There are two short essays in this section. Originally they were in one file and, according to the print preview, were between three and four typewritten pages long. But to keep them short, I had to cut a few corners which aren't cut in the more detailed argument presented in Essay #3. So if you have objections anywhere along the line in these shorter essays, the chances are tolerably good that I've addressed them in the longer version.Until recently there were three essays in this section, but a few "cut and paste" operations have eliminated one of them (at the modest expense of another page of print to the first essay). On the other hand, this time I added a few simple diagrams. This increases the download time by a few seconds, and I thought it prudent to present each essay as a separate file. The original claim of three to four pages length is still correct for the main text, although the diagrams and new accompanying text increase the length on the print preview by two pages.
The first essay poses a few questions which, if answered correctly, show an assumption you can make about any large elastically expanding structure which changes it from "flat" to "positively curved" by incorporating an infinite "large" spatial dimension which has to be there (although we may not notice it in the local vicinity). The leading questions, if they stump you, are answered in the end of the essay via the "extra dimension theorem".
Then, assuming it's possible to apply the extra dimension theorem in a way that gives us a "large" fourth spatial dimension in the universe, how does the structure in this alternate model differ from the structure in the conventional model? The second essay begins with one of the typical representations of the conventional Friedmann universe, and then shows how it can be modified to accommodate this non-conventional model with another spatial dimension. The essay concludes with a (very) brief discussion of some oddly behaving distributions in deep space that might be explained via the existence of a "large" fourth spatial dimension.
Both essays have a common denominator in that they question whether the conventional cosmological wisdom implicitly assumed by Friedmann space - three spatial dimensions, plus one time dimension - is correct by showing an easily derived paradox in such space when time is reversed. This subtle, overlooked paradox in Friedmann space disappears if we modify the model to Riemannian space.
So the model being derived here is the "standard" Friedmann-Robertson-Walker (FRW) metric, aka Riemann's "closed cosmic hypersphere", which necessarily includes one more "hidden" spatial dimension in addition to the three we easily recognize. Although Riemann's hypersphere was first described over 150 years ago and its reconstitution as the "standard" FRW metric has been around for six or seven decades, apparently no one has bothered to ask a very simple question: how do we get it?
I'll try to give the simplest answer...
If you read either one of these essays and manage to remove the Euclidean blinders most of us have, there's a good chance that you'll say something like "Whoops...Why on earth did we accept those open-flat-closed Friedmann models since the 1930s without thinking them through?"
Sidebars will be presented in similar red copy, and what seems awkward spacing in the HTML format is to prevent the diagrams from being "dissected" if you print either one of these two short essays.
[There is a clearly defined line of logic which enables us to incorporate one more "hidden" spatial dimension into any elastically expanding structure provided we're willing to make a reasonable assumption in topology. The principle can be reduced to a very simple statement - I call it the extra dimension theorem - and is presented at the end of this essay. A few tutorial or pedagogical questions are posed first to see if you can deduce what the extra dimension theorem says, before it's formally presented.]
Suppose we are inhabitants of a spherical surface defined this way:
where x,y,z are the three spatial dimensions, r is the radius, and the time "t" shows those values can change with time. Note that, if r >>> 0, only two of the spatial dimensions are obvious to an inhabitant of the surface (a situation we'll consider in Step 2).
Next we'll assume the surface is expanding according to the Hubble parameter H (so many km/sec/mpc) and that light is confined to the surface. Any structure expanding by a given value of H - assumed to be constant for the moment - always has an inferred zero-dimensional collapsed position; it's the farthest position we can look toward in theory when t = 0, and the distance to it is c/H, where c is the speed of light. So we're assuming that we can look "around and around", always back in time along an increasingly smaller surface, to when it literally collapses to zero size at 1/H seconds ago.
Note that at the inferred collapsed position two types of space collapse to zero size, since both depend on the same radius r:
- the positively curved "surface" (circumference) on which our geodesics are confined, and
- the "interior" (radius) which is the space in which all surface positions really move with respect to the collapsed receding position, at right angles to the geodesics.
The surface of the sphere is referred to in topology as S2, while the interior (volume) is referred to as V3.
[The derivative of the volume in V3 (4pr3/3) with respect to the radius yields the positively curved area in S2 space (4pr2). This intriguing derivative relationship between the two types of space in topology is maintained in the next lower- and higher-dimensional structures (i.e., the circle and hypersphere, respectively) and is explained at the beginning of the first essay.]
