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7. Cosmology

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 quasi-static 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 is the cosmological constant. Before invoking the field equations we can consider the general form of a metric that is suitable for representing the large-scale 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) function with units of distance, and d is the total space differential. Recall that for a perfectly flat Euclidean space the differential line element is

where r2 = x2 + y2 + z2. 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. In other words, the Gaussian curvature of every two-dimensional tangent subspace has the same value at every point. Now suppose we embed a Euclidean three-dimensional space (x,y,z) in a four-dimensional space (w,x,y,z) whose metric is

where k is a fixed constant equal to either +1 or -1. If k = +1 the four-dimensional space is Euclidean, whereas if k = -1 it is pseudo-Euclidean (like the Minkowski metric). In either case the four-dimensional space is "flat", i.e., has zero Riemannian curvature. Now suppose we consider a three-dimensional subspace comprising a sphere (or pseudo-sphere), i.e., the locus of points satisfying the condition

From this we have w2 = (1 r2)/k = k kr2, and therefore

Substituting this into the four-dimensional line element above gives the metric for the three-dimensional sphere (or pseudo-sphere)

Taking this as the spatial part of our overall spacetime metric (2) that satisfies the Cosmological Principle, 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 re-discovered by Robertson and Walker

(independently) in 1935, so it is now often referred to as the Robertson-Walker 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 + 3kr2), 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 Robertson-Walker metric has the form

As noted above, Einstein originally assumed R(t) = constant, i.e., he envisioned a static un-changing universe. He also assumed the matter in the universe was roughly "stationary" at each point with respect to these cosmological coordinates, so the only non-zero component of the stress-energy tensor in these coordinates is Ttt = where 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 non-zero cosmological constant , because if we assume a static universe and the Cosmological Principle, the field equations of general relativity can only be satisfied if the density 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 = 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 he 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 Robertson-Walker 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 non-zero cosmological constant required a non-zero density of matter. Shortly after Einstein's paper appeared, de Sitter gave a counter-example (see Section 7.6), i.e., he described a model universe that had a non-zero 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 de Sitter's idea of a dynamic universe, and 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 Robertson-Walker 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 Robertson-Walker metric (3), we can compute the components of the Ricci tensor and scalar and substitute these along with the simple uniform stress-energy 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 left-hand side is equal to times the left-hand side of the second condition, which equals zero, so the right hand side must also vanish, i.e., the derivative of (8/3)GR(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 (8/3)G 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 = 0 this is identical to the gravitational separation equation (2) 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 = 0) are formally identical to the gravitational free-fall solutions described in Section 4.3. Solving equation (4) (with = 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 = 0, the expansion of the universe is slowing down, albeit only slightly for the case k = -1. However, if we allow a non-zero cosmological constant , 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 with enough precision, we could decide on empirical grounds the sign of k, and whether or not is zero. Thus, assuming the maximal topologies (and the large-scale validity of general

relativity), we could determine whether the universe is open or closed, and whether it will expand forever or eventually re-contract. Unfortunately, none of the parameters is known with enough precision to distinguish between these possibilities. One source of uncertainty is in our estimates of the mass density 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 non-zero cosmological constant (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 quasi-Euclidean 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 quasi-Euclidean, 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 initial-value equations if they are to yield a well-determined and unique 4-geometry." 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 solution of the field equations are those with

where R is the radius of the universe, is the mean density of matter, and is the gravitational constant. This much is consistent with modern treatments, which agree that at any given epoch in a Friedmann universe with constant non-negative 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 quasi-Euclidean, and its radius of curvature therefore infinite, then 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 quasi-Euclidean.

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 "quasi-Euclidean", 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 possess 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 is 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 (8/3)G is greater than H2, 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 (8/3)G exceeds H2, whereas if our current estimate of is less than this threshold, it will always be possible that there is still more "missing matter" that we have not yet identified. Of course, 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, thereby automatically proving that the universe is open. 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 energy. In such a universe the "background" inertial field can in no way be associated with the matter and energy content of the universe. From this standpoint, general relativity can never gives 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 mass-energy. It is determined primarily by whatever absolute boundary conditions we choose to impose, independent of the distribution of mass-energy. Einstein believed that such boundary conditions were inherently non-relativistic, 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, this explicitly singles out one particular frame of reference as the absolute inertial frame of the universe, which, as Einstein said, "is contrary to the spirit of the relativity principle". This 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 mass-energy 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 mass-energy, 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 mass-energy 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 non-linear) 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 well-determined (and, we now know, a unique) 4-geometry". In view of this, we might propose the somewhat outlandish argument that the (apparent) uniqueness of 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. 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". 7.2 The Formation and Growth of Black Holes

It is a light thing for the shadow to go down ten degrees: nay, but let the shadow return backward ten degrees. 2 Kings 20

One of the most common questions about black holes is how they can exist if it takes infinitely long (from the perspective of an outside observer) for anything to reach the event horizon. The usual response is to explain that although the Schwarzschild coordinates are ill-behaved at the event horizon, the intrinsic structure of spacetime itself is well-behaved in that region, and an infalling object passes through the event horizon in finite proper time of the object. This is certainly an accurate description of the

Schwarzschild structure (as discussed in Section 6.4), but it doesn't fully address the question, which can be summarized in terms of the following two seemingly contradictory facts:

(1) An event horizon can grow in finite coordinate time only if the masscontained inside the horizon increases in finite coordinate time.

(2) According to the Schwarzschild metric, nothing crosses the event horizon in finite coordinate time.

Item (1) is a consequence of the fact that, as in Newtonian gravity, the field contributed by a (static) spherical shell on its interior is zero, so an event horizon can't be expanded by accumulating mass on its exterior. Nevertheless, if mass accumulates near the exterior of a black hole's event horizon the gravitational radius of the combined system must eventually (in finite coordinate time) increase far enough to encompass the accumulated mass, leading unavoidably to the conclusion that matter from the outside must reach the interior in finite coordinate time, which seems to directly conflict with Item 2 (and certainly seems inconsistent with the "frozen star" interpretation). To resolve this apparent paradox requires a careful examination of the definition of a black hole, which leads directly to several interesting results, such as the fact that if two black holes merge, then their event horizons are contiguous, and have been so since they were formed. The matter content of a black hole is increased when it combines with another black hole, but in such a case we obviously aren't dealing with a simple "one-body problem", so the spherically symmetrical Schwarzschild solution is not applicable. Lacking an exact solution of the field equations for the two-body problem, we can at least get a qualitative idea of the process by examining the "trousers" topology shown below:

As we progress through the sequence of external time slices the first event horizon appears at A, then another appears at B, then at C, and then A and B merge together. The "surfaces" of the trousers represent future null infinity (I+) of the external region, consistent with the definition of black holes as regions of spacetime that are not in the causal past of future null infinity. (If the universe is closed, the "ceiling" from which these "stalactites" descend is at some finite height, and our future boundary is really just a single surface. In such a universe these protrusions of future infinity are not true "event horizons", making it difficult to give a precise definition of a black hole. In this

discussion we assume an infinite open universe.) The "interior" regions enclosed by these surfaces are, in a sense, beyond the infinite future of our region of spacetime. If we regard a small test object as a point particle with zero radius then it's actually a black hole too, and the process of "falling in" to a "macro" black hole would simply be the trousers operation of merging the two I+ surfaces together, just like the merging of two macro black holes. On this basis the same interpretation would apply to the original formation of a macro black hole, by the coalescing of the I+ surfaces represented by the individual particles of the original collapsing star. Thus, we can completely avoid the "paradox" of black hole formation by considering all particles of matter to already be black holes. According to this view, it makes no sense to talk about the "interior" of a black hole, any more than it makes sense to talk about what's "outside" the universe, because the surface of a black hole is a boundary (future null infinity) of the universe. Unfortunately, it isn't at all clear that small particles of matter can be regarded as black holes surrounded by their own microscopic event horizons, so the "trousers" approach may not be directly applicable to the accumulation of small particles of "naked matter" (i.e., matter not surrounded by an event horizon). We'd like an explanation for the absorption of matter into a black hole that doesn't rely on this somewhat peculiar model of matter. To reconcile the Schwarszchild solution with the apparent paradox presented by items (1) and (2) above, it's worthwhile to recall from Chapter 6.4 what a radial freefall path really looks like in simple Schwarzschild geometry. We saw that the radial position of a test particle starting at radius r = 10m and t = 0 (for example) as a function of the particle’s proper time is a simple cycloid right down to r = 0, whereas if the same trajectory is described in terms of Schwarzschild coordinate time, the infalling object traverses through infinite coordinate time in order to reach the event horizon, and then traverses back through an infinite range of coordinate times until reaching r = 0 (in the interior) in a net coordinate time that is not too different from the elapsed proper time. In other words, the object goes infinitely far into the "future" (of coordinate time), and then infinitely far back to the "present" (also in coordinate time), and since these two segments must always occur together, we can "re-normalize" the round trip and just deal with the net change in coordinate time (for any radius other than precisely r = 2m). Admittedly it’s unorthodox to attribute any physical significance to the Schwarzschild time coordinate of a particle passing through the event horizon, and to some extent this is justified, because this coordinate is, after all, just a labeling of events. Moreover, as discussed in Section 6.4, the mapping of the Schwarzschild time coordinate from outside to inside the event horizon is formally arbitrary. Symmetrical drawings of the fully extended Kruskal solution, with a white hole as well as a black hole, tend to disguise this ambiguity, because they strongly suggest a symmetrical alignment of the time coordinates outside and inside the horizon, but this symmetry isn’t required by the equations. This interpretation is permitted by the equations, so it may still be worth considering. Also, the Schwarzschild time coordinate is physically significant in the sense that it is the unique time coordinate in terms of which the spherically symmetrical

solution is static, i.e., the metric coefficients are independent of time. (In other words, the time coordinate is a Killing vector field.) A more mundane objection to this interpretation is that a single infalling object occupies two different places (one inside and one outside the event horizon) at the same coordinate time. However, this shouldn't be surprising, because worldlines need not be single-valued in terms of arbitrary curvilinear coordinates. Still, it might seem that this "dual presence" opens the door to time-travel paradoxes. For example, we can observe the increase in the gravitational radius at some finite coordinate time, when the particle that caused the increase has still not yet crossed the event horizon (using the terms "when" and "not yet" in the sense of coordinate time), so it might seem that we have the opportunity to retrieve the particle before it crosses the horizon, thus preventing the increase that triggered our retrieval! However, if we carefully examine the path of the particle, both outside and inside the event horizon, we find that by the time it has gotten "back" close to our present coordinate time on the interior branch, the exterior branch is past the point of last communication. Even a photon could not catch up with it prior to crossing the horizon. The "backward" portion of the particle's trajectory through coordinate time inside the horizon ends just short of enabling any causality paradoxes. (It's apparent from these considerations that classical relativity must be a strictly deterministic theory - in which each worldline can be treated as already existing in its entirety - because we could construct genuine paradoxes in a non-deterministic theory.) It’s also worth noticing that the two strategies described above for explaining the formation and growth of black holes are essentially the same. In both cases the event horizon "reaches back" to us all the way from future null infinity. In a sense, that's why the infalling geodesics in Schwarzschild space go to infinity at the event horizon. To show the correspondence more clearly, we can turn the figure in Section 6.4 on end (so the coordinate time axis is vertical) and then redraw the constant-t lines as curves so as to accurately represent the absolute spacetime intervals. The result is shown below for a small infalling test particle:

Notice that the infalling worldline passes through all the Schwarzschild time slices t as it crosses the event horizon. Now suppose we take a longer view of this, beginning all the way back at the point of formation of the black hole, and suppose the infalling mass is significant relative to the original mass m. The result looks like this:

This shows how the stalactite reaches down from future infinity, and how the infalling mass passes through this infinity - but in finite proper time - to enter the interior of the black hole, and the event horizon expands accordingly. This figure is based on the actual spacetime intervals, and shows how the lines of constant Schwarzschild time t wrap around the exterior of the event horizon down to the point of formation, where they enter the interior of the black hole and "expand" back close to the region where they originated on the outside. One thing that sometimes concerns people when they look at a radial free-fall plot in Schwarzschild coordinates is related to the left hand side of the ballistic trajectory. Does the symmetry of the figure imply that we could launch a particle from r = 0, have it climb up to 5m, and then drop back down? No, because the light cones have tipped over at 2m, so the timelike and spacelike axes are reversed. Inside the event horizon the effective time axis points parallel to "r". As a result, although the left hand trajectory in the region above 2m is possible, the portion for r less than 2m is not; it's really just the time-reversed version of the right hand side. (We could also imagine a topology in which all inward and outward trajectories are realized (Kruskal space), but there is no known mechanism that would generate such a structure.) Still, it's valid to ask "how did we decide which way was forward in time inside the event horizon?" The only formal requirement seems to be that our choice be consistent for any given event horizon, always increasing r or always decreasing r. If we make one choice of sign convention we have a "white hole" spewing objects outward into our universe, whereas if we make the opposite choice we have a black hole, drawing things inward.

