7.6  Cosmological Coherence

 

Our main “difference in creed” is that you have a specific belief and I am a skeptic.

                                                                                                                Willem de Sitter, 1917

 

Almost immediately after Einstein arrived at the final field equations of general relativity, the very foundation of his belief in those equations was shaken, first by appearance of Schwarzschild’s exact solution of the one-body problem. This was disturbing to Einstein because at the time he held the Machian belief that inertia must be attributable to the effects of distant matter, so he thought the only rigorous global solutions of the field equations would require some suitable distribution of distant matter. Schwarzschild’s solution represents a well-defined spacetime extending to infinity, with ordinary inertial behavior for infinitesimal test particles, even though the only significant matter in this universe is the single central gravitating body. That body influences the spacetime in its vicinity, but the metric throughout spacetime is primarily determined by the spherical symmetry, leading to asymptotically flat spacetime at great distances from the central body. This seems rather difficult to reconcile with “Mach’s Principle”, but there was worse to come, and it was Einstein himself who opened the door.

 

In an effort to conceive of a static cosmology with uniformly distributed matter he found it necessary to introduce another term to the field equations, with a coefficient called the cosmological constant. (See Section 5.8.) Shortly thereafter, Einstein received a letter from the astronomer Willem de Sitter, who pointed out a global solution of the modified field equations (i.e., with non-zero cosmological constant) that is entirely free of matter, and yet that possesses non-trivial metrical structure. This thoroughly un-Machian universe was a fore-runner of Gödel’s subsequent cosmological models containing closed-timelike curves. After a lively and interesting correspondence about the shape of the universe, carried on between a Dutch astronomer and a German physicist at the height of the first world war, de Sitter published a paper on his solution, and Einstein published a rebuttal, claiming (incorrectly) that “the De Sitter system does not look at all like a world free of matter, but rather like a world whose matter is concentrated entirely on the [boundary]”. The discussion was joined by several other prominent scientists, including Weyl, Klein, and Eddington, who all tried to clarify the distinction between singularities of the coordinates and actual singularities of the manifold/field. Ultimately all agreed that de Sitter was right, and his solution does indeed represent a matter-free universe consistent with the modified field equations.

 

We’ve seen that the Schwarzschild metric represents the unique spherically symmetrical solution of the original field equations of general relativity - assuming the cosmological constant, denoted by l in Section 5.8, is zero.  If we allow a non-zero value of l, the Schwarzschild solution generalizes to

 

 

To avoid upsetting the empirical successes of general relativity, such as the agreement with Mercury’s excess precession, the value of l must be extremely small, certainly less than 10-40 m-2, but not necessarily zero. If l is precisely zero, then the Schwarzschild metric goes over to the Minkowski metric when the gravitating mass m equals zero, but if l is not precisely zero the Schwarzschild metric with zero mass is

 

 

where L is a characteristic length related to the cosmological constant by L2 = 3/l. This is one way of writing the metric of de Sitter spacetime. Just as Minkowski spacetime is a solution of the original vacuum field equation Rmn = 0, so the de Sitter metric is a solution of the modified field equations Rmn = lgmn. Since there is no central mass in this case, it may seem un-relativistic to use polar coordinates centered on one particular point, but it can be shown that – just as with the Minkowski metric in polar coordinates – the metric takes the same form when centered on any point.

 

The metric (1) can be written in a slightly different form in terms of the radial coordinate r defined by

 

 

Noting that dr/L = cos(r/L)dr, the de Sitter metric is

 

 

Interestingly, with a suitable change of coordinates, this is actually the metric of the surface of a four-dimensional pseudo-sphere in five-dimensional Minkowski space. Returning to equation (1), let x,y,z denote the usual three orthogonal spatial coordinates such that x2 + y2 + z2 = r2, and suppose there is another orthogonal spatial coordinate W and a time coordinate T defined by

 

 

For any values of x,y,z,t we have

 

 

so this locus of events comprises the surface of a hyperboloid, i.e., a pseudo-sphere of “radius” L. In other words, the spatial universe for any given time T is the three-dimensional surface of the four-dimensional sphere of squared radius L2 + T2. Hence the space shrinks to a minimum radius L at time T = 0 and then expands again as T increases, as illustrated below (showing only two of the spatial dimensions).

 

 

Assuming the five-dimensional spacetime x,y,z,W,T has the Minkowski metric

 

 

we can determine the metric on the hyperboloid surface by substituting the squared differentials (dT)2 and (dW)2

 

 

into the five-dimensional metric, which gives equation (1). The accelerating expansion of the space for a positive cosmological constant can be regarded as a consequence of a universal repulsive force. The radius of the spatial sphere follows a hyperbolic trajectory similar to the worldlines of constant proper acceleration discussed in Section 2.9. To show that the expansion of the de Sitter spacetime can be seen as exponential, we can put the metric into the “Robertson-Walker form” (see Section 7.1) by defining a new system of coordinates  such that

 

 

where

 

It follows that

 

where

 

Substituting into the metric (1) gives the exponential form

 

 

This the characteristic length R(t) for this metric is the simple exponential function. (This form of the metric covers only part of the manifold.) Equations (1), (2), and (3) are the most common ways of expressing de Sitter’s metric, but in the first letter that de Sitter wrote to Einstein on this subject he didn’t give the line element in any of these familiar forms. We can derive his original formulation beginning with (1) if we define new coordinates  related to the r,t coordinates of (1) by

 

 

Incidentally, the t coordinate is the “relativistic difference” between the advanced and retarded combinations of the barred coordinates, i.e.,

 

 

The differentials in (1) can be expressed in terms of the barred coordinates as

 

 

where the partials are

 

 

and

 

 

Making these substitutions and simplifying, we get the “Cartesian” form of the metric that de Sitter presented in his first letter to Einstein

 

 

where dW denotes the angular components, which are unchanged from (1). These expressions have some purely mathematical features of interest. For example, the line element is formally similar to the expressions for curvature discussed in Section 5.3. Also, the denominators of the partials of t are, according to Heron’s formula, equal to 16A2 where A is the area of a triangle with edge lengths .