Figure 1: "Around and Around". H, the Hubble parameter, is assumed to be constant and describes the rate of expansion on this elastically expanding spherical surface. The distance to the collapsed central position - via several circuits around the surface - is c/H, where "c" is the speed of light. The dark arrows show how the structure (in "S2") appears to an observer at "A" who looks out along the surface in one direction. The red arrows (in "V3") show what the movement of the surface positions - such as B, C, and D, in addition to A - really is with respect to the collapsed central site.Question: Suppose r >>> 0, and there is absolutely no clue suggesting curvature in the local vicinity. Before we conclude that the surface is "flat" on the global scale, there is an assumption we could make about the collapsed receding position that would automatically add another "hidden" spatial dimension and change the elastic structure from "flat" to (positively) "curved"?
What is that assumption?
Thus we can look past all loci when their receding velocity v is less than c, yet all loci - including our own - are topologically defined to exist at the collapsed receding t = 0 site, toward which we inevitably look (without ever being able to "see" it physically, since this position is receding at the speed of light, and is dimensionless anyway).
Suppose we were absolutely convinced that we occupied an elastic "flat" surface, expanding at H, with only two spatial dimensions and one time dimension (x,y,t). In other words, we're assuming initially that the z-dimension does not exist since we don't notice it directly.
Nevertheless, there is a topological assumption we can make about the inferred collapsed receding site which tells us unambiguously that the expanding structure has to have that z-dimension at right angles to the x and y dimensions we recognize without difficulty. In other words, we can deduce that we really occupy a "finite but unbounded" spherical surface that we're looking around at least once. This deduction gives us a model in which S2 space is a "brane" on (or derivative of) higher-dimensional V3 space.
What is the topological assumption that gives us the z-dimension?
Figure 2: Flat or Curved? In this diagram the observer at A assumes that, since the surface appears "flat" in the local vicinity, it is likewise flat on the global scale.(Nearly the same) Question: Again, is there any way this person could deduce the necessary existence of higher dimensional space? The Hubble parameter, H, is assumed to have the same constant value that it had in Step 1. So there is still an inferred collapsed dimensionless position that the observer at A looks toward in virtually every direction, that position is still at a distance of c/H, and (as before) he can infer the existence of the collapsed state at 1/H seconds ago.
A simple geodesic is shown on the surface, leaving A and going past B, C, and D. By definition, A, B, C, and D (as well as all other loci on the surface) must reside at the time (t) = 0 dimensionless position, the "edge" of the apparent ellipse. How can we look "further than" or "beyond" the loci at B, C, and D and still have them be topologically defined to exist at the collapsed receding site when t = 0.
Hint: Can we look toward the collapsed site and have everything collapse to zero size in the same space in which our geodesics occur? Remember: nothing is defined away from the collapsed site when t = 0 and we're looking toward that site at nature's speed limit. So how do all the loci we look past - including our own - get there as we necessarily look back in time along the surface?
We apparently occupy a universe with elastic space expanding via three easily three recognizable spatial dimensions and one time dimension in (x,y,z,t) that may or may not be curved gravimetrically by the density. H = ~70 km/sec/mpc, although that is the value only in the current era. H over time is affected by a deceleration parameter (slowdown caused by density), and also by an acceleration parameter (speedup caused by "dark energy"; this is the inferred "cosmological constant").
Still, the "age" of the universe, which is determined by H and the two other parameters, is fairly well established between 13 and 14 billion years, meaning this is the length of time the universe has been expanding from the original t = 0 position. (Other methods, such as the age of the oldest globular clusters, yield a comparable estimate.)
But even though it seems to be a Euclidean universe with three spatial dimensions and one time dimension, there is a topological assumption we can make about the collapsed receding site - i.e., the singularity - some 13 to 14 billion light years away which tells us unambiguously that universe is not Euclidean, and that there has to be one more "hidden" spatial dimension associated with the universe, at right angles to the three spatial dimensions we know are there.
That assumption does the following simultaneously:
- it gives a topologist "Riemannian space",
- it gives the theoretical physicist a universe with a "3D-brane" (aka "Riemannian surface") embedded in "4D bulk"; most, but not all, physical phenomena occur in the "3D brane" - and
- it gives a cosmologist a model in the inventory known as the "standard" Friedmann-Robertson-Walker (FRW) metric whose structure is defined this way:
Equation 2: x2(t) + y2(t) + z2(t) + w2(t) = r2(t) where x,y,z r, and t are as defined previously, and w is that ever elusive (and "large") fourth spatial dimension.