The question of whether we should expect to find as many white holes as black holes in the universe is still a subject of lively debate. In the forgoing reference was made to mass accumulating "near" the horizon, but we need to be careful about the concepts of nearness. The intended meaning in the above context was that the mass is (1) exterior to the event horizon, and (2) within a small increment r of the horizon, where r is the radial Schwarzschild coordinate. I've also assumed spherical symmetry so that the Schwarzschild solution and Birkhoff's uniqueness proof apply (meaning that the spacetime in the interior of an empty spherically symmetrical shell is necessarily flat). Of course, in terms of the spacelike surfaces of simultaneity of an external particle, the event horizon is always infinitely far away, or, more accurately, the horizon doesn't intersect with any external spacelike surface, with the exception of the single degenerate time&space-like surface precisely at 2m, where the external time and space surfaces close on each other like scissors (and then swap roles in the interior). So in terms of these coordinates the particle is infinitely far from the horizon right up to the instant it crosses the horizon! And this is the same "instant" that every other infalling object crosses the horizon, although separated by great "distances". (This isn't really so strange. Midnight tonight is infinitely far from us in this same sense, because it is no finite spatial distance away, and it will remain so until the instant we reach it. Likewise the event horizon is ahead of us in time, not in space.) Incidentally, I should probably qualify my dismissal of the "frozen star" interpretation, because there's a sense in which it's valid, or at least defensible. Remember that historically the two most common conceptual models for general relativity have been the "geometric interpretation" (as exemplified by Misner/Thorne/Wheeler's "Gravitation") and the "field interpretation" (as in Weinberg's "Gravitation and Cosmology"). These two views are operationally equivalent outside event horizons, but they tend to lead to different conceptions of the limit of gravitational collapse. According to the field interpretation, a clock runs increasingly slowly as it approaches the event horizon (due to the strength of the field), and the natural "limit" of this process is that the clock just asymptotically approaches "full stop" (i.e., running at a rate of zero) as it approaches the horizon. It continues to exist for the rest of time, but it's "frozen" due to the strength of the gravitational field. Within this conceptual framework there's nothing more to be said about the clock's existence. This leads to the "frozen star" conception of gravitational collapse. In contrast, according to the geometric interpretation, all clocks run at the same rate, measuring out real distances along worldlines in spacetime. This leads us to think that, rather than slowing down as it approaches the event horizon, the clock is following a shorter and shorter path to the future. In fact, the path gets shorter at such a rate that it actually reaches (our) future infinity in finite proper time. Now what? If we believe the clock is still running just like every other clock (and there's no local pathology of the spacetime) then it seems natural to extrapolate the clock's existence right past our future infinity and into another region of spacetime. Obviously this implies that the universe has a "transfinite topology", which some people find troubling, but there's nothing logically

contradictory about it (assuming the notion of an infinite continuous universe is not itself logically contradictory). In both of these interpretations we find that an object goes to future infinity (of coordinate time) as it approaches an event horizon, and its rate of proper time as a function of coordinate time goes to zero. The difference is that the field interpretation is content to truncate its description at the event horizon, while the geometric interpretation carries on with its description right through the event horizon and down to r = 0 (where it too finally gives up). What, if anything, is gained by extrapolating the worldlines of infalling objects through the event horizon? One obvious gain is that it offers a prediction of what would be experienced by an infalling observer. Since this represents a worldline that we could, in principle, follow, and since the formulas of relativity continue to make coherent predictions along those worldlines, there doesn't seem to be any compelling reason to truncate our considerations at the horizon. After all, if we limit our view of the universe to just the worldlines we have followed, or that we intend to follow, we end up with a very oddly shaped universe. On the other hand, the "frozen star" interpretation does have the advantage of simplifying the topology, i.e., it allows us to exclude event horizons separating transfinite regions of spacetime. More importantly, by declining to consider the fate of infalling worldlines through the event horizon, we avoid dealing with the rather awkward issue of a genuine spacetime singularity at r = 0. Therefore, if the "frozen star" interpretation gave equivalent predictions for all externally observable phenomena, and was logically consistent, it would probably be the preferred view. The question is, does the concept of a "frozen star" satisfy those two conditions? We saw above that the idea of a frozen star as an empty region around which matter "bunches up" outside an event horizon isn't viable, because if nothing ever passes from the exterior to the interior of an event horizon (in finite coordinate time) we cannot accommodate infalling matter. Either the event horizon expands or it doesn't, and in either case we arrive at a contradiction unless the value of m inside the horizon increases, and does so in finite coordinate time. The "trousers topology" described previously is, in some ways, the best of both worlds, but it relies on a somewhat dubious model of material particles as micro singularities in spacetime. We've also seen how the analytical continuation of the external free-fall geodesics into the interior leads to an apparently self-consistent picture of black hole growth in finite coordinate time, and this picture turns out to be fairly isomorphic to the trousers model. (Whether it's isomorphic to the truth is another question.) It may be worthwhile to explicitly describe the situation. Consider a black hole of mass m. The event horizon has radius r = 2m in Schwarzschild coordinates. Now suppose a large concentric spherical dust cloud of total mass m surrounds the black hole is slowly pulled to within a shell of radius, say, 2.1m. The mass of the combined system is 2m, giving it a gravitational radius of r = 4m, and all the matter is now within r = 4m, so there must be, according to the unique spherically symmetrical solution of the field equations, an event horizon at r = 4m. Evidently the dust has somehow gotten inside the event horizon. We might think that although the event horizon has expanded to 4m, maybe the dust is being held "frozen" just outside the horizon at, say, 4.1m. But that can't be true because then

there would be only 1m of mass inside the 4m radius, and the horizon would collapse. Also, this would imply that any dust originally inside 4m must have been pushed outward, and there is no known mechanism for that to happen. One possible way around this would be for the density of matter to be limited (by some mechanism we don't understand) to just sub-critical. In other words, each spherical region of radius r would be limited to just less than r/2 mass. It might be interesting to figure out the mass density profile necessary to be just shy of having an event horizon at every radius r (possibly inverse square?), but the problem with this idea is that there just isn't any known force that would hold the matter in this configuration. By all the laws we know it would immediately collapse. Of course, it's easy to posit some kind of Pauli-like gravitational "exclusion principle" which would simply prohibit two particles of matter from occupying the same "gravitational state". After all, it's the electron and nucleon exclusion principles that yield the white dwarf and neutron star configurations, respectively. The only reason we end up with black holes is because the universe seems to be one exclusion principle short. Thus, barring any "new physics", there is nothing to prevent an event horizon from forming and expanding, and this implies that the value of m inside the horizon increases in finite coordinate time, which conflicts with the "frozen star" interpretation. The preceding discussion makes clear the fact that general relativity is not a relational theory. Schwarzschild spacetime represents a cosmology with a definite preferred frame of reference, the one associated with the time-independent metric components. (Einstein at first was quite disappointed when he learned that the field equations have such an explicitly non-Machian solution, i.e., a single mass in an otherwise empty infinite universe). Of course, we introduced the preferred frame ourselves by imposing spherical symmetry in the first place, but it's always necessary to impose some boundary or initial value conditions, and these conditions (in an open infinite universe) unavoidably single out a particular frame of reference (as discussed further in Section 7.7). That troubled Einstein greatly, and was his main reason for arguing that the universe must be closed, because only in that context can we claim that the entire metric is in some sense fully determined by the distribution of mass-energy. However, there is no precise definition of a black hole in a closed universe, so for the purposes of this discussion we're committed to a cosmology with an arbitrarily preferred frame. To visualize how this preferred frame effectively governs the physics in Schwarzschild space, consider the following schematic of a black hole:

The star collapsed at point "a", and formed an event horizon of radius 2m in Schwarzschild coordinates. How far is the observer at "O" from the event horizon? If we trace along the spacelike surface "t = now" we find that the black hole doesn't exist at time t = now, which is to say, it is nowhere on the t = now timeslice. The event horizon is in the future of every external timeslice, all the way to future infinity. In fact, the event horizon is part of future null infinity. Nevertheless, the black hole clearly affects the physics on the timeslice t = now. For example, if the "observer" at O looks toward the "nearby star", his view will be obstructed, i.e., the star will be eclipsed, because the observer is effectively in the shadow of the infinite future. The size of this shadow will increase as the size of the event horizon increases. Thus we can derive knowledge of a black hole from the shadow it casts (like an eclipse), noting that the outline of a shadow isn't subject to speed-of-light restrictions, so there's nothing contradictory about being able to detect the presence and growth of a black hole region in finite coordinate time. Moreover, if the observer is allowed to fall freely, he will go mostly leftward (and slightly up) toward r = 0, quickly carrying him through all future timeslices (which are infinitely compressed around the event horizon) and into the interior. In doing so, he causes the event horizon to expand slightly.

7.3 Falling Into and Hovering Near A Black Hole

Unless the giddy heaven fall,And earth some new convulsion tear,And, us to join, the world should allBe cramped into a planisphere. As lines so loves oblique may wellThemselves in every angle greet;

But ours, so truly parallel,Though infinite, can never meet. Therefore the love which us doth bind,But Fate so enviously debars,Is the conjunction of the mind,And opposition of the stars.

Andrew Marvell (1621-1678) The empirical evidence for the existence of black holes – or at least something very much like them has become impressive, although it is arguably still largely circumstantial. Indeed, most relativity experts, while expressing high confidence (bordering on certainty) in the existence of black holes, nevertheless concede that since any electromagnetic signal reaching us must necessarily have originated outside any putative black holes, it may always be possible to imagine that they were produced by some mechanism just short of a black hole. Hence we may never acquire, by electromagnetic signals, definitive proof of the existence of black holes – other than by falling into one. (It’s conceivable that gravitational waves might provide some conclusive external evidence, but no such waves have yet been detected.) Of course, there are undoubtedly bodies in the universe whose densities and gravitational intensities are extremely great, but it isn’t self-evident that general relativity remains valid in these extreme conditions. Ironically, considering that black holes have become one of the signature predictions of general relativity, the theory’s creator published arguments purporting to show that gravitational collapse of an object to within its Schwarzschild radius could not occur in nature. In a paper published in 1939, Einstein argued that if we consider progressively smaller and smaller stationary systems of particles revolving around each other under their mutual gravitational attraction, the particles would need to be moving at the speed of light before reaching the critical density. Similarly Karl Schwarzschild had computed the behavior of a hypothetical stationary star of uniform density, and found that the pressure must go to infinity as the star shrank toward the critical radius. In both cases the obvious conclusion is that there cannot be any stationary configurations of matter above the critical density. Some scholars have misinterpreted Einstein’s point, claiming that he was arguing against the existence of black holes within the context of general relativity. These scholars underestimate both Einstein’s intelligence and his radicalism. He could not have failed to understand that sub-light particles (or finite pressure in Schwarchild’s star) meant unstable collapse to a singular point of infinite density – at least if general relativity holds good. Indeed this was his point: general relativity must fail. Thus we are not surprised to find him writing in “The Meaning of Relativity”

For large densities of field and matter, the field equations and even the field variables which enter into them have no real significance. One may not therefore assume the validity of the equations for very high density of field and matter… The present relativistic theory of gravitation is based on a separation of the

concepts of “gravitational field” and of “matter”. It may be plausible that the theory is for this reason inadequate for very high density of matter…

These reservations were not considered to be warranted by other scientists at the time, and even less so today, but perhaps they can serve to remind us not to be too dogmatic about the validity of our theories of physics, especially when extrapolated to very extreme conditions that have never been (and may never be) closely examined. Furthermore, we should acknowledge that, even within the context of general relativity, the formal definition of a black hole may be impossible to satisfy. This is because, as discussed previously, a black hole is strictly defined as a region of spacetime that is not in the causal past of any point in the infinite future. Notice that this refers to the infinite future, because anything short of that could theoretically be circumvented by regions that are clearly not black holes. However, in some fairly plausible cosmological models the universe has no infinite future, because it re-collapses to a singularity in finite coordinate time. In such a universe (which, for all we know, could be our own), the boundary of any gravitationally collapsed region of spacetime would be contiguous with the boundary of the ultimate collapse, so it wouldn’t really be a separate black hole in the strict sense. As Wald says, "there appears to be no natural notion of a black hole in a closed Robertson-Walker universe which re-collapses to a final singularity", and further, "there seems to be no way to define a black hole in a closed universe, because it requires going to infinity, but there is no infinity in a closed universe." It’s interesting that this is essentially the same objection that is often raised by people when they first hear about black holes, i.e., they reason that if it takes infinite coordinate time for any object to cross an event horizon, and if the universe is going to collapse in a finite coordinate time, then it’s clear that nothing can possess the properties of a true black hole in such a universe. Thus, in some fairly plausible cosmological models it's not strictly possible for a true black hole to exist. On the other hand, it is possible to have an approximate notion of a black hole in some isolated region of a closed universe, but of course many of the interesting transfinite issues raised by true (perhaps a better name would be "ideal") back holes are not strictly applicable to an "approximate" black hole. Having said this, there is nothing to prevent us from considering an infinite open universe containing full-fledged black holes in all their transfinite glory. I use the word “transfinite” because ideal black holes involve singular boundaries at which the usual Schwarzschild coordinates for the external field of a gravitating body go to infinity - and back - as discussed in the previous section. There are actually two distinct kinds of "spacetime singularities" involved in an ideal black hole, one of which occurs at the center, r = 0, where the spacetime manifold actually does become unequivocally singular and the field equations are simply inapplicable (as if trying to divide a number by 0). It's unclear (to say the least) what this singularity actually means from a physical standpoint, but oddly enough the "other" kind of singularity involved in a black hole seems to shield us from having to face the breakdown of the field equations. This is because it seems (although it has not been proved) to be a characteristic of all realistic spacetime singularities in general relativity that they are invariably enclosed within an event

horizon, which is a peculiar kind of singularity that constitutes a one-way boundary between the interior and exterior of a black hole. This is certainly the case with the standard black hole geometries based on the Schwarzschild and Kerr solutions. The proposition that it is true for all singularities is sometimes called the Cosmic Censorship Conjecture. Whether or not this conjecture is true, it's a remarkable fact that at least some (if not all) of the singular solutions of Einstein's field equations automatically enclose the singularity inside an event horizon, an amazing natural contrivance that effectively shields the universe from direct two-way exposure to any regions in which the metric of spacetime breaks down. Perhaps because we don't really know what to make of the true singularity at r = 0, we tend to focus our attention on the behavior of physics near the event horizon, which, for a non-rotating black hole, resides at the radial location r = 2m, where the Schwarzschild coordinates become singular. Of course, a singularity in a coordinate system doesn't necessarily represent a pathology of the manifold. (Consider traveling due East at the North Pole). Nevertheless, the fact that no true black hole can exist in a finite universe shows that the coordinate singularity at r = 2m is not entirely inconsequential, because it does (or at least can) represent a unique boundary between fundamentally separate regions of spacetime, depending on the cosmology. To understand the nature of this boundary, it's useful to consider hovering near the event horizon of a black hole. The components of the curvature tensor at r = 2m are on the order of 1/m2, so the spacetime can theoretically be made arbitrarily "flat" (Lorentzian) at that radius by making m large enough. Thus, for an observer "hovering" at a value of r that exceeds 2m by some arbitrarily small fixed ratio, the downward acceleration required to resist the inward pull can be arbitrarily small for sufficiently large m. However, in order for the observer to be hovering close to 2m his frame must be tremendously "boosted" in the radial direction relative to an in-falling particle. This is best seen in terms of a spacetime diagram such as the one below, which show the future light cones of two events located on either side of a black hole's event horizon.

