infinite) and "closed" (finite) universes with the cases k = −1 and k = +1 respectively, we are actually assuming the "maximal topology" for the given metric in each case. 
Based on the RobertsonWalker metric (3), we can compute the components of the Ricci tensor and scalar and substitute these along with the simple uniform stressenergy tensor into the field equations (1) to give the conditions on the scale function R = R(t): 


where dots signify derivatives with respect to t. As expected, if R(t) is constant, these equations reduce to the ones that appeared in Einstein's original 1917 paper, whereas with variable R(t) we have a much wider range of possible solutions. 
It may not be obvious that these two equations have a simultaneous solution, but notice that if we multiply the first condition through by R(t)^{3} and differentiate with respect to t, we get 

The lefthand side is equal to times the lefthand side of the second condition, which equals zero, so the right hand side must also vanish, i.e., the derivative of (8p/3)GrR(t)^{3} must equal zero. This implies that there is a constant C such that 

With this stipulation, the two conditions are redundant, i.e., a solution of one is guaranteed to be a solution of the other. Substituting for (8p/3)Gr in the first condition and multiplying through by R(t)^{3}, we arrive at the basic differential equation for the scale parameter of a Friedmann universe 

Incidentally, if we multiply through by R(t), differentiate with respect to t, divide through by , and differentiate again, the constants k and C drop out, and we arrive at 

With l = 0 this is identical to the gravitational separation equation (3) in Section 4.2, showing that the cosmological scale parameter R(t) is yet another example of a naturally occurring spatial separation that satisfies this differential equation. It follows that the admissible functions R(t) (with l = 0) are formally identical to the gravitational freefall solutions described in Section 4.3. Solving equation (4) (with l = 0) for and switching to normalized coordinates T = t/C and X = R/C, we get 

Accordingly as k equals −1, 0, or +1, integration of this equation gives 



A plot of these three solutions is shown below. 

In all three cases with l = 0, the expansion of the universe is slowing down, albeit only slightly for the case k = 1. However, if we allow a nonzero cosmological constant l, there is a much greater variety of possible solutions to Friedmann's equation (2), including solutions in which the expansion of the universe is actually accelerating exponentially. Based on the cosmic scale parameter R and its derivatives, the three observable parameters traditionally used to characterize a particular solution are 



In terms of these parameters, the constants appearing in the Friedmann equation (4) can be expressed as 

In principle if astronomers could determine the values of H, q, and s with enough precision, we could decide on empirical grounds the sign of k, and whether or not l is zero. Thus, assuming the maximal topologies (and the largescale validity of general relativity), we could determine whether the universe is open or closed, and whether it will expand forever or eventually recontract. Unfortunately, none of the parameters is known with enough precision to distinguish between these possibilities. 
One source of uncertainty is in the estimates of the mass density r of the universe. Given the best current models of star masses, and the best optical counts of stars in galaxies, and the apparent density of galaxies, we estimate an overall mass density that is only a small fraction of what would be required to make k = 0. However, there are reasons to believe that much (perhaps most) of the matter in the universe is not luminous. (For example, the observed rotation of individual galaxies indicates that they ought to fly apart unless there is substantially more mass in them than is visible to us.) This has led physicists and astronomers to search for the "missing mass" in various forms. 
Another source of uncertainty is in the values of R and its derivatives. For example, in its relatively brief history, Hubble's constant has undergone revisions of an order of magnitude, both upwards and downwards. In recent years the Hubble space telescope and several modern observatories on Earth seem to have found strong evidence that the expansion of the universe is actually accelerating. If so, then it could be accounted for in the context of general relativity only by a nonzero cosmological constant l (on a related question, see Section 7.6), with the implication that the universe is infinite and will expand forever (at an accelerating rate). 
Nevertheless, the idea of a closed finite universe is still of interest, partly because of the historical role it played in Einstein's thought, but also because it remains (arguably) the model most compatible with the spirit of general relativity. In an address to the Berlin Academy of Sciences in 1921, Einstein said 
I must not fail to mention that a theoretical argument can be adduced in favor of the hypothesis of a finite universe. The general theory of relativity teaches that the inertia of a given body is greater as there are more ponderable masses in proximity to it; thus it seems very natural to reduce the total effect of inertia of a body to action and reaction between it and the other bodies in the universe... From the general theory of relativity it can be deduced that this total reduction of inertia to reciprocal action between masses  as required by E. Mach, for example  is possible only if the universe is spatially finite. On many physicists and astronomers this argument makes no impression... 
This is consistent with the approach taken in Einstein's 1917 paper. Shortly thereafter he presented (in "The Meaning of Relativity", 1922) the following three arguments against the conception of infinite space, and for the conception of a bounded, or closed, universe: 
(1) From the standpoint of the theory of relativity, to postulate a closed universe is very much simpler than to postulate the corresponding boundary condition at infinity of the quasiEuclidean structure of the universe. 
(2) The idea that Mach expressed, that inertia depends on the mutual attraction of bodies, is contained, to a first approximation, in the equations of the theory of relativity; it follows from these equations that inertia depends, at least in part, upon mutual actions between masses. Thereby Mach's idea gains in probability, as it is an unsatisfactory assumption to make that inertia depends in part upon mutual actions, and in part upon an independent property of space. But this idea of Mach's corresponds only to a finite universe, bounded in space, and not to a quasiEuclidean, infinite universe. From the standpoint of epistemology it is more satisfying to have the mechanical properties of space completely determined by matter, and this is the case only in a closed universe. 
(3) An infinite universe is possible only if the mean density of matter in the universe vanishes. Although such an assumption is logically possible, it is less probable than the assumption of a finite mean density of matter in the universe. 
Along these same lines, Misner, Thorne, and Wheeler ("Gravitation") comment that general relativity "demands closure of the geometry in space as a boundary condition on the initialvalue equations if they are to yield a welldetermined and unique 4geometry." Interestingly, when they quote Einstein's reasons in favor of a closed universe they omit the third without comment, although it reappears (with a caveat) in the subsequent "Inertia and Gravitation" of Ciufolini and Wheeler. As we've seen, Einstein was initially under the mistaken impression that the only cosmological solutions of the field equations are those with 