What is the assumption in topology that necessarily incorporates an "extra" hidden spatial dimension into an elastically expanding universe? That changes a Friedmann universe in V3 into a higher-dimensional S3/V4 universe?
Figure 3: Friedmann Space or Riemannian Space? The apparent ellipse in Figure 2 is also in this figure. Whatever its nature is - a two-dimensional plane or a three-dimensional spherical surface - the usual assumption is that it's a "slice" of a higher-dimensional structure. Conventional cosmology assumes that the structure in Figure 2 is a "flat" Euclidean plane with no intrinsic curvature and that the higher-dimensional structure in which it resides is likewise "flat".But by now you should begin to realize that this may not be correct, as it is possible to define the collapsed receding position - called the "singularity" - in such a way that the universe will necessarily have the ab initio positive curvature caused by a "large" fourth spatial dimension in the FRW metric. Ever since the 1930s, the consensus in cosmology has been that the universe is basically "Euclidean", perhaps with minor deviations from this global assumption due to gravimetric considerations (i.e., density). There is, however, a subtle "tweaking" of the singularity in topology which nullifies this long held assumption.
Have you figured out yet what that "tweaking" is?
In this model a spherical surface is literally "embedded" in higher-dimensional space, when one of the terms on the left side of the equation is zero. And if any two terms on the left side of Equation 2 are zero, what remains is a "geodesic" defined exclusively along the two non-zero dimensions, likewise "embedded" in higher-dimensional space.
In topological terms, we have a universe in which (most of) our observations are confined to an expanding positively curved S3 "brane" (along the circumference) and this lies embedded in higher-dimensional V4 space, which is where the expansion really occurs (along the radius).
[The inclusion of the w term in Equation 2 projects the topological behavior of S2/V3 into S3/V4. The formal expressions defining S3 and V4 are mentioned at the beginning of Essay #1, and what "projects" means is explained in Appendices 2 and 3. For the moment, it suffices to state that whatever happens on S2 also happens on S3, since the expression defining S2 remains extant (or "embedded") in S3.]
Hint: Can everything collapse to zero size if the global expansion - or contraction, when time is reversed - occurs in the same space in which our geodesics occur? (The reasoning here expands into another dimension the Hint at the end of the Second Step above.)
The leading questions thus far are intended to lead you to the following statement, which I call the "extra dimension theorem":
If all loci along the geodesic on any elastically expanding structure when t > 0 are topologically defined to exist at the collapsed receding t = 0 site, then the elastic structure we observe via such geodesics necessarily appears positively curved (Sn) and includes one more spatial dimension (Vn + 1), along whose radius the global expansion occurs.
The inclusion of all loci at the collapsed site indicates that, when time is reversed, they reach this unique position along an alternate "route" - that is, along the radius of a hidden dimension, in a different type of space - since there is no adequate explanation in classical physics showing how they can reach it at the same time and in the same space in which our light geodesics are confined.
No matter what the latest cosmological fad is, the perennial positively curved "finite but unbounded" universe has always been lurking out there in the background. This "extra dimension theorem" simply permits us to use a topological definition of the singularity which changes space with three spatial dimensions (conventional Friedmann models) into space with four spatial dimensions (FRW metric).
Georg Riemann first described this space in his doctoral thesis in 1851, and so S3/V4 space is called "Riemannian space" and the structure itself is called a "Riemannian hypersphere". Curiously, for the first few decades of the twentieth century, Einstein favored a universe with Riemannian space in S3/V4 until he became a supporter of the V3 Friedmann models in the 1930s.
[Although it was presented in the Introduction, this article in 1932 shows when Einstein changed his mind. Scroll down until you get to "Einstein and De Sitter Return to Euclidean Idea of Cosmos".]
Just as there are global differences between a flat plane in V2 and a spherical surface in S2 which we will notice if we look out far enough, there are easily detectable differences between V3 and S3 space for the same reason. The argument I'm making in this series of essays is that there are enough anomalies and dramatic changes in trend lines of several distributions in deep space to give us some clues that Riemann's S3/V4 "closed cosmic hypersphere" might be the right model, after all. (The beginning of the discussion about easily predictable observed behavior is near the end of the second condensed essay.)
As a minimum, now you know how to define the singularity topologically in a way that gives you Riemann's S3/V4 hypersphere, which in all likelihood is something you didn't know a few minutes ago.
An intriguing question is whether astronomers, physicists, and cosmologists have been using the topological definition of the singularity which favors Riemannian space (rather than conventional Friedmann space) since the 1930s without realizing that's what they've been doing.
E-mail me at any time with comments or questions.