In this drawing r is the radial Schwarzschild coordinate and t' is an Eddington-Finkelstein mapping of the Schwarzschild time coordinate, i.e.,

The right-hand ray of the cone for the event located just inside the event horizon is tilted just slightly to the left of vertical, whereas the cone for the event just outside 2m is tilted just slightly to the right of vertical. The rate at which this "tilt" changes with r is what determines the curvature and acceleration, and for a sufficiently large black hole this rate can be made negligibly small. However, by making this rate small, we also make the outward ray more nearly "vertical" for a radial coordinate r that exceeds 2m by any given ratio greater than 1, which implies that the hovering observer's frame needs to be even more "boosted" relative to the local frame of an observer falling freely from infinity. The gravitational potential, which need not be changing very steeply at r = 2m, has nevertheless changed by a huge amount relative to infinity. We must be very deep in a potential hole in order for the light cones to be tilted that far, even though the rate at which the tilt has been increasing can be arbitrarily slow. This just means that for a super-massive black hole they started tilting at a great distance. As can be seen in the diagram, relative to the frame of a particle falling in from infinity, a hovering observer must be moving outward at near light velocity. Consequently his axial distances are tremendously contracted, to the extent that, if the value of r is normalized to his frame of reference, he is actually a great distance (perhaps even light-years) from the r = 2m boundary, even though he is just 1 inch above r = 2m in terms of the Schwarzschild coordinate r. Also, the closer he tries to hover, the more radial boost he needs to hold that value of r, and the more contracted his radial distances become. Thus he is living in a thinner and thinner shell of r, but from his own perspective there's a world of room. Assuming he brought enough rocket fuel to accelerate himself up to this "hovering frame" at that radius 2m + r (or actually to slow himself down to a hovering frame), he would thereafter just need to resist the local acceleration of gravity to maintain that frame of reference. Quantitatively, for an observer hovering at a small Schwarzschild distance r above the horizon of a black hole, the radial distance r' to the event horizon with respect to the observer's local coordinates would be

which approaches as r goes to zero. This shows that as the observer hovers closer to the horizon in terms of Schwarzschild coordinates, his "proper distance" remains relatively large until he is nearly at the horizon. Also, the derivative of r' with respect to

r in this range is , which goes to infinity as r goes to zero. (These relations pertain to a truly static observer, so they don’t apply when the observer is moving from one radial position to another, unless he moves sufficiently slowly.) Incidentally, it's amusing to note that if a hovering observer's radial distance contraction factor at r was 12m/r instead of the square root of that quantity, his scaled distance to the event horizon at a Schwarzschild distance of r would be r' = 2m + r. Thus when he is precisely at the event horizon his scaled distance from it would be 2m, and he wouldn’t achieve zero scaled distance from the event horizon until arriving at the origin r = 0 of the Schwarzschild coordinates. This may seem rather silly, but it’s actually quite similar to one of Einstein’s proposals for avoiding what he regarded as the unpleasant features of the Schwarzschild solution at r = 2m. He suggested replacing the radial coordinate r with

= , and noted that the Schwarzschild solution expressed in terms of this coordinate behaves regularly for all values of . Whether or not there is any merit in this approach, it clearly shows how easily we can “eliminate” poles and singularities simply by applying coordinates that have canceling zeros or by restricting the domain of the variables. However, we shouldn’t assume that every arbitrary system of coordinates has physical significance. What "acceleration of gravity" would a hovering observer feel locally near the event horizon of a black hole? In terms of the Schwarzschild coordinate r and the proper time of the particle, the path of a radially free-falling particle can be expressed parametrically in terms of the parameter by the equations

where R is the apogee of the path (i.e., the highest point, where the outward radial velocity is zero). These equations describe a cycloid, with = 0 at the top, and they are valid for any radius r down to 0. We can evaluate the second derivative of r with respect to as follows

At = 0 the path is tangent to the hovering worldline at radius R, and so the local gravitational acceleration in the neighborhood of a stationary observer at that radius equals m/R2, which implies that if R is approximately 2m the acceleration of gravity is about 1/(4m). Thus the acceleration of gravity in terms of the coordinates r and is finite at the event horizon, and can be made arbitrarily small by increasing m. However, this acceleration is expressed in terms of the Schwarzschild radial parameter r, whereas the hovering observer’s radial distance r' must be scaled by the “gravitational

boost” factor, i.e., we have dr' = dr/(12m/r)1/2. Substituting this expression for dr into the above formula gives the proper local acceleration of a stationary observer

This value of acceleration corresponds to the amount of rocket thrust an observer would need in order to hold position, and we see that it goes to infinity as r goes to 2m. Nevertheless, for any ratio r/(2m) greater than 1 we can still make this acceleration arbitrarily small by choosing a sufficiently large m. On the other hand, an enormous amount of effort would be required to accelerate the rocket into this hovering condition for values of r/(2m) very close to 1. This amount of “boost” effort cannot be made arbitrarily small, because it essentially amounts to accelerating (outwardly) the rocket to nearly the speed of light relative to the frame of a free-falling particle from infinity. Interestingly, as the preceding figure suggests, an outward going photon can hover precisely at the event horizon, since at that location the outward edge of the light cone is vertical. This may seem surprising at first, considering that the proper acceleration of gravity at that location is infinite. However, the proper acceleration of a photon is indeed infinite, since the edge of a light cone can be regarded as hyperbolic motion with acceleration “a” in the limit as “a” goes to infinity, as illustrated in the figure below.

Also, it remains true that for any fixed r above the horizon we can make the proper acceleration arbitrarily small by increasing m. To see this, note that if r = 2m + r for a sufficiently small increment r we have m/r ~ 1/2, and we can bring the other factor of r into the square root to give

Still, these formulas contain a slight "mixing of metaphors", because they refer to two different radial parameters (r' and r) with different scale factors. To remedy this, we can

define the locally scaled radial increment r' = as the hovering observer’s “proper” distance from the event horizon. Then, since r = r 2m, we have r'

and so r = . Substituting this into the formula for the proper local acceleration gives the proper acceleration of a stationary observer at a "proper distance" r' above the event horizon of a (non-rotating) object of mass m is given by

Notice that as (r'/M) becomes small the acceleration approaches -1/(2r'), which is the asymptotic proper acceleration at a small "proper distance" r' from the event horizon of a large black hole. Thus, for a given proper distance r' the proper acceleration can't be made arbitrarily small by increasing m. Conversely, for a given proper acceleration g our hovering observer can't be closer than 1/(2g) of proper distance, even as m goes to infinity. For example, the closest an observer can get to the event horizon of a super-massive black hole while experiencing no more than 1g proper acceleration is about half a light-year of proper distance. At the other extreme, if (r'/m) is very large, as it is in normal circumstances between gravitating bodies, then this acceleration approaches m/(r'2, which is just Newton's inverse-square law of gravity in geometrical units. We've seen that the amount of local acceleration that must be overcome to hover at a radial distance r increases to infinity at r = 2m, but this doesn't imply that the gravitational curvature of spacetime at that location becomes infinite. The components of the curvature tensor depend to some extent on the choice of coordinate systems, so we can't simply examine the components of R to ascertain whether the intrinsic curvature is actually singular at the event horizon. For example, with respect to the Schwarzschild coordinates the non-zero components of the covariant curvature tensor are

along with the components related to these by symmetry. The two components relating the radial coordinate to the spherical surface coordinates are singular at r = 2m, but this is again related to the fact that the Schwarzschild coordinates are not well-behaved on this manifold near the event horizon. A more suitable system of coordinates in this region (as noted by Misner, et al) is constructed from the basis vectors

where = . With respect to this "hovering" orthonormal system of coordinates the non-zero components of the curvature tensor (up to symmetry) are

Of course, it isn’t possible to hover precisely at (or inside) the event horizon, but remarkably, if we transform to the orthonormal coordinates of a free-falling particle, the curvature components remain unchanged. Plugging in r = 2m, we see that these components are all proportional to 1/m2 at the event horizon, so the intrinsic spacetime curvature at r = 2m is finite. Indeed, for a sufficiently large mass m the curvature can be made arbitrarily mild at the event horizon. If we imagine the light cone at a radial coordinate r extremely close to the horizon (i.e., such that r/(2m) is just slightly greater than 1), with its outermost ray pointing just slightly in the positive r direction, we could theoretically boost ourselves at that point so as to maintain a constant radial distance r, and thereafter maintain that position with very little additional acceleration (for sufficiently large m). But, as noted above, the work that must be expended to achieve this hovering condition from infinity cannot be made arbitrarily small, since it requires us to accelerate to nearly the speed of light. Having discussed the prospects for hovering near a black hole, let's review the process by which an object may actually fall through an event horizon. If we program a space probe to fall freely until reaching some randomly selected point outside the horizon and then accelerate back out along a symmetrical outward path, there is no finite limit on how far into the future the probe might return. This sometimes strikes people as paradoxical, because it implies that the in-falling probe must, in some sense, pass through all of external time before crossing the horizon, and in fact it does, if by "time" we mean the extrapolated surfaces of simultaneity for an external observer. However, those surfaces are not well-behaved in the vicinity of a black hole. It's helpful to look at a drawing like this:

This illustrates schematically how the analytically continued surfaces of simultaneity for external observers are arranged outside the event horizon of a black hole, and how the in-falling object's worldline crosses (intersects with) every timeslice of the outside world prior to entering a region beyond the last outside timeslice. The dotted timeslices can be modeled crudely as simple "right" hyperbolic branches of the form tj T = 1/R. We just repeat this same -y = 1/x shape, shifted vertically, up to infinity. Notice that all of these infinitely many time slices curve down and approach the same asymptote on the left. To get to the "last timeslice" an object must go infinitely far in the vertical direction, but only finitely far in the horizontal (leftward) direction. The key point is that if an object goes to the left, it crosses every single one of the analytically continued timeslice of the outside observers, all the way to their future infinity. Hence those distant observers can always regard the object as not quite reaching the event horizon (the vertical boundary on the left side of this schematic). At any one of those slices the object could, in principle, reverse course and climb back out to the outside observers, which it would reach some time between now and future infinity. However, this doesn't mean that the object can never cross the event horizon (assuming it doesn't bail out). It simply means that its worldline is present in every one of the outside timeslices. In the direction it is traveling, those time slices are compressed infinitely close together, so the in-falling object can get through them all in finite proper time (i.e., its own local time along the worldline falling to the left in the above schematic). Notice that the temporal interval between two definite events can range from zero to infinity, depending on whose time slices we are counting. One observer's time is another observer's space, and vice versa. It might seem as if this degenerates into chaos, with no absolute measure for things, but fortunately there is an absolute measure. It's the absolute invariant spacetime interval "ds" between any two neighboring events, and the absolute distance along any specified path in spacetime is just found by summing up all the "ds" increments along that path. For any given observer, a local absolute increment ds can be projected onto his proper time axis and local surface of simultaneity, and these projections can be called dt, dx, dy, and dz. For a sufficiently small region around the

observer these components are related to the absolute increment ds by the Minkowski or some other flat metric, but in the presence of curvature we cannot unambiguously project the components of extended intervals. The only unambiguous way of characterizing extended intervals (paths) is by summing the incremental absolute intervals along a given path. An observer obviously has a great deal of freedom in deciding how to classify the locations of putative events relative to himself. One way (the conventional way) is in terms of his own time-slices and spatial distances as measured on those time slices, which works fairly well in regions where spacetime is flat, although even in flat spacetime it's possible for two observers to disagree on the lengths of objects and the spatial and temporal distances between events, because their reference frames may be different. However, they will always agree on the ds between two events. The same is true of the integrated absolute interval along any path in curved spacetime. The dt,dx,dy,dz components can do all sorts of strange things, but observers will always agree on ds. This suggests that rather than trying to map the universe with a "grid" composed of time slices and spatial distances on those slices, an observer might be better off using a sort of "polar" coordinate system, with himself at the center, and with outgoing geodesic rays in all directions and at all speeds. Then for each of those rays he measures the total ds between himself and whatever is "out there". This way of "locating" things could be parameterized in terms of the coordinate system [, , , s] where and are just ordinary latitude and longitude angles to determine a direction in space, is the velocity of the outgoing ray (divided by c), and s is the integrated ds distance along that ray as it emanates out from the origin to the specified point along a geodesic path. (Incidentally, these are essentially the coordinates Riemann used in his 1854 thesis on differential geometry.) For any event in spacetime the observer can now assign it a location based on this system of coordinates. If the universe is open, he will find that there are things which are only a finite absolute distance from him, and yet are not on any of his analytically continued time slices! This is because there are regions of spacetime where his time slices never go, specifically, inside the event horizon of a black hole. This just illustrates that an external observer's time slices aren't a very suitable set of surfaces with which to map events near a black hole, let alone inside a black hole. For this reason it's best to measure things in terms of absolute invariant distances rather than time slices, because time slices can do all sorts of strange things and don't necessarily cover the entire universe, assuming an open universe. Why did I specify an open universe? The schematic above depicted an open universe, with infinitely many external time slices, but if the universe is closed and finite, there are only finitely many external time slices, and they eventually tip over and converge on a common singularity, as shown below

