7. Cosmology 1 Is the Universe Closed?

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 K_n 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.