 

If the cosmological constant was zero (meaning that L was infinite) all the dynamic solutions of the field equations with matter predict a slowing rate of expansion, but in 1998 two independent groups of astronomers reported evidence that the expansion of the universe is actually accelerating. If these findings are correct, then some sort of repulsive force is needed in models based on general relativity. This has led to renewed interest in the cosmological constant and de Sitter spacetime, which is sometimes denoted as dS4. If the cosmological constant is negative the resulting spacetime manifold is called anti-de Sitter spacetime, denoted by AdS4. In the latter case, we still get a hyperboloid, but the time coordinate advances circumferentially around the surface. To avoid closed time-like curves, we can simply imagine “wrapping” sheets around the hyperboloid.

 

As discussed in Section 7.1, the characteristic length R(t) of a manifold (i.e., the time-dependent coefficient of the spatial part of the manifold) satisfying the modified Einstein field equations (with non-zero cosmological constant) varies as a function of time in accord with the Friedmann equation

 

 

where dots signify derivatives with respect to a suitable time coordinate, C is a constant, and k is the curvature index, equal to either -1, 0, or +1. The terms on the right hand side are akin to potentials, and it’s interesting to note that the first two terms correspond to the two hypothetical forms of gravitation highlighted by Newton in the Principia. (See Section 8.2 for more on this.) As explained in Section 7.1, the Friedmann equation implies that R satisfies the equation

 

 

which shows that, if l = 0, the characteristic cosmological length R is a solution of the “separation equation” for non-rotating gravitationally governed distances, as given by equation (2) of Section 4.2. Comparing the more general gravitational separation from Section 4.2 with the general cosmological separation, we have

 

 

which again highlights the inverse square and the direct proportionalities that caught Newton’s attention. It’s interesting that with m = 0 the left-hand expression reduces to the purely inertial separation equation, whereas with l = 0 the right hand expression reduces to the (non-rotating) gravitational separation equation. We saw that the “homogeneous” forms of these equations are just special cases of the more general relation

 

 

where subscripts denote derivatives with respect to a suitable time coordinate. Among the solutions of this equation, in addition to the general co-inertial separations, non-rotating gravitational separations, and rotating-sliding separations, are sinusoidal functions and exponential functions. Historically this led to the suspicion, long before the recent astronomical observations, that there might be a class of exponential cosmological distances in addition to the cycloidal and parabolic distances. In other words, there could be different classes of observable distances, some very small and oscillatory, some larger and slowing, and some – the largest of all – increasing at an accelerating rate. This is illustrated in the figure below.

 

 

Of course, according to all conventional metrical theories, including general relativity, the spatial relations between material objects (on any chosen temporal foliation) conform to a single three-dimensional manifold. Assuming homogeneity and isotropy, it follows that all the cosmological distances between objects are subject to the ordinary metrical relations such as the triangle inequality. This greatly restricts the observable distances. On the other hand, our assumption that the degrees of freedom are limited in this way is based on our experience with much smaller distances. We have no direct evidence that cosmological distances are subject to the same dependencies. As an example of how concepts based on limited experience can be misleading, recall how special relativity revealed that the metric of our local spacetime fails to satisfy the axioms of a metric, including the triangle inequality. The non-additivity of relative speeds was not anticipated based on human experience with low speeds. Likewise for three “co-linear” objects A,B,C, it’s conceivable that the distance AC is not the simple sum of the distances AB and BC. The feasibility of regarding separations (rather than particles) as the elementary objects of nature was discussed in Section 4.1.

 

One possible observational consequence of having distances of several different classes would be astronomical objects that are highly red-shifted and yet much closer to us than the standard Hubble model would imply based on their redshifts. (Of course, even if this view was correct, it might be the case that all the exponential separations have already passed out of view.) Another possible consequence would be that some observable distances would be increasing at an accelerating rate, whereas others of the same magnitude might be decelerating.

 

The above discussion shows that the idea of at least some cosmological separations increasing at an accelerating rate can (and did) arise from completely a priori considerations. Of course, as long as a single coherent expansion model is adequate to explain our observations, the standard GR models of a smooth manifold will remain viable. Less conventional notions such as those discussed above would only be called for only if we begin to see conflicting evidence, e.g., if some observations strongly indicate accelerating expansion while others strongly indicate decelerating expansion.

 

The cosmological constant is hardly ever discussed without mentioning that (according to Gamow) Einstein called it his “biggest blunder”, but the reasons for regarding this constant as a “blunder” are seldom discussed. Some have suggested that Einstein was annoyed at having missed the opportunity to predict the Hubble expansion, but in his own writings Einstein argued that “the introduction of [the cosmological constant] constitutes a complication of the theory, which seriously reduces its logical simplicity”. He also wrote “If there is no quasi-static world, then away with the cosmological term”, adding that it is “theoretically unsatisfactory anyway”. In modern usage the cosmological term is usually taken to characterize some feature of the vacuum state, and so it is a fore-runner of the extremely complicated vacua that are contemplated in the “string theory” research program. If Einstein considered the complication and loss of logical simplicity associated with a single constant to be theoretically unsatisfactory, he would presumably have been even more dissatisfied with the nearly infinite number of possible vacua contemplated in current string research. Oddly enough, the de Sitter and anti-de Sitter spacetimes play a prominent role in this research, especially in relation to the so-called AdS/CFT conjecture involving conformal field theory.

 

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