In this context the sequence of tj slices eventually does include the vertical slices. Thus, in a closed universe an external observer's time slices do cover the entire universe, which is why there really is no true event horizon in a closed universe. An observer could use his analytically continued time slices to map all events if he wished, although they would still make an extremely somewhat ill-conditioned system of coordinates near an approximate black hole. One common question is whether a man falling (feet first) through an even horizon of a black hole would see his feet pass through the event horizon below him. As should be apparent from the schematics above, this kind of question is based on a misunderstanding. Everything that falls into a black hole falls in at the same local time, although spatially separated, just as everything in your city is going to enter tomorrow at the same time. We generally have no trouble seeing our feet as we pass through midnight tonight, although it is difficult one minute before midnight trying to look ahead and see your feet one minute after midnight. Of course, for a small black hole you will have to contend with tidal forces that may induce more spatial separation between your head and feet than you'd like, but for a sufficiently large black hole you should be able to maintain reasonable point-to-point co-moving distances between the various parts of your body as you cross the horizon. On the other hand, we should be careful not to understate the physical significance of the event horizon, which some authors have a tendency to do, perhaps in reaction to earlier over-estimates of its significance. Section 6.4 includes a description of a sense in which spacetime actually is singular at r = 2m, even in terms of the proper time of an in-falling particle, but it turns out to be what mathematicians call a "removable singularity", much like the point x = 0 on the function sin(x)/x. Strictly speaking this "curve" is undefined at that point, but by analytic continuation we can "put the point back in", essentially by just defining sin(x)/x to be 1 at x = 0. Whether nature necessarily adheres to analytic continuation in such cases is an open question. Finally, we might ask what an observer would find if he followed a path that leads across an event horizon and into a black hole. In truth, no one really knows how seriously to

take the theoretical solutions of Einstein's field equations for the interior of a black hole, even assuming an open infinite universe. For example, the "complete" Schwarzschild solution actually consists of two separate universes joined together at the black hole, but it isn't clear that this topology would spontaneously arise from the collapse of a star, or from any other known process, so many people doubt that this complete solution is actually realized. It's just one of many strange topologies that the field equations of general relativity would allow, but we aren't required to believe something exists just because it's a solution of the field equations. On the other hand, from a purely logical point of view, we can't rule them out, because there aren't any outright logical contradictions, just some interesting transfinite topologies.

7.4 Curled-Up Dimensions

I do not mind confessing that I personally have often found relief from the dreary infinities of homaloidal space in the consoling hope that, after all, this other may be the true state of things. William Kingdon Clifford, 1873

The simplest cylindrical space can be represented by the perimeter of a circle. This one-dimensional space with the coordinate X has the natural embedding in two-dimensional space with orthogonal coordinates (x1,x2) given by the circle formulas

From the derivatives dx1/dX = sin(X/R) and dx2/dX = cos(X/R) we have the Pythagorean identity (dx1)2 + (dx2)2 = (dX)2. The length of this cylindrical space is 2R. We can form the Cartesian product of n such cylindrical spaces, with radii R1, R2, ..,Rn respectively, to give an n-dimensional space that is cylindrical in all directions, with a total "volume" of

For example, a three-dimensional space that is everywhere locally Euclidean and yet cylindrical in all directions can be constructed by embedding the three spatial dimensions in a six-dimensional space according to the parameterization

so the spatial Euclidean line element is

giving a Euclidean spatial metric in a closed three-space with total volume (2)3R1R2R3. Subtracting from this an ordinary temporal component gives an everywhere-locally-Lorentzian spacetime that is cylindrical in the three spatial directions, i.e.,

However, this last step seems half-hearted. We can imagine a universe cylindrical in all directions, temporal as well as spatial, by embedding the entire four-dimensional spacetime in a manifold of eight dimensions, two of which are purely imaginary, as follows:

This leads again to the locally Lorentzian four-dimensional metric (1), but now all four of the dimensions X,Y,Z,T are periodic. So here we have an everywhere-locally-Lorentzian manifold that is closed and unbounded in every spatial and temporal direction. Obviously this manifold contains closed time-like worldlines, although they circumnavigate the entire universe. Whether such a universe would appear (locally) to possess a directional causal structure is unclear. We might imagine that a flat, closed, unbounded universe of this type would tend to collapse if it contained any matter, unless a non-zero cosmological constant is assumed. However, it's not clear what "collapse" would mean in this context. For example, it might mean that the Rn parameters would shrink, but they are not strictly dynamical parameters of the model. The four-dimensional field equations of general relativity operate only on X,Y,Z,T, so we have no context within which the Rn parameters could "evolve". Any "change" in Rn would imply some meta-time parameter , so that all the Rn coefficients in the embedding formulas would actually be functions Rn(). Interestingly, the local flatness of the cylindrical four-dimensional spacetime is independent of the value of R(), so if our "internal" field equations are satisfied for one set of Rn values they would be satisfied for any other values. The meta-time and associated meta-dynamics would be independent of the internal time T for a given observer unless we imagine some "meta field equations" relating to the internal parameters X,Y,Z,T. We might even speculate that these meta-equations would allow (require?) the values of Rn to be "increasing" versus , and therefore indirectly versus our internal time T = f(), in order to ensure stability. (One interesting question raised by these considerations locally flat n-dimensional spaces embedded in flat 2n-dimensional

spaces is whether every orthogonal basis in the n-space maps to an orthogonal basis in the 2n-space according to a set of formulas formally the same as those shown above, and, if not, whether there is a more general mapping that applies to all bases.) The above totally-cylindrical spacetime has a natural expression in terms of "octonion space", i.e., the Cayley algebra whose elements are two ordered quaternions

Thus each point (X,Y,Z,T) in four-dimensional spacetime represents two quaternions

To determine the absolute distances in this eight-dimensional manifold we again consider the eight coordinate differentials, exemplified by

(using the rule for total differentials) so the squared differentials are exemplified by

Adding up the eight squared differentials to give the square of the absolute differential interval leads again to the locally Lorentzian four-dimensional metric

Naturally it isn't necessary to imagine an embedding of our hypothesized closed dimensions in a higher-dimensional space, but it can be helpful for visualizing the structure. One of the first suggestions for closed cylindrical dimensions was made by Theodor Kaluza in 1919, in a paper communicated to the Prussian Academy by Einstein in 1921. The idea proposed by Kaluza was to generalize relativity from four to five dimensions. The introduction of the fifth dimension increases the number of components of the Riemann metric tensor, and it was hoped that some of this additional structure would represent the electromagnetic field on an equal footing with the gravitational field on the "left side" of Einstein's field equations, instead of being lumped into the stress-energy tensor T. Kaluza showed that, at least in the weak field limit for low velocities, we can arrange for a five dimensional manifold with one cylindrical dimension such that geodesic paths correspond to the paths of charged particles under the combined influence

of gravitational and electromagnetic fields. In 1926 Oskar Klein proved that the result was valid even without the restriction to weak fields and low velocities. The fifth dimension seems to have been mainly a mathematical device for Kaluza, with little physical significance, but subsequent researchers have sought to treat it as a real physical dimension, and more recent "grand unification theories" have postulated field theories in various numbers of dimensions greater than four (though none with fewer than four, so far as I know). In addition to increasing the amount of mathematical structure, which might enable the incorporation of the electromagnetic and other fields, many researchers (including Einstein and Bergmann in the 1930's) hoped the indeterminacy of quantum phenomena might be simply the result of describing a five-dimensional world in terms of four-dimensional laws. Perhaps by re-writing the laws in the full five dimensions quantum mechanics could, after all, be explained by a field theory. Alas, as Bergmann later noted, "it appears these high hopes were unjustified". Nevertheless, theorists ever since have freely availed themselves of whatever number of dimensions seemed convenient in their efforts to devise a fundamental "theory of everything". In nearly all cases the extra dimensions are spatial and assumed to be closed with extremely small radii in terms of macroscopic scales, thus explaining why it appears that macroscopic objects exist in just three spatial dimensions. Oddly enough, it is seldom mentioned that we do, in fact, have six extrinsic relational degrees of freedom, consisting of the three open translational dimensions and the closed orientational dimensions, which can be parameterized (for example) by the Euler angles of a frame. Of course, these three dimensions are not individually cylindrical, nor do they commute, but at each point in three-dimensional space they constitute a closed three-dimensional manifold isomorphic to the group of rotations. It's also worth noting that while translational velocity in the open dimensions is purely relativistic, angular velocity in the closed dimensions is absolute, and there is no physical difficulty in discerning a state of absolute non-rotation. This is interesting because, even though a closed cylindrical space may be locally Lorentzian, it is globally absolute, in the sense that there is a globally distinguished state of motion with respect to which an inertial observer's natural surfaces of simultaneity are globally coherent. In any other state of motion the surfaces of simultaneity are helical in time, similar to the analytically continued systems of reference of observers at rest on the perimeter of a rotating disk. To illustrate, consider two possible worldlines of a single particle P in a one-dimensional cylindrical space as shown in the spacetime diagrams below.

The cylindrical topology of the space is represented by identifying the worldline AB with the worldline CD. Now, in the left-hand figure the particle P is stationary, and it emits pulses of light in both directions at event a. The rightward-going pulse passes through event c, which is the same as event b, and then it proceeds from b to d. Likewise the leftward-going pulse goes from a to b and then from c to d. Thus both pulses arrive back at the particle P simultaneously. However, if the particle P is in absolute motion as shown in the right-hand figure, the rightward light pulse goes from a to c and then from c’ to d2, whereas the leftward pulse goes from a to b and then from b’ to d1, so in this case the pulses do not arrive back at particle P simultaneously. The absolutely stationary worldlines in this cylindrical space are those for which the diverging-converging light cones remain coherent. (In the one-dimensional case there are discrete absolute speeds greater than zero for which the leftward and rightward pulses periodically re-converge on the particle P.) Of course, for a different mapping between the events on the line AB and the events on the line CD we would get a different state of rest. The worldlines of identifiable inertial entities establish the correct mapping. If we relinquish the identifiability of persistent entities through time, and under completed loops around the cylindrical dimension, then the mapping becomes ambiguous. For example, we assume particle P associates the pulses absorbed at event d with the pulses emitted at event a, although this association is not logically necessary.

