Supplemental Material

Abstract

Recently two physicists, Lisa Randall and Raman Sundrum, proposed an "uncompactified" fourth spatial dimension with a "3-brane" on its surface in which we observe most, but not all, physical activity. The discussion here describes what assumptions are necessary to derive a universe with this "uncompactified" fourth spatial dimension.

Introduction

The Lisa Randall - Raman Sundrum suggestion in 1999 of an "uncompactified" fourth spatial dimension was not the first time it's been mentioned in this manner.¹ Although the positively curved space now associated with a four-dimensional hypersphere was described in Georg Riemann's doctoral thesis in 1851, the feasibility of another dimension was taken more seriously when an unknown Prussian mathematician, Theodor Kaluza, wrote a letter to Albert Einstein in 1919 noting that the existence of a fourth spatial dimension would unify gravity and electromagnetic radiation.

Einstein's tardy reply in 1921 agreed with Kaluza's suggestion.

But if there was a fourth spatial dimension, why didn't we notice it?

One of Kaluza's former students, Oskar Klein, postulated in 1926 that a fourth spatial dimension might be "curled up small", that is, too small to detect in subatomic packets of space. These were known later as "Kaluza-Klein bottles" and eventually became the foundation for modern string theory. Thus the common assumption among physicists for decades has been that the fourth spatial dimension is "compactified", that is, the way it's assumed to exist in string theory.

Since Randall and Sundrum are proposing an alternate "uncompactified" state, a rather intriguing question in both cosmology and topology is what are the assumptions which yield a universe that includes a fourth spatial dimension in this manner.

A. Riemannian Space

Although many topologists are familiar with the space described in Riemann's four-dimensional (4D) hypersphere, it's not known very well outside this select group. So the following table introduces this space by listing Riemann's 4D hypersphere with its cousins - the circle and sphere.

"N" is the number of terms in the formula and is also the true number of spatial dimensions in the structure. The next two columns are self explanatory. The Vⁿ column is the expression for the sum of the points less than or equal to the radius r; it's the topological "volume" of the structure in all n dimensions, even though "volume" in our Euclidean world is accurate only when n = 3.

S^{n - 1}, the "surface", is intriguing because the expression here is always the derivative with respect to r of the formula in the same row of the Vⁿ column. Again, "surface" is accurate in Euclidean terms only when n - 1 = 2 (or n = 3), although S refers topologically to this type of space no matter what value n assumes. The formula here refers to the sum of all points in n - 1 dimensions equal to the radius r.

The unusual topological characteristics of an S-type structure have been referred to for years in a variety of ways: (1) "finite but unbounded", because it has finite size but no formal boundary; (2) "multiply-connected", since there is more than one way to reach another position; and (3) "curves back on itself", because an apparently straight path from a position in S-type space is a circular route - called a "geodesic" - and returns to the original position from the opposite direction.
Thus an S-type structure has a positive global curvature in such a manner that, if r = constant, a route around the circle/sphere/hypersphere is always 2pr. On the other hand, if r >>> 0 - that is, r is very large - we have no way of knowing from local observations whether the structure we occupy is Vⁿ (presumably orthogonal) or Sⁿ (globally curved).

Also it's important to understand the concept of "embeddedness" in S-type structures, which means that under certain circumstances - when one or more terms equals zero - the lower-dimensional structure is "embedded" (still there) in the higher-dimensional structure. Thus whatever happens on any curved "surface" in S^{n - 1} also happens on the comparably curved "surface" in Sⁿ.

And even though 2,300 years of Euclidean geometry probably make a curved "Riemannian surface" in S³ difficult to understand at first, the familiar Euclidean relationships that S¹ has with V² and S² has with V³ are virtually duplicated in the Riemannian relationship that S³ has with V⁴. For example, just as a spherical surface in S² is the "skin" of a sphere in V³, a comparably curved volume of space in S³ is also the "skin" of Riemann's hypersphere in V⁴. (Randall and Sundrum are designating the surface in S³/V⁴ a "3-brane", which is the same notion.)

