Hyperspheres, Hyperspace,

and the Fourth Spatial Dimension

 
 

Subtitle: A Look at the Universe as a Higher-Dimensional Brane World

 

A Collection of Links, Table of Contents, and Condensed Arguments

(The last link is for those in a hurry.)

by

Michael R. Feltz

 

Updates in May, 2005

 
 
 

Introduction



John Adams:

            All that part of Creation that lies within our observation is liable to change.



Ortega y Gasset:

            Our firmest convictions are apt to be the most suspect, they mark our limitations and our bounds... Obstinately to insist on carrying on within the same familiar horizon betrays weakness and a decline of vital energies.



J. B. S. Haldane:

            The universe is not only stranger than we imagine, it is stranger than we can imagine.




            Is space curved?

           The three well known Friedmann models provide one way to answer the question about curvature: whether space in the universe is curved or not depends on the density (so much matter per unit volume). If the density is equal to a certain critical value - the estimates are typically equivalent to one or two hydrogen atoms per cubic meter - there is no curvature: the three spatial dimensions we clearly recognize remain orthogonal at all eras; this is the Euclidean "flat" model.

            Should the density be less than critical density, the universe is "open", and its (infinite) space is negatively curved. A slice of this model is saddle-shaped, and the sum of the three angles of a triangle on this surface is a little less than 180º . On the other hand, if the density is greater than the critical value, the universe is "closed" and its (finite) space is positively curved. A comparable slice resembles the surface of an expanding sphere, and the sum of three angles of a triangle here is a little greater than 180º.

            The three Friedmann models have had enormous appeal since the 1930s because they are so easy to understand. Also the trio seems to cover all possibilities in that they predict the ultimate fate of the universe: it will expand forever only if the density is less than or equal to the critical value. Otherwise the days of an expanding universe are numbered because this high density model will expand only to certain maximum size and then contract into a "big crunch" - "big bang" in reverse - at some future date at least tens of billions of years from today.

            For the last few decades, the preferred model has been Friedmann's "flat" model with critical density - space is also infinite in this model, but just barely - and several recent probes seem to provide ample evidence favoring this model.

            On the other hand, several researchers caution "not so fast", as not all the latest data is consistent with this model. For instance, the title of this two-page pdf file is "Precision Cosmology? Not Just Yet..." The text includes a sprinkle of phrases like "...important issues remain", "there remains room for radical alternatives", "...the underlying model may be wrong", and "...even greater technological advances...will uncover further cosmological surprises".

            There seems to be a strange dichotomy these days in that those who favor the conventional flat universe point with some justification to data from the latest satellite probes which overwhelmingly support this model, while those who have been skeptical all along argue that not all the results in other arenas are consistent with the current favorite. The estimates of key cosmological parameters by different methods do not overlap as well as they ought to, if the conventional flat model is correct.

[From time to time I'll insert comments in the format like this which can be considered metaphorically a "tributary" to the "mainstream" or an addition to older text. These are the kind of changes that would occur in a book that goes through several editions.

The flat universe lies precisely at the boundary between the closed and open models. By the time we consider the customary margins of error, any parameter that seems to confirm the flat universe does not necessarily rule out the other two models. Curiously, many of the "best guesses" these days seem to tilt slightly toward the closed model, which means the universe is more likely to have positive curvature than negative curvature, if it has any curvature at all. Nevertheless, none of the models can be accepted with complete confidence yet, and the warning about "precision cosmology" is probably more deserved than current researchers are willing to concede.

The "flat" Friedmann universe without any curvature is still the favorite, but with a somewhat narrower margin than it had, say, ten years ago. For example, consider the following quote from an article ("Cosmic Soccer Ball? Theory Already Takes Sharp Kicks") written by Dennis Overbye in the 2003 October 9 New York Times:

The evidence for and against a finite universe resides in a radio map of the baby universe produced last February by a NASA satellite, the Wilkinson Microwave Anisotropy Probe.* It shows that 400,000 years after the Big Bang, the event in which space and time emerged, the universe was laced with faint waves and ripples, which are the origin of modern galaxies and other cosmic structures. In an infinite universe, according to theory, waves of all size should appear in the sky, but in the Wilkinson data there was a cutoff: no waves larger than about 60 degrees across appeared in the sky.

If the universe were a musical instrument, it would be inexplicably missing its low notes, perhaps, some cosmologists have suggested, because it is too small to play them. The universe is finite rather than infinite, they speculate. Like a violin that cannot produce deep cello notes, the universe cannot produce waves larger than itself.