7.5 Packing Universes In Spacetime

All experience is an arch wherethroughGleams that untraveled world whose margin fadesForever and forever when I move. Tennyson, 1842

One of the interesting aspects of the Minkowski metric is that every lightcone (in principle) contains infinitely many nearly-complete lightcones. Consider just a single

spatial dimension in which an infinite number of point particles are moving away from each other with mutual velocities as shown below:

Each particle finds itself mid-way between its two nearest neighbors, which are receding at nearly the speed of light, so that each particle can be regarded as the origin of a nearly-complete lightcone. On the other hand, all of these particles emanate from a single point, and the entire infinite set of points (and nearly-complete lightcones) resides within the future lightcone of that single point. More formally, a complete lightcone in a flat Lorentzian xt plane comprises the boundary of all points reachable from a given point P along world lines with speeds less than 1 relative to any and every inertial worldline through P. Also, relative to any specific inertial frame W we can define an "-complete lightcone" as the region reachable from P along world lines with speeds less than (1) relative to W, for some arbitrarily small > 0. A complete lightcone contains infinitely many epsilon-complete lightcones, as illustrated above by the infinite linear sequence of particles in space, each receding with a speed of (1) relative to its closest neighbors. Since we can never observe something infinitely red-shifted, it follows that our observable universe can fit inside an -complete lightcone just as well as in a truly complete lightcone. Thus a single lightcone in infinite flat Lorentzian spacetime encompasses infinitely many mutually exclusive -universes. If we arbitrarily select one of the particles as the "rest" particle P0, and number the other particles sequentially, we can evaluate the velocities of the other particles with respect to the inertial coordinates of P0, whose velocity is v0 = 0. If each particle has a mutual velocity u relative to each of its nearest neighbors, then obviously P1 has a speed v1 = u. The speed of P2 is u relative to P1, and its speed relative to P0 is given by the relativistic speed composition formula v2 = (v1 + u)/(uv1 + 1). In general, the speed of Pk can be computed recursively based on the speed of Pk-1 using the formula

This is just a linear fractional function, so we can use the method described in Section 2.6 to derive the explicit formula

Similarly, in full 3+1 dimensional spacetime we can consider packing -complete lightspheres inside a complete lightsphere. A flash of light at point P in flat Lorentzian

spacetime emanates outward in a spherical shell as viewed from any inertial worldline through P. We arbitrarily select one such worldline W0 as our frame of reference, and let the slices of simultaneity relative to this frame define a time parameter t. The points of the worldline W0 can be regarded as the stationary center of a 3D expanding sphere at each instant t. On any given time-slice t we can set up orthogonal space coordinates x,y,z relative to W0 and normalize the units so that the radius of the expanding lightsphere at time t equals 1. In these terms the boundary of the lightsphere is just the sphere

Now let W1 denote another inertial worldline through the point P with a velocity v = v1 relative to W0, and consider the region R1 surrounding W1 consisting of the points reachable from P with speeds not exceeding u = u1 relative to W1. The region R1 is spherical and centered on W1 relative to the frame of W1, but on any time-slice t (relative to W0) the region R1 has an ellipsoidal shape. If v is in the z direction then the cross-sectional boundary of R1 on the xy plane is given parametrically by

as ranges from 0 to 2. The entire boundary is just the surface of rotation of this ellipse about the z axis. If v1 has a magnitude of (1 ) for some arbitrarily small > 0, and if we set u1 = |v1|, then as goes to zero the boundary of the region R1 approaches the limiting ellipsoid

Similarly if W2 is an inertial worldline with speed |v2| = |v1| in the negative z direction relative to W0, then the boundary of the region R2 consisting of the points reachable from P with speeds not exceeding u2 = |v2| approaches the limiting ellipsoid

The regions R1 and R2 are mutually exclusive, meeting only at the point of contact [0,0,0]. Each of these regions can be called an "-complete" lightsphere. Interestingly, beginning with R1 and R2 we can construct a perfect tetrahedral packing of eight epsilon-complete lightspheres by placing six more spheres in a hexagonal ring about the z axis with centers in the xy plane, such that each sphere just touches R1 and R2 and its two adjacent neighbors in the ring. Each of these six spheres represents a region reachable from P with speeds less than u1 relative to one of six worldlines whose speeds

are (1 4) relative to W0. The normalized boundaries of these six ellipsoids on a time-slice t are given by

for k = 0,1,..,5. In the limit as epsilon goes to zero the hexagonal cluster of e-spheres touching any given e-sphere becomes vanishingly small with respect to the given sphere's frame of reference, so we approach the condition that this hexagonal pattern tessellates the entire surface of each e-sphere in a perfectly symmetrical tetrahedral packing of identical epsilon-complete lightspheres. A cross-sectional side-view and top-view of this configuration are shown below.

These considerations show that we can regard a single light cone as a cosmological model, taking advantage of the complete symmetry in Minkowski spacetime. Milne was the first to discuss this model in detail. He postulated a cloud of particles expanding in flat spacetime from a single event O, with a distribution of velocities such that the mutual velocities between neighboring particles was the same for every particle, just as in the one-dimensional case described at the beginning of this section. With respect to any particular system of inertial coordinates t,x,y,z whose origin is at the event O, the cloud of particles is spherically symmetrical with radially outward speed v = r/t. The density of the particles is also spherically symmetrical, but it is not isotropic. To determine the density with respect to the inertial coordinates t,x,y,z, we first consider the density in the radial direction at a point on the x axis at time t. If we let u denote the mutual speed between neighboring particles, then the speed vn of the nth particle away from the center is

where xn is the radial distance of the nth particle along the x axis. Solving for n gives

Differentiating with respect to x gives the density of particles in the x directions

This confirms that the one-dimensional density at the spatial origin drops in proportion to 1/t. Also, by symmetry, the densities in the transverse directions y and z at any point are

given by this same expression as a function of the proper time = t at that point

This shows that the densities in the transverse directions are less than in the radial

direction by a factor of . Neglecting the anisotropy, the number of particles in a volume element dxdydz at a radial distance r from the spatial origin at time t is proportional to

This distribution applies to every inertial system or coordinates with origin at O, so this cosmology looks the same, and is spherically symmetrical, with respect to the rest frame of each individual particle.

The above analysis was based on a foliation of spacetime into slices of constant-t for some particular system of inertial coordinates, but this is not the only possible foliation, nor even the most natural. From a cosmological standpoint we might adopt as our time coordinate at each point the proper time of uniform worldline extending from O to that point. This would give hyperboloid spacelike surfaces consisting of the locus of all the points with a fixed proper age from the origin event O. One of these spacelike slices is illustrated by the " = k" line in the figure below.

Rindler points out that if = k is the epoch at which the density of the expanding cloud drops low enough so that matter and thermal radiation decouple, we should expect at the present event "p" to be receiving an isotropic and highly red-shifted "background radiation" along the dotted lightlike line from that de-coupling surface as shown in the figure. As our present event p advances into the future we expect to see a progressively more red-shifted (i.e., lower temperature) background radiation. This simplistic model gives a surprisingly good representation of the 3K microwave radiation that is actually observed. It's also worth noting that if we adopt the hyperboloid foliation the universe of this expanding cloud is spatially infinite. We saw in Section 1.7 that the absolute radial distance along this surface from the spatial center to a point at r is

where r2 = x2 + y2 + z2 in terms of the inertial coordinates of the central spatial point. Furthermore, we can represent this hyperboloid spatial surface as existing over the flat

Euclidean xy plane with the elevation . By making the elevation imaginary, we capture the indefinite character of the surface. In the limit near the origin we can expand h to give

So, according to the terminology of Section 5.3, we have a surface tangent to the xy plane at the origin with elevation given by h = ax2 + bxy + cy2 where a = c = i/2 and b = 0. Consequently the Gaussian curvature of this spatial surface is K = 4ac b2 = -1/2. By symmetry the same analysis is applicable at every point on the surface, so this surface has constant negative curvature. This applies to any two-dimensional spatial tangent plane in the three-dimensional space at each point for constant . We can also evaluate the metric on this two-dimensional spacelike slice, by writing the total differential of h

Squaring this and adding the result to (dx)2 + (dy)2 gives the line element for this surface in terms of the tangent xy plane coordinates projected onto the surface

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 in Section 5.8, is zero. If we allow a non-zero value of , 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 must be extremely small, certainly less than 10-40 m-2, but not necessarily zero. If is precisely zero, then the Schwarzschild metric goes over to the Minkowski metric when the gravitating mass m equals zero, but if 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/. 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 R = 0, so the de Sitter metric is a solution of the modified field equations R = g. 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 defined by

Noting that dr/L = cos(/L)d, 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 d 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 = 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 = 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.

7.7 Boundaries and Symmetries

Whether Heaven move or Earth,Imports not, if thou reckon right. John Milton, 1667

Each point on the surface of an ordinary sphere is perfectly symmetrical with every other point, but there is no difficulty imagining the arbitrary (random) selection of a single point on the surface, because we can define a uniform probability density on this surface. However, if we begin with an infinite flat plane, where again each point is perfectly symmetrical with every other point, we face an inherent difficulty, because there does not exist a perfectly uniform probability density distribution over an infinite surface. Hence, if we select one particular point on this infinite flat plane, we can't claim, even in principle, to have chosen from a perfectly uniform distribution. Therefore, the original empty infinite flat plane was not perfectly symmetrical after all, at least not with respect to our selection of individual points. This shows that the very idea of selecting a point from a pre-existing perfectly symmetrical infinite manifold is, in a sense, self-contradictory. Similarly the symmetry of infinite Minkowski spacetime admits no distinguished position or frame of reference, but the introduction of an inertial particle not only destroys the symmetry, it also contradicts the premise that the points of the original manifold were perfectly symmetrical, because the non-existence of a uniform probability density distribution over the possible positions and velocities implies that the placement of the particle could not have been completely impartial. Even if we postulate a Milne cosmology (described in Section 7.5), with dust particles emanating from a single point at uniformly distributed velocities throughout the future null cone (note that this uniform distribution isn't normalized as a probability density, so it can't be use make a selection), we still arrive at a distinguished velocity frame at each point. We could retain perfect Minkowskian symmetry in the presence of matter only by postulating a "super-Milne" cosmology in which every point on some past spacelike slice is an equivalent source of infinitesimal dust particles emanating at all velocities distributed uniformly throughout the respective future null cones of every point. In such a cosmology this same condition would apply on every time-slice, but the density would be infinite, because each point is on the surface of infinitely many null cones, and we would have infinitely dense flow of particles in all directions at every point. Whether this could correspond to any intelligible arrangement of physical entities is unclear. The asymmetry due to the presence of an infinitesimal inertial particle in flat Minkowski spacetime is purely circumstantial, because the spacetime is considered to be unaffected by the presence of this particle. However, according to general relativity, the presence of any inertial entity disturbs the symmetry of the manifold even more profoundly, because it implies an intrinsic curvature of the spacetime manifold, i.e., the manifold takes on an intrinsic shape that distinguishes the location and rest frame of the particle. For a single non-rotating uncharged particle the resulting shape is Schwarzschild spacetime, which obviously exhibits a distinguished center and rest frame (the frame of the central mass). Indeed, this spacetime exhibits a preferred system of coordinates, namely those for which the metric coefficients are independent of the time coordinate.

Still, since the field variables of general relativity are the metric coefficients themselves, we are naturally encouraged to think that there is no a priori distinguished system of reference in the physical spacetime described by general relativity, and that it is only the contingent circumstance of a particular distribution of inertial entities that may distinguish any particular frame or state of motion. In other words, it's tempting to think that the spacetime manifold is determined solely by its "contents", i.e., that the left side of Guv = 8Tuv is determined by the right side. However, this is not actually the case (as Einstein and others realized early on), and to understand why, it's useful to review what is involved in actually solving the field equations of general relativity as an initial-value problem. The ten algebraically independent field equations represented by Guv = 8Tuv involve the values of the ten independent metric coefficients and their first and second derivatives with respect to four spacetime coordinates. If we're given the values of the metric coefficients throughout a 3D spacelike "slice" of spacetime at some particular value of the time coordinate, we can directly evaluate the first and second derivatives of these components with respect to the space coordinates in this "slice". This leaves only the first and second derivatives of the ten metric with respect to the time coordinate as unknown quantities in the ten field equations. It might seem that we could arbitrarily specify the first derivatives, and then solve the field equations for the second derivatives, enabling us to "integrate" forward in time to the next timeslice, and then repeat this process to predict the subsequent evolution of the metric field. However, the structure of the field equations does not permit this, because four of the ten field equations (namely, G0v = 8T0v with v = 0,1,2,3) contain only the first derivatives with respect to the time coordinate x0, so we can't arbitrarily specify the guv and their first derivatives with respect to x0 on a surface of constant x0. These ten first derivatives, alone, must satisfy the four G0v conditions on any such surface, so before we can even pose the initial value problem, we must first solve this subset of the field equations for a viable set of initial values. Although these four conditions constrain the initial values, they obviously don't fully determine them, even for a given distribution of Tuv. Once we've specified values of the guv and their first derivatives with respect to x0 on some surface of constant x0 in such a way that the four conditions for G0v are satisfied, the four contracted Bianchi identities ensure that these conditions remain satisfied outside the initial surface, provided only that the remaining six equations are satisfied everywhere. However, this leaves only six independent equations to govern the evolution of the ten field variables in the x0 direction. As a result, the second derivatives of the guv with respect to x0 appear to be underdetermined. In other words, given suitable initial conditions, we're left with a four-fold ambiguity. We must arbitrarily impose four more conditions on the system in order to uniquely determine a solution. This was to be expected, because the metric coefficients depend not only on the absolute shape of the manifold, but also on our choice of coordinate systems, which represents four degrees of freedom. Thus, the field equations actually determine an equivalence class of solutions, corresponding to all the ways in which a given absolute metrical manifold can be expressed in various coordinate systems. In order to actually generate a solution of the initial value problem, we need to impose four "coordinate conditions" along with the six

"dynamical" field equations. The conditions arise from any proposed system of coordinates by expressing the metric coefficients g0v in terms of these coordinates (which can always be done for any postulated system of coordinates), and then differentiating these four coefficients twice with respect to x0 to give four equations in the second derivatives of these coefficients. Notwithstanding the four-fold ambiguity of the dynamical field equations, which is just a descriptive rather than a substantive ambiguity, it's clear that the manifold is a definite absolute entity, and its overall characteristics and evolution are determined not only by the postulated Tuv and the field equations, but also by the conditions specified on the initial timeslice. As noted above, these conditions are constrained by the field equations, but are by no means fully determined. We are still required to impose largely arbitrary conditions in order to fix the absolute background spacetime. This state of affairs was disappointing to Einstein, because he recognized that the selection of a set of initial conditions is tantamount to stipulating a preferred class of reference systems, precisely as in Newtonian theory, which is "contrary to the spirit of the relativity principle" (referring presumably to the relational ideas of Mach). As an example, there are multiple distinct vacuum solutions of the field equations, some with gravitational waves and even geons (temporarily) zipping around, and some not. Even more ambiguity arises when we introduce mass, as Gödel showed with his cosmological solutions in which the average mass of the universe is rotating with respect to the spacetime background. These examples just highlight the fact that general relativity can no more dispense with the arbitrary stipulation of a preferred class of reference systems (the inertial systems) than could Newtonian mechanics or special relativity. This is clearly illustrated by Schwarzschild spacetime, which (according to Birkhoff's theorem) is the essentially unique spherically symmetrical solution of the field equations. Clearly this cosmological model, based on a single spherically symmetrical mass in an otherwise empty universe, is "contrary to the spirit of the relativity principle", because as noted earlier there is an essentially unique time coordinate for which the metric coefficients are independent of time. Translation along a vector that leaves the metric formally unchanged is called an isometry, and a complete vector field of isometries is called a Killing vector field. Thus the Schwarzschild time coordinate t constitutes a Killing vector field over the entire manifold, making it a highly distinguished time coordinate, no less than Newton's absolute time. In both special relativity and Newtonian physics there is an infinite class of operationally equivalent systems of reference at any point, but in Schwarzschild spacetime there is an essentially unique global coordinate system with respect to which the metric coefficients are independent of time, and this system is related in a definite way to the inertial class of reference systems at each point. Thus, in the context of this particular spacetime, we actually have a much stronger case for a meaningful notion of absolute rest than we do in Newtonian spacetime or special relativity, both of which rest naively on the principle of inertia, and neither of which acknowledges the possibility of variations in the properties of spacetime from place to place (let alone under velocity transformations).