B. The Spherical Big Bang Model

Next, here is a brief summary of what we know, or at least think we know, about the universe:

1. Elastic space in the local vicinity is expanding according to the Hubble parameter, H, whose most likely value is probably between 50 km/sec/mpc and 75 km/sec/mpc. (Mpc = megaparsec or 3.26 million light years.) The "age" of the universe is the length of time it has been expanding, and the range of H implies an age for the universe between 13 and 20 billion years.
2. Note the way H is defined: km/sec/mpc. Two terms - km and mpc - are length; they cancel each other out, leaving a dimensionless constant. Thus H = constant/sec and, if we assume H remains the same over the billions of years of expansion, the age of the universe is simply 1/H (in seconds). Here H indicates how long ago any two points, now separating, were together.
3. The Friedmann models assumed that the rate of expansion was decelerating (and how much depended on the density). If H was higher in the past, this implied a younger universe than the rate of expansion in the current era would suggest. Then, surprisingly, in early 1998 it was announced that the luminosity of supernovae in distant galaxies indicated that the expansion was accelerating, not decelerating. If H was lower in the past, then we inhabit an older universe.
4. Light travels at 186,000 miles/sec (or 300,000 km/sec) and is designated by "c".

Since the universe is expanding and light travels at a finite velocity, we actually look back in time, into a significantly smaller universe that has not had as much time to expand as it has in the current era. If the red shift z of a receding object occurs in a spherical universe, the relative size of the universe in that earlier epoch, compared to its size where we are, is 1/(1 + z).

But the universe cannot appear to shrink in size indefinitely: the theoretical limit of this shrinkage is a collapsed dimensionless "singularity" which we look toward in all directions somewhere between 13 and 20 billion light years away, depending on H and how it changes. The singularity has an infinite red shift and so the universe must have zero size according to the 1/(1 + z) ratio. If H = constant through all time, the distance to the singularity is c/H and the lookback time is 1/H seconds.

The universe certainly looks "spherical" in V³ and there is plenty of evidence supporting the orthogonally flat Friedmann model. In fact, some of the most convincing evidence is very recent.²

C. The Standard Model Modified Into Riemann's "Closed Cosmic Hypersphere"

Although the cosmological community is rather satisfied with a presumably flat Friedmann universe in V³ and sees no compelling reason to place an alternate model on the table, is there a way to derive Riemann's hypersphere in S³/V⁴ which includes an "uncompactified" fourth spatial dimension? If this is approached primarily as a problem in topology, there's an easy way to do it.

To begin, suppose we have a spherical surface defined by:

The t in parentheses is time, which begins when the surface starts to expand and indicates that the radius (r) and the other values (x,y,z) may change as we look or travel around the spherical surface.

We'll adopt the following conventions/assumptions: (1) a position on the surface in (x,y,z,t) will be designated a "locus" (plural: loci) whose movement is restricted exclusively to the hidden radius, r; (2) light is confined to the surface; (3) the observed rate of expansion along the S² surface is defined by H, the Hubble parameter (and H obviously seems higher along the multiply-connected S² surface than it is along the hidden radius in V³); and (4) r >>> 0, which means that r, the hidden radius, is so large that it's impossible to detect the positive global curvature in the local vicinity.

One thing we do know is that we look back in time for 1/H seconds toward the collapsed receding site at a distance of c/H away, perhaps via many circuits around the surface (though we don't know that's what we're doing). But remember: as we look back in time along the surface each locus, including our own, moves toward the central location along an unsuspected radius at right angles to the surface. That's certainly not the impression we get because we surface inhabitants are convinced that all loci are constrained to movement along the two spatial dimensions we easily recognize, presumably in V².

If we could see all the way to the collapsed receding site (which we can't because it's dimensionless and receding at the speed of light) our line of sight and all surface loci, including our own, would arrive at this dimensionless location at the same instant, when t = 0 at 1/H seconds ago.