* Link obviously not in the original hard copy. Near the bottom of this file is a link titled "Results Promote Winning Theories", which provides the latest cosmological parameters according to WMAP.

While the results of the Wilkinson Microwave Anistropy Probe are the latest cosmological "gospel" and a host of articles have been confidently published about them, a few researchers are starting to comment that different "notes" (a different type of power spectrum) seem to appear in the northern and southern hemispheres. And this recent article by Frod Hansen et al. is a challenge to the long held assumption of isotropy (things should look the same in all directions), a characteristic the universe is supposed to have.

And then there's this abstract by Richard Lieu and Jonathan Mittaz which points out that what we observe in the microwave background is distorted by foreground galaxies. (Scroll down to the diagrams near the bottom of the file.) When they make allowances for this distortion, the universe seems to be "supercritical", which is another way of saying "closed" or "positively curved".

This diagram is at the site of one of the teams analyzing the behavior of supernovae. If the results fall along the diagonal line, the universe is flat. The majority of researchers consider the proximity of the CMB (cosmic microwave background) to the diagonal line to be rather convincing evidence for a flat universe. But the Lieu/Mittaz paper above suggests that the CMB data are probably a little higher than shown on the diagram, and therefore further into the closed region.

Also, while the supernovae in this diagram do not rule out any model, they seem to favor a closed universe. However, the values quoted most often are those at the intersection of the supernovae with the CMB, rather than the "mean" (i.e., at the center of the margin of error for the supernovae).]


            Another reason for renewed interest in a positively curved universe is that in recent years several theoretical physicists have begun to suspect that gravity might be affected by at least one more "large" spatial dimension, although there is considerable divergence among the physicists about how "large" that dimension is:

  1. Large Finite Fourth Spatial Dimension: Two physicists, Lisa Randall and Raman Sundrum, are the modern movers of extra-dimensional theory, as explained in this article. A fourth spatial dimension in the Randall-Sundrum model is finite: scroll down about midway for Professor Randall's clarification in this file. Although not mentioned in either link, the maximum size of a finite large fourth spatial dimension is assumed to be around one millimeter (and probably much smaller than that, though still many orders of magnitude larger than has been hypothesized in string theory).

  2. Large Infinite Fourth Spatial Dimension: Other physicists suggest that the fourth spatial dimension might be infinitely large, and a typical example is this abstract on "brane world cosmology". And if that extra dimension exists in this manner, there may be discernible astrophysical implications.
[Both abstracts above refer to "embedding" (which I will discuss generically in the first essay). The traditional type of "embedding" referred to most often is a spherical surface "embedded" in higher-dimensional space. Surprisingly, this model is easy to derive via a simple argument in topology, and its derivation suggests a way to prove that this type of "embedding" actually occurs.]

In the first version each point in space is surrounded by a 4D bubble with a maximum radius of no more than one millimeter. If the second idea with one more large spatial dimension is correct, then the physicists have backed into an alternate model in the cosmological inventory known as the "standard" Friedmann-Robertson-Walker (FRW) metric. But the FRW metric is what a cosmologist calls it; a topologist would refer to this model as a "Riemannian hypersphere", while a theoretical physicist would consider it to be one of the variations of a "brane world".

            But no matter what name is used, this model incorporates a fourth spatial dimension which, while not having the same character as the first three spatial dimensions, exists in a way which causes the three obvious spatial dimensions to become globally and positively curved. If physical phenomena do not behave the way they're supposed to in our "brane world", such anomalies can be explained by assuming that some of the behavior occurs in higher-dimensional space (called "bulk").

            To repeat: if a fourth spatial dimension exists, it cannot have the same character as the first three spatial dimensions or our physical world would not behave the way it does. Gravity and light intensity, which vary as the inverse square of the distance in three-dimensional (3D) space, will change as the inverse cube of the distance in four-dimensional (4D) space if the fourth spatial dimension behaves exactly like the first three.

            Nothing like this has been observed, and so we can safely assume that a fourth spatial dimension does not have a character comparable to the (x,y,z) dimensions with which we're familiar.

            So what alternate modes of existence can a fourth spatial dimension assume?

            For decades, only two variations were thought possible. In recent years, a third possibility has been added to the list, and there has always been a fourth way of considering this possibility in a formal mathematical context:

  1. A very small fourth spatial dimension.

While the foundations for non-Euclidean geometry were laid by several excellent mathematicians in the 19th century - such as Gauss, Bolyai, Riemann, Lobachevsky, Clifford, Minkowski, and Poincaré - the possibility of extra-dimensional space received the largest impetus in the early 20th century through a brilliant insight of an unknown Prussian mathematician named Theodor Kaluza. In 1919 Kaluza wrote a letter to Albert Einstein postulating that, should there be a fourth spatial dimension, then Einstein's theory of gravity and Maxwell's equations for electromagnetic radiation could be combined.