The unique physical significance of the Schwarzschild time coordinate is also shown by the fact that Fermat's principle of least time applies uniquely to this time coordinate. To see this, consider the path of a light pulse traveling through the solar system, regarded as a Schwarzschild geometry centered around the Sun. Naturally there are many different parameterizations and time coordinates that we could apply to this geometry, and in general a timelike geodesic extremizes d (not dt for whatever arbitrary time coordinate t we might be using), and of course a spacelike geodesic extremizes ds (again, not dt). However, for light-like paths we have d = ds = 0 by definition, so the path is confined to null surfaces, but this is not sufficient to pick out which null path will be followed. So, starting with a line element of the form

where and represent the usual Schwarzschild coordinates, we then set d = 0 for light-like paths, which reduces the equation to

This is a perfectly good metrical (not pseudo-metrical) space, with a line element given by dt, and in fact by extremizing (dt)2 we get the paths of light. Note that this only works because gtt, grr , g, g all happen to be independent of this time coordinate, t, and also because gtr = gt = gt = 0. If and only if all these conditions apply, we reduce to a simple line element of dt on the null surfaces, and Fermat's Principle applies to the parameter t. Thus, in a Schwarzschild universe, this works only when using the essentially unique Schwarzschild coordinates, in which the metric coefficients are independent of the time coordinate. Admittedly the Schwarzschild geometry is a highly simplistic and symmetrical cosmology, but it illustrates how the notion of an absolute rest frame can be more physically meaningful in a relativistic spacetime than in Newtonian spacetime. The spatial configuration of Newton's absolute space is invariant and the Newtonian metric is independent of time, regardless of which member of the inertial class of reference systems we choose, whereas Schwarzschild spacetime is spherically symmetrical and its metric coefficients are independent of time only with respect to the essentially unique Schwarzschild system of coordinates. In other words, Newtonian spacetime is operationally symmetrical under translations and uniform velocities, whereas the spacetime of general relativity is not. The curves and dimples in relativistic spacetime automatically destroy symmetry under translation, let alone velocity. Even the spacetime of special relativity is (marginally) less relational (in the Machian sense) than Newtonian spacetime, because it combines space and time into a single manifold that is only partially ordered, whereas Newtonian spacetime is totally ordered into a continuous sequence of spatial instants. Noting that Newtonian spacetime is explicitly less relational than Galilean spacetime, it can be argued that the actual evolution of spacetime theories historically has been from the purely kinematically relational spacetime of Copernicus to

inertial relativity of Galileo and special relativity to the purely absolute spacetime of general relativity. At each stage the meaning of relativity has been refined and qualified. We might suspect that the distinguished "Killing-time" coordinate in the Schwarzschild cosmology is exceptional - in the sense that the manifold was designed to satisfies a very restrictive symmetry condition - and that perhaps more general spacetime manifolds do not exhibit any preferred directions or time coordinates. However, for any specific manifold we must apply some symmetry or boundary conditions sufficient to fix the metrical relations of the manifold, which unavoidably distinguishes one particular system of reference at any given point. For example, in the standard Friedmann models of the universe there is, at each point in the manifold, a frame of reference with respect to which the rest of the matter and energy in the universe has maximal spherical symmetry, which is certainly a distinguished system of reference. Still, we might imagine that these are just more exceptional cases, and that underneath all these specific examples of relativistic cosmologies that just happen to have strongly distinguished systems of reference there lies a purely relational theory. However, this is not the case. General relativity is not a relational theory of motion. The spacetime manifold in general relativity is an absolute entity, and it's clear that any solution of the field equations can only be based on the stipulation of sufficient constraints to uniquely determine the manifold, up to inertial equivalence, which is precisely the situation with regard to the Newtonian spacetime manifold. But isn't it possible for us to invoke general relativity with very generic boundary conditions that do not commit us to any distinguished frame of reference? What if we simply stipulate asymptotic flatness at infinity? This is typically the approach taken when modeling the solar system or some other actual configuration, i.e., we require that, with a suitable choice of coordinates, the metric tensor approaches the Minkowski metric at spatial infinity. However, as Einstein put it, the specifications of "these boundary conditions presuppose a definite choice of the system of reference". In other words, we must specify a suitable choice of coordinates in terms of which the metric tensor approaches the Minkowski metric, but this specification is tantamount to specifying the absolute spacetime (up to inertial equivalence, as always) in Newtonian physics. The well-known techniques for imposing asymptotic flatness at "conformal infinity", such as discussed by Wald, are not exceptions, because they place only very mild constraints on the field solution in the finite region of the manifold. Indeed, the explicit purpose of such constructions is to establish asymptotic flatness at infinity while otherwise constraining the solution as little as possible, to facilitate the study gravitational waves and other phenomena in the finite region of the manifold. These phenomena must still be "driven" by the imposition of conditions that inevitably distinguish a particular frame of reference at one or more points. Furthermore, to the extent that flatness at conformal infinity succeeds in imposing an absolute reference for gravitational "potential" and the total energy of an isolated system, it still represents an absolute background that has been artificially imposed.

Since the condition of flatness at infinity is not sufficient to determine a solution, we must typically impose other conditions. Obviously there are many physically distinct ways in which the metric could approach flatness as a function of radial spatial distance from a given region of interest, and one of the most natural-seeming and common approaches, consistent with local observation, is to assume a spherically symmetrical approach to spatial infinity. This tends to seem like a suitably frame-independent assumption, since spatial spherical symmetry is frame-independent in Newtonian physics. The problem, of course, is that in relativity the concept of spherical symmetry automatically distinguishes a particular frame of reference - not just a class of frames, but one particular frame. For example, if we choose a system of reference that is moving toward Sirius at 0.999999c, the entire distribution of stars and galaxies in the universe is drastically shrunk (spatially) along that direction, and if we define a spherically symmetrical asymptotic approach to flatness at spatial infinity in these coordinates we will get a physically different result (e.g., for solar system calculations) than if we define a spherically symmetrical asymptotic approach to flatness with respect to a system of coordinates in which the Sun is at rest. It's true that the choice of coordinate systems is arbitrary, but only until we impose physically meaningful conditions on the manifold in terms of those coordinates. Once we do that, our choice of coordinate systems acquires physical significance, because the physical meaning of the conditions we impose is determined largely by the coordinates in terms of which they are expressed, and these conditions physically influence the solution. Of course, we can in principle define any boundary conditions in conjunction with any set of coordinates, i.e., we could take the rest frame of a near-light-speed cosmic particle to work out orbital mechanics of our Solar system by (for example) specifying an asymptotic approach to flatness at spatial infinity in a highly elliptical pattern, but the fact remains that this approach give a uniquely spherical pattern only with respect to the Sun's rest frame. Whenever we pose a Cauchy initial-value problem, the very act of specifying timeslices (a spacelike foliation) and defining a set of physically recognizable conditions on one of these surfaces establishes a distinguished reference system at each point. These individual local frames need not be coherent, nor extendible, nor do we necessarily require them to possess specific isometries, but the fact remains that the general process of actually applying the field equations to an initial-value problem involves the stipulation of a preferred space-time decomposition at each point, since the tangent plane of the timeslice at each point singles out a local frame of reference, and we are assigning physically meaningful conditions to every point on this surface in terms that unavoidably distinguish this frame. More generally, whenever we apply the field equations in any particular situation, whether in the form of an initial-value problem or in some other form, we must always specify sufficient boundary conditions, initial conditions, and/or symmetries to uniquely determine the manifold, and in so doing we are positing an absolute spacetime just as surely (and just as arbitrarily) as Newton did. It's true that the field equations themselves would be compatible with a wide range of different absolute spacetimes, but this ambiguity, from a predictive standpoint, is a weakness rather than a strength of the theory, since, after all, we live in one definite universe, not infinitely many arbitrary ones.

Indeed, when taken as a meta-theory in this sense, general relativity does not even give unique predictions for things like the twins paradox, etc, unless the statement of the question includes the specification of the entire cosmological boundary conditions, in which case we're back to a specific absolute spacetime. It was this very realization that led Einstein at one point to the conviction that the universe must be regarded as spatially closed, to salvage at least a semblance of unique for the cosmological solution as a function of the mass energy distribution. (See Section 7.1.) However, the closed Friedmann models are not currently in favor among astronomers, and in any case the relational uniqueness that can be recovered in such a universe is more semantic than substantial. Moreover, the strategy of trying to obviate arbitrary boundary conditions by selecting a topology without boundaries generally results in a topologically distinguished system of reference at any point. For example, in cylindrical coordinates (assuming the space is everywhere locally Lorentzian) there is only one frame in which the surfaces of simultaneity of an inertial observer coherent. In all other frames, if we follow a surface of simultaneity all the way around the closed dimension we find that it doesn't meet up with itself. Instead, we get a helical pattern (if we picture just a single cylindrical spatial dimension versus time). It may seem that we can disregard peculiar boundary conditions involving waves and so on, but if we begin to rule out valid solutions of the field equations by fiat, then we're obviously not being guided by the theory, but by our prejudices and preferences. Similarly, in order to exclude "unrealistic" cosmological solutions of the field equations we must impose energy conditions, i.e., we find that it's necessary to restrict the class of allowable Tuv tensor fields, but this again is not justified by the field equations themselves, but merely by our wish to force them to give us "realistic" solutions. It would be an exaggeration to say that we get out of the field equations only what we put into them, but there's no denying that a considerable amount of "external" information must be imposed on them in order to give realistic solutions.

7.8 Global Interpretations of Local Experience

I have been standing all my life in thedirect path of a battery of signals,the most accurately transmitted, mostuntranslatable language in the universe…I am an instrument… trying to translate pulsations into images for the relief of the bodyand the reconstruction of the mind. Adrienne Rich, 1971