Given that we know we're going to be looking toward the collapsed site receding at the speed of light at 1/H seconds ago (regardless of what the structure is) and that there's not the slightest clue indicating anything other than just two orthogonal spatial dimensions apparently in V², what assumption would allow us to deduce that an undetected third spatial dimension actually exists? That the surface really has positive global curvature in S² caused by the presence of a "hidden" third spatial dimension?

If the elastic structure is expanding according to the Hubble parameter H, the following "extra dimension theorem" (EDT) provides the answer:

If all loci when t > 0 also exist at the farther collapsed site when t = 0 at 1/H seconds ago that we necessarily look toward, then what appears at first to be an orthogonal Vⁿ structure is actually a positively curved Sⁿ structure with an undetected spatial dimension.

No matter how many loci we look past as we look out along the surface at the speed of light, if those loci also exist at the collapsed receding site (which is the farthest defined position we can look toward at 1/H seconds ago) those loci must approach the collapsed position along a shorter route than the surface our line of sight is confined to, along the radius of an undetected spatial dimension.

What the EDT tells us in this instance is that we have the option to define the collapsed receding site in a certain way that let's us know we're on a globally curved S² spherical surface that we're looking around at least once rather than an orthogonally flat V² plane. And if it's only once, then the route toward this collapsed site has the generic character of a spiral, that is, looking along a spherical surface which collapses to zero size when we look around it precisely one time.

An intriguing deduction of the EDT is that looking toward the collapsed receding site is equivalent to looking toward our "back side", the proper description of encountering our locus again along a route that seems "straight", even if our locus is defined to exist at this site with all other loci.

Next question: within the framework of the conventional big bang model how do we get a universe with multiply-connected space and whose global structure in S³/V⁴ is described by:

The assumptions/conditions following Equation 1 are applicable here, with the following exceptions: (1) a locus on the surface is now designated by (x,y,z,w,t) but is still restricted to exclusive movement along a radius we don't notice; (2) we can't see all the way back to the collapsed singularity for reasons already mentioned, plus the fact that the early universe is opaque to light transmission for the first 300,000 years or so; and (3) the lookback time to the singularity is not exactly 1/H seconds, since we need to make an adjustment for the changing values of H over time. For instance, if the expansion is accelerating, the lookback time exceeds 1/H seconds.

The way to get Riemann's hypersphere and an "uncompactified" fourth spatial dimension is simply to reapply the EDT theorem and assume that all loci when t > 0 in the observable universe also exist at the singularity when t = 0, at the beginning of time. To reiterate: any two loci don't separate until t > 0; at t = 0 those two loci are always together, as mentioned in the second item of Section B.

This line of reasoning in topology implies that a fourth spatial dimension, suggested by both Riemann and Kaluza, is actually implicit in the Hubble parameter. We always look toward the singularity at the receding t = 0 position which necessarily includes all loci until t > 0, thus leading us to conclude (via the EDT) that a fourth spatial dimension exists in the "uncompactified" mode of S³/V⁴.

Assuming that we look around a Riemannian hypersphere just once as we look toward the singularity, this path has the same characteristic spiral route referred to on the spherical surface, because whatever happens in S² also happens in S³. And the Hubble parameter H will seem higher if measured along the multiply-connected S³ surface than it is along the hidden radius in V⁴, for the same reason given earlier. Likewise, in this scenario looking toward the singularity is equivalent to looking toward our "back side", because that's what happens on the embedded spherical surface in S².

Although a V³ spherical universe has always been favored, the EDT shows a way to define the singularity which yields Riemann's four-space hypersphere (five-space, with time) in S³/V⁴. Its characteristics were mentioned in Section A: finite but unbounded, multiply-connected, with space that curves back on itself, and it includes the Randall/Sundrum "uncompactified" fourth spatial dimension.

Footnotes:

1. L. Randall and R. Sundrum, Phys. Rev. Lett., #83, 4690 (6 December 1999)

2. M.White et al, Boomerang Results, 11 pages, 3 figures, submitted to ApJL on April 27, 1999