[The Kaluza link above mentions this as the fifth dimension, after three spatial dimensions and one time dimension. Today Kaluza's idea is usually referred to as the fourth spatial dimension.]

It took Einstein until 1921 before he replied to Kaluza's remarkable suggestion with a belated agreement.

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

A Swedish physicist, Oskar Klein, had also been thinking independently about a fourth spatial dimension during the 1920s. When he became aware of Kaluza's little known work on the same topic, Klein proposed a solution still accepted in modern physics: a fourth spatial dimension did exist, but in packets of space too small for us to detect; it was "curled up small". A "compactified" unit of such space is usually called the Kaluza-Klein bottle, and it is the basis for today's string theory in physics.

Today many physicists believe that the Kaluza-Klein bottle contains more dimensions than just the fourth one; usually ten spatial dimensions are assumed (three big dimensions and seven small ones in the Kaluza-Klein bottle), sometimes even twenty six!

[There are four forces in physics: gravity, magnetism , and strong and weak nuclear forces. One of the most enduring questions in physics is why the latter two are so much stronger than the first two. (Michelle Thaller mentions in this article in the Christian Science Monitor that the force keeping two protons together is 1040 time stronger than the force of gravity between them.) The incredible difference may be due to the fact that several more dimensions exist at the infinitesimally small distances over which the strong and weak nuclear forces operate, which is one of the better reasons for string theory.]

 

  1. A very large fourth spatial dimension.

An alternative mode for a fourth spatial dimension, much less emphasized by the physicists on the string theory bandwagon, is described in a typical undergraduate topology course. To understand its nature in topology, consider a spherical surface: although its curved spherical surface always seems flat in the local vicinity, there is a third less obvious spatial dimension which causes the two easily recognized spatial dimensions to lose their apparent orthogonal relationship over larger areas of the surface. The real motion of a position on an expanding spherical surface is confined to a radius at right angles to the surface from the inaccessible center. But to the inhabitants on its surface, the positions seem constrained to move along only the two spatial dimensions they recognize without trouble.

The same thing happens in Riemann's four-space hypersphere, but in another dimension. Three spatial dimensions are obvious to its inhabitants and seem to exist at right angles to each other. Not noticed in local observations by its inhabitants, a "hidden" fourth spatial dimension causes the three clearly recognizable spatial dimensions to lose their apparent orthogonal relationship eventually. On a global scale the four-space hypersphere has positively curved space, although it doesn't seem to be curved in the local vicinity any more than the lower-dimensional spherical surface does.

When a "finite but unbounded" universe with a fourth spatial dimension is referred in the popular literature, it is this model - the one studied in a typical topology course - that is referred to, and I will tell you how to derive this model within the framework of the "big bang theory". As there is some indication of a positively curved universe these days, perhaps this model might be the best "fit" for the latest data.

[The two most likely models of a "closed" or "positively curved" universe are (1) the Friedmann universe with greater than critical density and (2) the FRW metric with a large fourth spatial dimension. Today any clues that seem to favor a positively curved universe are almost always interpreted with respect to the closed Friedmann model. But perhaps they are hints for the FRW metric, whose existence depends on a simple assumption in topology we can make about the singularity (i.e., the universe as it was at the beginning of space and time).

Since the most reasonable assumptions associated with each model indicate that the FRW metric has a higher degree of curvature than the closed Friedmann model, surprisingly high estimates of the density - such as those currently derived from supernovae - probably tend to favor the FRW metric rather than the closed Friedmann universe.]

 

  1. A "medium" (or "medium large") fourth spatial dimension.

As mentioned above, a third possibility has emerged in recent years by theoretical physicists concerned with gravity. The way gravity behaves on a large scale is well known and is assumed to behave as the "reciprocal squared" of the distance.

But does it behave this way at all distances and for less massive bodies, such as subatomic particles?

For instance, two very small objects - say, two spheres one-tenth the size of a BB pellet - might not necessarily behave the same way because they are so light that physicists don't know for certain that the "reciprocal squared" of the distance works the same way as it does for large, heavy objects. While most of the gravitational behavior occurs in the three-dimensional space with which we're familiar (the "brane", shortened from "membrane"), perhaps a small part of it occurs in underlying higher-dimensional space ("bulk"). And we could infer the existence of "bulk" by any change in the inverse square rule: that is, the exponent might not be exactly two in our "brane world".