The usual interpretation of general relativity is based on a conceptual framework consisting of primary entities – such as particles and non-gravitational fields – embedded in an extensive differentiable manifold of space and time. The theory is presented in the

form of differential equations, interpreted as giving a description of the local metrical properties of the manifold around any specific point. However, the physically meaningful statements derived from the theory refer to properties of the manifold over extended regions. To produce these statements, the differential equations are integrated (under certain constraints) to give a single coherent extended region of a manifold that everywhere satisfies those equations. This enables us to infer the extended spatio-temporal configurations of fields and particles, from which we derive predictions about observable interactions, which are ultimately reducible to the events of our experience. One question that naturally arises is whether the usual interpretation (or any interpretation) is uniquely singled out by our experience, or whether the same pattern of raw experiences might be explainable within some other, possibly quite different, conceptual framework. In one sense the answer is obvious. We can always accommodate any sequence of perceptions within an arbitrary ontology merely by positing a suitable theory of appearances separate from our presumed ontology. This approach can be traced back to ancient philosophers such as Parmenides, who taught that motion, change, and even plurality are merely appearances, while the reality is an unchanging unity. Our experience of dreams (for example) shows that the direct correspondence between our perceptions and the events of the external world can always be doubted. Of course, a solipsistic approach to the interpretation of experiences is somewhat repugnant, and need not be taken too seriously, but it nevertheless serves to remind us (if we needed reminding) that the link between our sense perceptions and the underlying external structure is always ambiguous, and any claim that our experiences do (or can) uniquely single out one specific ontology is patently false. There is always a degree of freedom in the selection of our model of the presumed external objective reality. In more serious models we usually assume that the processes of perception are "of the same kind" as the external processes that we perceive, but we still bifurcate our models into two parts, consisting of (1) an individual's sense impressions and interior experiences, such as thoughts and dreams, and (2) a class of objective exterior entities and events, of which only a small subset correspond to any individual's direct perceptions. Even within this limited class of models, the task of inferring (2) from (1) is not trivial, and there is certainly no a priori requirement that a given set of local experiences uniquely determines a particular global structure. Even if we restrict ourselves to the class of naively realistic models consistent with the observable predictions representable within general relativity, there remains an ambiguity in the conceptual framework. The situation is complicated by the fact that the field equations of general relativity, by themselves, permit a very wide range of global solutions if no restrictions are placed on the type of boundary conditions, initial values, and energy conditions that are allowed, but most of these solutions are (presumably) unphysical. As Einstein said, "A field theory is not yet completely determined by the system of field equations". In order to extract realistic solutions (i.e., solutions consistent with our experiences) from the field equations we must impose some constraints on the boundary or global topology, and on the

allowable form of the “source term”, i.e., energy conditions. In this sense the field equations do not represent a complete theory, because these restrictions can’t be inferred from the field equations; they are auxiliary assumptions that must simply be imposed on the basis of external considerations. This incompleteness is a characteristic of any physical law that is expressed as a set of differential equations, because such equations generally possess a vast range of possible formal solutions, and require one or more external principle or constraint to yield definite results. The more formal flexibility that our theory possesses, the more inclined we are to ask whether the actual physical content of the theory is contained in the rational "laws" or the circumstantial conditions that we impose. For example, consider a theory consisting of the assertion that certain aspects of our experience can be modeled by means of a suitable Turing machine with suitable initial data. This is a very flexible theoretical framework, since by definition anything that is computable can be computed from some initial data using a suitable Turing machine. Such a theory undeniably yields all applicable and computable results, but of course it also (without further specification) encompasses infinitely many inapplicable results. An ideal theoretical framework would be capable of representing all physical phenomena, but no unphysical phenomena. This is just an expression of the physicist's desire to remove all arbitrariness from the theory. However, as the general theory of relativity stands at present, it does not yield unique predictions about the overall global shape of the manifold. Instead, it simply imposes certain conditions on the allowable shapes. In this sense we can regard general relativity as a meta-theory, rather than a specific theory. So, when considering the possibility of alternative interpretations (or representations) of general relativity, we need to decide whether we are trying to find a viable representation of all possible theories that reside within the meta-theory of general relativity, or whether we are trying to find a viable representation of just a single theory that satisfies the requirements of general relativity. The physicist might answer that we need only seek representations that conform with those aspects of general relativity that have been observationally verified, whereas a mathematician might be more interested in whether there are viable alternative representations of the entire meta-theory. First we should ask whether there are any viable interpretations of general relativity as a meta-theory. This is a serious question, because one plausible criterion for viability is that we can analytically continue all worldlines without leading to any singularities or physical infinities. In other words, an interpretation is considered to be not viable if the representation "breaks down" at some point due to an inability to diffeomorphically continue the solution within that representation. The difficulty here is that even the standard interpretation of general relativity in terms of curved spacetime leads, in some circumstances, to inextendible worldlines and singularities in the field. Thus if we take the position that such attributes are disqualifying, it follows that even the standard interpretation of general relativity in terms of an extended spacetime manifold is not viable.

One possible approach to salvaging the geometrical interpretation would be to adopt, as an additional component of the theory, the principle that the manifold must be free of singularities and infinities. Indeed this principle was often suggested by Einstein, who wrote

It is my opinion that singularities must be excluded. It does not seem reasonable to me to introduce into a continuum theory points (or lines, etc.) for which the field equations do not hold... Without such a postulate the theory is much too vague.

He even hoped that the exclusion of singularities might (somehow) lead to an understanding of atomistic and quantum phenomena within the context of a continuum theory, although he acknowledged that he couldn't say how this might come about. He believed that the difficulty of determining exact singularity-free global solutions of non-linear field equations prevents us from assessing the full content of a non-linear field theory such as general relativity. (He recognized that this was contrary to the prevailing view that a field theory can only be quantized by first being transformed into a statistical theory of field probabilities, but he regarded this as "only an attempt to describe relationships of an essentially nonlinear character by linear methods".) Another approach, more in the mainstream of current thought, is to simply accept the existence of singularities, i.e., not consider them as a disqualifying feature of an interpretation. According to theorems of Penrose, Hawking, and others, it is known that the existence of a trapped surface (such as the event horizon of a black hole) implies the existence of inextendible worldlines, provided certain energy conditions are satisfied and we exclude closed timelike curves. Therefore, a great deal of classical general relativity and its treatment of black holes, etc., is based on the acceptance of singularities in the manifold, although this is often accompanied with a caveat to the effect that in the vicinity of a singularity the classical field equations may give way to quantum effects. In any case, since the field equations by themselves undeniably permit solutions containing singularities, we must either impose some external constraint on the class of realistic solutions to exclude those containing singularities, or else accept the existence of singularities. Each of these choices has implications for the potential viability of alternative interpretations. In the first case we are permitted to restrict the range of solutions to be represented, which means we really only need to seek representations of specific theories, rather than of the entire meta-theory represented by the bare field equations. In the second case we need not rule out interpretations based on the existence of singularities, inextendible worldlines, or other forms of "bad behavior". To illustrate how these considerations affect the viability of alternative interpretations, suppose we attempt to interpret general relativity in terms of a flat spacetime combined with a universal force field that distorts rulers and clocks in just such a way as to match the metrical relations of a curved manifold in accord with the field equations. It might be argued that such a flat-spacetime formulation of general relativity must fail at some point(s) to diffeomorphically map to the corresponding curved-manifold if the latter

possesses a non-trivial global topology. For example, the complete surface of a sphere cannot be mapped diffeomorphically to the plane. By means of sterographic projection from the North Pole of a sphere to a plane tangent to the South Pole we can establish a diffeomorphic mapping to the plane of every point on the sphere except the North Pole itself, which maps to a "point at infinity". This illustrates the fact that when mapping between two topologically distinct manifolds such as the plane and the surface of a sphere, there must be at least one point where the mapping is not well-behaved. However, this kind of objection fails to rule out physically viable alternatives to the curved spacetime interpretation (assuming any viable interpretation exists), and for several reasons. First, we may question whether the mapping between the curved spacetime and the alternative manifold needs to be everywhere diffeomorphic. Second, even if we accede to this requirement, it's important to remember that the global topology of a manifold is sensitive to pointwise excisions. For example, although it is not possible to diffeomorphically map the complete sphere to the plane, it is possible to map the punctured sphere, i.e., the sphere minus one point (such as the North Pole in the sterographic projection scheme). We can analytically continue the mapping to include this point by simply adding a "point at infinity" to the plane - without giving the extended plane intrinsic curvature. Of course, this interpretation does entail a singularity at one point, where the universal field must be regarded as infinitely strong, but if we regard the potential for physical singularities as disqualifying, then as noted above we have no choice but to allow the imposition of some external principles to restrict the class of solutions to global manifolds that are everywhere "well-behaved". If we also disallow this, then as discussed above there does not exist any viable interpretation of general relativity. Once we have allowed this, we can obviously posit a principle to the effect that only global manifolds which can be diffeomorphically mapped to a flat spacetime are physically permissible. Such a principle is no more in conflict with the field equations than are any of the well-known "energy conditions", the exclusion of closed timelike loops, and so on. Believers in one uniquely determined interpretation may also point to individual black holes, whose metrical structure of trapped surfaces cannot possibly be mapped to flat spacetime without introducing physical singularities. This is certainly true, but according to theorems of Penrose and Hawking it is precisely the circumstance of a trapped surface that commits the curved-spacetime formulation itself to a physical singularity. In view of this, we are hardly justified in disqualifying alternative formulations that entail physical singularities in exactly the same circumstances. Another common objection to flat interpretations is that even for a topologically flat manifold like the surface of a torus it is impossible to achieve the double periodicity of the closed torriodal surface, but this objection can also be countered, simply by positing a periodic flat universe. Admittedly this commits us to distant correlations, but such things cannot be ruled out a priori (and in fact distant correlations do seem to be a characteristic of the universe from the standpoint of quantum mechanics, as discussed in Section 9).

More generally, as Poincare famously summarized it, we can never observe our geometry G in a theory-free sense. Every observation we make relies on some prior conception of physical laws P which specify how physical objects behave with respect to G. Thus the universe we observe is not G, but rather U = G + P, and for any given G we can vary P to give the observed U. Needless to say, this is just a simplified schematic of the full argument, but the basic idea is that it's simply not within the power of our observations to force one particular geometry upon us (nor even one particular topology), as the only possible way in which we could organize our thoughts and perceptions of the world. We recall Poincare's famous conventionalist dictum "No geometry is more correct than any other - only more convenient". Those who claim to "prove" that only one particular model can be used to represent our experience would do well to remember John Bell's famous remark that the only thing "proved" by such proofs is lack of imagination. The interpretation of general relativity as a field theory in a flat background spacetime has a long history. This approach was explored by Feynman, Deser, Weinberg, and others at various times, partly to see if it would be possible to quantize the gravitational field in terms of a spin-2 particle, following the same general approach that was successful in quantizing other field theories. Indeed, Weinberg's excellent "Gravitation and Cosmology" (1972) contained a provocative paragraph entitled "The Geometric Analogy", in which he said

Riemann introduced the curvature tensor R to generalize the [geometrical] concept of curvature to three or more dimensions. It is therefore not surprising that Einstein and his successors have regarded the effects of a gravitational field as producing a change in the geometry of space and time. At one time it was even hoped that the rest of physics could be brought into a geometric formulation, but this hope has met with disappointment, and the geometric interpretation of the theory of gravitation has dwindled to a mere analogy, which lingers in our language in terms like "metric", "affine connection", and "curvature", but is not otherwise very useful. The important thing is to be able to make predictions about the images on the astronomer's photographic plates, frequencies of spectral lines, and so on, and it simply doesn't matter whether we ascribe these predictions to the physical effect of a gravitational field on the motion of planets and photons or to a curvature of space and time.

The most questionable phrase here is the claim that, aside from providing some useful vocabulary, the geometric analogy "is not otherwise very useful". Most people who have studied general relativity have found the geometric analogy to be quite useful as an aid to understanding the theory, and Weinberg can hardly have failed to recognize this. I suspect that what he meant (in context) is that the geometric framework has not proven to be very useful in efforts to unify gravity with the rest of physics. The idea of "bringing the rest of physics into a geometric formulation" refers to attempts to account for the other forces of nature (electromagnetism, strong, and weak) in purely geometrical terms as attributes of the spacetime manifold, as Einstein did for gravity. In other words, eliminate the concept of "force" entirely, and show that all motion is geodesic in some suitably defined spacetime manifold. This is what is traditionally called a "unified field

theory", and led to Weyl's efforts in the 20's, and the Kluza-Klein theories, and Einstein's anti-symmetric theories, and so on. As Weinberg said, those hopes have (so far) met with disappointment. In another sense, one might say that all of physics has been subsumed by the geometric point of view. We can obviously describe baseball, music, thermodynamics, etc., in geometrical terms, but that isn't the kind of geometrizing that is being discussed here. Weinberg was referring to attempts to make the space-time manifold itself account for all the "forces" of nature, as Einstein had made it account for gravity. Quantum field theory works on a background of space-time, but posits other ingredients on top of that to represent the fields. Obviously we're free to construct a geometrical picture in our minds of any gauge theory, just as we can form a geometrical picture in any arbitrary kind of "space", e.g., the phase space of a system, but this is nothing like what Einstein, Weyl, Kaluza, Weinberg, etc. were talking about. The original (and perhaps naive) hope was to eliminate all other fields besides the metric field of the spacetime manifold itself, to reduce physics to this one primitive entity (and its metric). It's clear that (1) physics has not been geometrized in the sense that Weinberg was talking about, viz, with the spacetime metric being the only ontological entity, and (2) in point of fact, some significant progress toward the unification of the other "forces" of nature has indeed been made by people (such as Weinberg himself) who did so without invoking the geometric analogy. Many scholars have expressed similar views to those of Poincare and Weinberg regarding the essential conventionality of geometry. In considering the question "Is Spacetime Curved?" Ian Roxburgh described the curved and flat interpretations of general relativity, and concluded that "the answer is yes or no depending on the whim of the answerer. It is therefore a question without empirical content, and has no place in physical inquiry." Thus he agreed with Poincare that our choice of geometry is ultimately a matter of convenience. Even if we believe that general relativity is perfectly valid in all regimes (which most people doubt), it's still possible to place a non-geometric interpretation on the "photographic plates and spectral lines" if we choose. The degree of "inconvenience" is not very great in the weak-field limit, but becomes more extreme if we're thinking of crossing event horizons or circumnavigating the universe. Still, we can always put a non-geometrical interpretation onto things if we're determined to do so. (Ironically, the most famous proponent of the belief that the geometrical view is absolutely essential, indeed a sine qua non of rational thought, was Kant, because the geometry he espoused so confidently was non-curved Euclidean space.) Even Kip Thorne, who along with Misner and Wheeler wrote the classic text Gravitation espousing the geometric viewpoint, admits that he was once guilty of curvature chauvinism. In his popular book "Black Holes and Time Warps" he writes

Is spacetime really curved? Isn't it conceivable that spacetime is actually flat, but the clocks and rulers with which we measure it... are actually rubbery? Wouldn't... distortions of our clocks and rulers make truly flat spacetime appear to be curved? Yes.