where R is the radius of the universe, r is the mean density of matter, and k is the gravitational constant. This much is consistent with modern treatments, which agree that at any given epoch in a Friedmann universe with constant nonnegative curvature the radius is inversely proportional to the square root of the mean density. On the basis of (5) Einstein continued 
If the universe is quasiEuclidean, and its radius of curvature therefore infinite, then r would vanish. But it is improbable that the mean density of matter in the universe is actually zero; this is our third argument against the assumption that the universe is quasiEuclidean. 
However, in the 2nd edition of "The Meaning of Relativity" (1945), he added an appendix, "essentially nothing but an exposition of Friedmann's idea", i.e., the idea that "one can reconcile an everywhere finite density of matter with the original form of the equations of gravity [without the cosmological term] if one admits the time variability of the metric distances...". In this appendix he acknowledged that in a dynamic model, as described above, it is perfectly possible to have an infinite universe with positive density of matter, provided that k = −1. It's clear that Einstein originally had not seriously considered the possibility of a universe with positive mass density but overall negative curvature. In the first edition, whenever he mentioned the possibility of an infinite universe he referred to the space as "quasiEuclidean", which I take to mean "essentially flat". He regarded this open infinite space as just a limiting case of a closed spherical universe with infinite radius. He simply did not entertain the possibility of a hyperbolic (k = −1) universe. (It's interesting that Riemann, too, excluded spaces of negative curvature from his 1854 lecture, without justification.) His basic objection was evidently that a spacetime with negative curvature possesses an inherent structure independent of the matter it contains, and he was unable to conceive of any physical source of negative curvature. This typically entails "ad hoc" boundary conditions at infinity, precisely what's required in an open universe, which Einstein regarded as contrary to the spirit of relativity. 
At the end of the appendix in the 2nd edition, Einstein conceded that it comes down to an empirical question. If (8p/3)Gr is greater than H^{2}, then the universe is closed and spherical; otherwise it is open and flat or pseudospherical (hyperbolic). He also makes the interesting remark that although we might possibly prove the universe is spherical, "it is hardly imaginable that one could prove it to be pseudospherical". His reasoning is that in order to prove the universe is spherical, we need only identify enough matter so that (8p/3)Gr exceeds H^{2}, whereas if our current estimate of r is less than this threshold, it will always be possible that there is still more "missing matter" that we have not yet identified. At this stage Einstein was assuming a zero cosmological constant, so it may not have occurred to him that it might someday be possible to determine empirically that the expansion of the universe is accelerating. 
Ironically, already in 1919 Einstein had written privately to Lorentz acknowledging the weakness of the idea that the universe must be closed. 
My view that the state of the ether (i.e., the g_{mn}’s) must be determined by the matter alone has nothing compelling about it. That is why one cannot argue for the closure of the world with as much certainty as I have done. 
His use of the word “ether” for the gravitational/inertial field was in deference to Lorentz, but even aside from that, this is a remarkable statement, considering how strongly Einstein subsequently continued to promote the concept of a closed universe, based on the Machian idea that the metrical field “must be determined by the matter alone”. Even after he later disavowed “Mach’s principle”, he went on arguing in favor of a closed universe. 
Ultimately, was there any merit in Einstein's skepticism toward the idea of an "open" universe? Even setting aside his third argument, the first two still carry some weight with some people, especially those who are sympathetic to Mach's ideas regarding the relational origin of inertia. In an open universe we must accept the fact that there are multiple, physically distinct, solutions compatible with a given distribution of matter and (nongravitational) energy. In such a universe the "background" inertial field can in no way be attributed to the matter and energy content of the universe. From this standpoint, general relativity can never give an unambiguous answer to the twins paradox (for example), because the proper time integral over a given path from A to B depends on the inertial field, and in an open universe this field cannot be inferred from the distribution of massenergy. It is determined primarily by whatever absolute boundary conditions we choose to impose, independent of the distribution of massenergy. Einstein believed that such boundary conditions were inherently nonrelativistic, because they require us to single out a specific frame of reference  essentially Newton's absolute space. (In later years a great deal of work has been done in attempting to develop boundary conditions "at infinity" that do not single out a particular frame. This is discussed further in Section 7.7.) 
The only alternative (in an open universe) that Einstein could see in 1917 was for the metric to degenerate far from matter in such a way that inertia vanishes, i.e., we would require that the metric at infinity go to something like 

Such a boundary condition would be the same with respect to any frame of reference, so it wouldn't single out any specific frame as the absolute inertial frame of the universe. Einstein pursued this approach for a long time, but finally abandoned it because it evidently implies that the outermost shell of stars must exist in a metric very different from ours, and as a consequence we should observe their spectral signatures to be significantly shifted. (At the time there was no evidence of any "cosmological shift" in the spectra of the most distant stars. We can only speculate how Einstein would have reacted to the discovery of quasars, the most distant objects known, which are in fact characterized by extreme redshifts and apparently extraordinary energies.) 
The remaining option that Einstein considered for an open asymptotically flat universe is to require that, for a suitable choice of the system of reference, the metric must go to 

at infinity. However, the imposition of this condition in a spherically symmetrical way (for example) singles out one particular frame of reference as the absolute inertial frame of the universe with no relational foundation. This, Einstein said, "is contrary to the spirit of the relativity principle", and it was the basis of his early view that general relativity is most compatible with a closed unbounded universe. The recent astronomical findings that seem to indicate an accelerating expansion have caused most scientists to abandon closed models, but there seems to be some lack of appreciation for the damage an open universe does to the epistemological strength of general relativity. As Einstein wrote in 1945, "the introduction of [the cosmological constant] constitutes a complication of the theory, which seriously reduces its logical simplicity". 
Of course, in both an open and a closed universe there must be boundary and/or initial conditions, but the question is whether the distribution of massenergy by itself is adequate to define the field, or whether independent boundary conditions are necessary to pin down the field. In a closed universe the "boundary conditions" can be more directly identified with the distribution of massenergy, whereas in an open universe they are necessarily quite independent. Thus a closed universe can claim to satisfy Mach's principle at least to some degree, whereas an open universe definitely can't. The seriousness of this depends on how seriously we take Mach's principle. Since we can just as well regard a field as a palpable constituent of the universe, and since the metric of spacetime itself is a field in general relativity, it can be argued that Mach's dualistic view is no longer relevant. However, the second issue is whether even the specification of the distribution of massenergy plus boundary conditions at infinity yields a unique solution. For Maxwell's equations (which are linear) it does, but for Einstein's equations (which are nonlinear) it doesn't. This is perhaps what Misner, et al, are referring to when they comment that "Einstein's theory...demands closure of the geometry in space ... as a boundary condition on the initial value equations if they are to yield a welldetermined (and, we now know, a unique) 4geometry". 
In view of this, one might argue that the (apparent) uniqueness of the metrical field supports the idea of a closed universe  at least within the context of general relativity. To put it more explicitly, if we believe the structure of the universe is governed by general relativity, and that the structure is determinate, then the universe must be closed. If the universe is not closed, then general relativity must be incomplete in the sense that there must be something other than general relativity determining which of the possible structures actually exists. Admittedly, completeness in this sense is a very ambitious goal for any theory, but it's interesting to recall the famous "EPR" paper in which Einstein criticized quantum mechanics on the grounds that it could not be a complete description of nature. (See Section 9.5.) He may well have had this on his mind when he pointed out how seriously the introduction of a cosmological constant undermines the logical simplicity of general relativity, which was always his criterion for evaluating the merit of any scientific theory. 
We can see him wrestling with this issue, even in his 1917 paper, where he notes that some people (such as de Sitter) have argued that we have no need to consider boundary conditions at infinity, because we can simple specify the metric at the spatial limit of the domain under consideration, just as we arbitrarily (or empirically) specify the inertial frames when working in Newtonian mechanics. But this clearly reduces general relativity to a rather weak theory that must be augmented by other principles and/or considerable amounts of arbitrary information in order to yield determinate results. Not surprisingly, Einstein was unenthusiastic about this alternative. As he said, "such a complete resignation in this fundamental question is for me a difficult thing. I should not make up my mind to it until every effort to make headway toward a satisfactory view had proved to be in vain". 