Thorne goes on to tell how, in the early 1970's, some people proposed a membrane paradigm for conceptualizing black holes. He says

When I, as an old hand at relativity theory, heard this story, I thought it ludicrous. General relativity insists that, if one falls into a black hole, one will encounter nothing at the horizon except spacetime curvature. One will see no membrane and no charged particles... the membrane theory can have no basis in reality. It is pure fiction. The cause of the field lines bending, I was sure, is spacetime curvature, and nothing else... I was wrong.

He goes on to say that the laws of black hole physics, written in accord with the membrane interpretation, are completely equivalent to the laws of the curved spacetime interpretation (provided we restrict ourselves to the exterior of black holes), but they are each heuristically useful in different circumstances. In fact, after he got past thinking it was ludicrous, Thorne spent much of the 1980's exploring the membrane paradigm. He does, however, maintain that the curvature view is better suited to deal with interior solutions of black holes, but isn't not clear how strong a recommendation this really is, considering that we don't really know (and aren't likely to learn) whether those interior solutions actually correspond to facts. Feynman’s lectures on gravitation, written in the early 1960’s, present a field-theoretic approach to gravity, while also recognizing the viability of Einstein’s geometric interpretation. Feynman described the thought process by which someone might arrive at a theory of gravity mediated by a spin-two particle in flat spacetime, analogous to the quantum field theories of the other forces of nature, and then noted that the resulting theory possesses a geometrical interpretation.

It is one of the peculiar aspect of the theory of gravitation that is has both a field interpretation and a geometrical interpretation… the fact is that a spin-two field has this geometrical representation; this is not something readily explainable – it is just marvelous. The geometric interpretation is not really necessary or essential to physics. It might be that the whole coincidence might be understood as representing some kind of gauge invariance. It might be that the relationships between these two points of view about gravity might be transparent after we discuss a third point of view, which has to do with the general properties of field theories under transformations…

He goes on to discuss the general notion of gauge invariance, and concludes that “gravity is that field which corresponds to a gauge invariance with respect to displacement transformations”. One potential source of confusion when discussing this issue is the fact that the local null structure of Minkowski spacetime makes it locally impossible to smoothly mimic the effects of curved spacetime by means of a universal force. The problem is that Minkowski spacetime is already committed to the geometrical interpretation, because it

identifies the paths of light with null geodesics of the manifold. Putting this together with some form of the equivalence principle obviously tends to suggest the curvature interpretation. However, this does not rule out other interpretations, because there are other possible interpretations of special relativity - notably Lorentz's theory - that don't identify the paths of light with null geodesics. It's worth remembering that special relativity itself was originally regarded as simply an alternate interpretation of Lorentz's theory, which was based on a Galilean spacetime, with distortions in both rulers and clocks due to motion. These two theories are experimentally indistinguishable - at least up to the implied singularity of the null intervals. In the context of Galilean spacetime we could postulate gravitational fields affecting the paths of photons, the rates of physical clocks, and so on. Of course, in this way we arrive at a theory that looks exactly like curved spacetime, but we interpret the elements of our experience differently. Since (in this interpretation) we believe light rays don't follow null geodesic paths (and in fact we don't even recognize the existence of null geodesics) in the "true" manifold under the influence of gravity, we aren't committed to the idea that the paths of light delineate the structure of the manifold. Thus we'll agree with the conventional interpretation about the structure of light cones, but not about why light cones have that structure. At some point any flat manifold interpretation will encounter difficulties in continuing its worldlines in the presence of certain postulated structures, such as black holes. However, as discussed above, the curvature interpretation is not free of difficulties in these circumstances either, because if there exists a trapped surface then there also exist non-extendable timelike or null geodesics for the curvature interpretation. So, the (arguably) problematical conditions for a "flat space" interpretation are identical to the problematical conditions for the curvature interpretation. In other words, if we posit the existence of trapped surfaces, then it's disingenuous for us to impugn the robustness of flat space interpretations in view of the fact that these same circumstances commit the curvature interpretation to equally disquieting singularities. It may or may not be the case that the curvature interpretation has a longer reach, in the sense that it's formally extendable inside the Schwarzschild radius, but, as noted above, the physicality of those interior solutions is not (and probably never will be) subject to verification, and they are theoretically controversial even within the curvature tradition itself. Also, the simplistic arguments proposed in introductory texts are easily seen to be merely arguments for the viability of the curvature interpretation, even though they are often mis-labeled as arguments for the necessity of it. There's no doubt that the evident universality of local Lorentz covariance, combined with the equivalence principle, makes the curvature interpretation eminently viable, and it's probably the "strongest" interpretation of general relativity in the sense of being exposed most widely to falsification in principle, just as special relativity is stronger than Lorentz’s ether theory. The curvature interpretation has certainly been a tremendous heuristic aid (maybe even indispensable) to the development of the theory, but the fact remains that it isn't the only possible interpretation. In fact, many (perhaps most) theoretical physicists today consider it likely that general relativity is really just an approximate consequence of some underlying structure, similar to how continuum fluid

mechanics emerges from the behavior of huge numbers of elementary particles. As was rightly noted earlier, much of the development of particle physics and more recently string theory has been carried out in the context of rather naive-looking flat backgrounds. Maybe Kant will be vindicated after all, and it will be shown that humans really aren't capable of conceiving of the fundamental world on anything other than a flat geometrical background. If so, it may tell us more about ourselves than about the world. Another potential source of confusion is the tacit assumption on the part of some people that the topology of our experiences is unambiguous, and this in turn imposes definite constraints on the geometry via the Gauss-Bonnet theorem. Recall that for any two-dimensional manifold M the Euler characteristic is a topological invariant defined as

where V, E, and F denote the number of vertices, edges, and faces respectively of any arbitrary triangulation of the entire surface. Extending the work that Gauss had done on the triangular excess of curves surfaces, Bonnet proved in 1858 the beautiful theorem that the integral of the Gaussian curvature K over the entire area of the manifold is proportional to the Euler characteristic, i.e.,

More generally, for any manifold M of dimension n the invariant Euler characteristic is

where k is the number of k-simplexes of an arbitrary "triangulation" of the manifold. Also, we can let Kn denote the analog of the Gaussian curvature K for an n-dimensional manifold, noting that for hypersurfaces this is just the product of the n principal extrinsic curvatures, although like K it has a purely intrinsic significance for arbitrary embeddings. The generalized Gauss-Bonnet theorem is then

where V(Sn) is the "volume" of a unit n-sphere. Thus if we can establish that the topology of the overall spacetime manifold has a non-zero Euler characteristic, it will follow that the manifold must have non-zero metrical curvature at some point. Of course, the converse is not true, i.e., the existence of non-zero metrical curvature at one or more points of the manifold does not imply non-zero Euler characteristic. The two-dimensional surface of a torus with the usual embedding in R3 not only has intrinsic curvature but is topologically distinct from R2, and yet (as discussed in Section 7.5) it can be mapped diffeomorphically and globally to an everywhere-flat manifold embedded in R4. This

illustrates the obvious fact that while topological invariants impose restrictions on the geometry, they don't uniquely determine the geometry. Nevertheless, if a non-zero Euler characteristic is stipulated, it is true that any diffeomorphic mapping of this manifold must have non-zero curvature at some point. However, there are two problems with this argument. First, we need not be limited to diffeomorphic mappings from the curved spacetime model, especially since even the curvature interpretation contains singularities and physical infinities in some circumstances. Second, the topology is not stipulated. The topology of the universe is a global property which (like the geometry) can only be indirectly inferred from local experiences, and the inference is unavoidably ambiguous. Thus the topology itself is subject to re-interpretation, and this has always been recognized as part-and-parcel of any major shift in geometrical interpretation. The examples that Poincare and others talked about often involved radical re-interpretations of both the geometry and the topology, such as saying that instead of a cylindrical dimension we may imagine an unbounded but periodic dimension, i.e., identical copies placed side by side. Examples like this aren't intended to be realistic (necessarily), but to convey just how much of what we commonly regard as raw empirical fact is really interpretative. We can always save the appearances of any particular apparent topology with a completely different topology, depending on how we choose to identify or distinguish the points along various paths. The usual example of this is a cylindrical universe mapped to an infinite periodic universe. Therefore, we cannot use topological arguments to prove anything about the geometry. Indeed these considerations merely extend the degrees of freedom in Poincare's conventionalist formula, from U = G + P to U = (G + T) + P, where T represents topology. Obviously the metrical and topological models impose consistency conditions on each other, but the two of them combined do not constrain U any more than G alone, as long as the physical laws P remain free. There may be valid reasons for preferring not to avail ourselves of any of the physical assumptions (such as a "universal force", let alone multiple copies of regions, etc.) that might be necessary to map general relativity to a flat manifold in various (extreme) circumstances, such as in the presence of trapped surfaces or other "pathological" topologies, but these are questions of convenience and utility, not of feasibility. Moreover, as noted previously, the curvature interpretation itself entails inextendable worldlines as soon as we posit a trapped surface, so topological anomalies hardly give an unambiguous recommendation to the curvature interpretation. The point is that we can always postulate a set of physical laws that will make our observations consistent with just about any geometry we choose (even a single monadal point!), because we never observe geometry directly. We only observe physical processes and interactions. Geometry is inherently an interpretative aspect of our understanding. It may be that one particular kind of geometrical structure is unambiguously the best (most economical, most heuristically robust, most intuitively appealing, etc), and any alternative geometry may require very labored and seemingly ad hoc "laws of physics" to

make it compatible with our observations, but this simply confirms Poincare's dictum that no geometry is more true than any other - only more convenient. It may seem as if the conventionality of geometry is just an academic fact with no real applicability or significance, because all the examples of alternative interpretations that we've cited have been highly trivial. For a more interesting example, consider a mapping (by radial projection) from an ordinary 2-sphere to a circumscribed polyhedron, say a dodecahedron. With the exception of the 20 vertices, where all the "curvature" is discretely concentrated, the surface of the dodecahedron is perfectly flat, even along the edges, as shown by the fact that we can "flatten out" two adjacent pentagonal faces on a plane surface without twisting or stretching the surfaces at all. We can also flatten out a third pentagonal face that joins the other two at a given vertex, but of course (in the usual interpretation) we can't fit in a fourth pentagon at that vertex, nor do three quite "fill up" the angular range around a vertex in the plane. At this stage we would conventionally pull the edges of the three pentagons together so that the faces are no longer coplanar, but we could also go on adjoining pentagonal surfaces around this vertex, edge to edge, just like a multi-valued "Riemann surface" winding around a pole in the complex plane. As we march around the vertex, it's as if we are walking up a spiral staircase, except that all the surfaces are laying perfectly flat. This same "spiral staircase" is repeated at each vertex of the solid. Naturally we can replace the dodecahedron with a polyhedron having many more vertices, but still consisting of nothing but flat surfaces, with all the "curvature" distributed discretely at a huge number of vertices, each of which is a "pole" of an infinite spiral staircase of flat surfaces. This structure is somewhat analogous to a "no-collapse" interpretation of quantum mechanics, and might be called a "no-curvature" interpretation of general relativity. At each vertex (cf. measurement) we "branch" into on-going flatness across the edge, never actually "collapsing" the faces meeting at a vertex into a curved structure. In essence the manifold has zero Euler characteristic, but it exhibits a non-vanishing Euler characteristic modulo the faces of the polyhedron. Interestingly, the term "branch" is used in multi-valued Riemann surfaces just as it's used in some descriptions of the "no-collapse" interpretation of quantum mechanics. Also, notice that the non-linear aspects of both theories are (arguably) excised by this maneuver, leaving us "only" to explain how the non-linear appearances emerge from this aggregate, i.e., how the different moduli are inter-related. To keep track of a particle we would need its entire history of "winding numbers" for each vertex of the entire global manifold, in the order that it has encountered them (because it's not commutative), as well as it's nominal location modulo the faces of the polyhedron. In this model the full true topology of the universe is very different from the apparent topology modulo the polyhedral structure, and curvature is non-existent on the individual branches, because every time we circle a non-flat point we simply branch to another level (just as in some of the no-collapse interpretations of quantum mechanics the state sprouts a new branch, rather than collapsing, each time an observation is made). Each time a particle crosses an edge between two vertices it's set of winding numbers is updated, and we end up with a combinatorial approach, based on a finite number of discrete poles

surrounded by infinitely proliferating (and everywhere-flat) surfaces. We can also arrange for the spiral staircases to close back on themselves after a suitable number of windings, while maintaining a vanishing Euler characteristic. For a less outlandish example of a non-trivial alternate interpretation of general relativity, consider the "null surface" interpretation. According to this approach we consider only the null surfaces of the traditional spacetime manifold. In other words, the only intervals under consideration are those such that g dx dx = 0. Traditional timelike paths are represented in this interpretation by zigzag sequences of lightlike paths, which can be made to approach arbitrarily closely to the classical timelike paths. The null condition implies that there are really only three degrees of freedom for motion from any given point, because given any three of the increments dx0, dx1, dx2, and dx3, the corresponding increment of the fourth automatically follows (up to sign). The relation between this interpretation and the conventional one is quite similar to the relation between special relativity and Lorentz's ether theory. In both cases we can use essentially the same equations, but whereas the conventional interpretation attributes ontological status to the absolute intervals dt, the null interpretation asserts that those absolute intervals are ultimately superfluous conventionalizations (like Lorentz's ether), and encourages us to dispense with those elements and focus on the topology of the null surfaces themselves.

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