[But the physicists concerned with gravity don't use "medium" or "medium large"; they use "large" (either finitely or infinitely "large"). The important thing to remember is that when a physicist uses "large" in this context, it is not necessarily the same way a topologist uses "large" to describe the fourth spatial dimension in a Riemannian hypersphere. The topologist always supposes there is no limit to the geometric size of the fourth spatial dimension; unless it's stated otherwise, the assumption is that the additional dimension is infinite. But many physicists put an upper limit on the size of a fourth spatial dimension, typically about one millimeter. Unfortunately, the same adjective - "large" - is used to describe the fourth spatial dimension in both contexts.]

 

  1. The Fourth Spatial Dimension as a Mathematical Artifice.

When Copernicus suggested that the sun, rather then the earth, was the center of the solar system, it was greeted with immense skepticism at first. However, even the skeptics agreed that it was acceptable to use the theory if it simplified the mathematical calculations of the planetary orbits. In other words, the heliocentric system could be used as a "mathematical artifice", even if they didn't think it was the right model.

In the same vein, many physicists and cosmologists don't think that a fourth spatial dimension exists as a geometric reality, at least, not in the "large" realm. But ever since Kaluza discovered that it makes sense to assume it's there somehow, they concede that a fourth spatial dimension exists only as a convenient "mathematical artifice", that is, another term included in some equations that represents only an imaginary "paper" dimension.

 

            It is interesting to note that Kaluza did not at first attach a "size" to the hypothetical extra dimension. Although the most popular cosmological model at the time was a Riemannian hypersphere and Kaluza's idea would have been consistent with a "large" dimension in that model, by the mid 1920s it appears the trend was toward a very "small" extra dimension.

            A little later in the early 1930s Albert Einstein and Willem de Sitter replaced the closed static (finite but unbounded) Riemannian universe with an open, infinite, expanding model, as explained in this article back in 1932. (Scroll down just a little until you get to "Einstein and De Sitter Return to Euclidean Idea of Cosmos".)

            But an intriguing historical question is what model would have emerged if the astronomical community had assumed Kaluza's insight was actually a large infinite fourth spatial dimension that would have confirmed Einstein's early intuition that the universe was most likely to be a "closed cosmic hypersphere" of the Riemannian variety?

            As amazing as it sounds after the better part of a century, this is still an open question and it is a question I will try to answer in these essays.

            I will not pretend there is anything new in this 4D structure; in fact, the four-space hypersphere was first described by the German mathematician, Georg Bernhard Riemann in his doctoral thesis in 1851. His mentor was the eminent Johann Karl Friedrich Gauss, unquestionably the greatest mathematician in first half of the nineteenth century and apparently the first scholar in the scientific era to suspect that the structure of the universe might not be Euclidean.

[Incidentally, the first Freudenthal reference in the Riemann link above - it's not quite halfway to the end of the file - describes what I am going to do in this series of essays:

It [Riemannian space] possesses shortest lines, now called geodesics, which resemble ordinary straight lines. In fact, at first approximation in a geodesic coordinate system such a metric is flat Euclidean, in the same way that a curved surface up to higher-order terms looks like its tangent plane. Beings living on the surface may discover the curvature of their world and compute it at any point as a consequence of observed deviations from Pythagoras' theorem.

This sets the stage for what I am about to discuss. Are there sufficient anomalies in deep space that might lead us to wonder if the conventional flat model is the right one? Are there "observed deviations" from what we expect in the most popular model?

For example, it has always been acknowledged theoretically that the universe might be "spherical", that is, on a global scale it behaves the same way a spherical surface does: it has positive curvature. All right, if that is possible, does it make sense to assume we can look as far as, and even beyond, the opposite side of a spherical surface? And what would happen if we were able to do this?

These two questions have very simple answers which, despite all the sophistication of modern cosmology and topology, seem to have been largely ignored. If you are a little patient, I think you will eventually agree with the French cosmologist, Jean-Pierre Luminet, who has written:

"...the subject of cosmic topology always remained confidential and widely ignored by the community of cosmologists."

Amen!

For decades, several terms and phrases - like "a finite but unbounded universe" and "space that curves back on itself" - have been bandied about with hardly any discussion or analysis about just exactly what they mean topologically and what assumptions are necessary to get a universe with these characteristics. While no one has a complete grasp of all the historical literature on any given topic, from what I've been able to gather over the years, this particular model - as simple as it is - has been underanalyzed and underexplored, perhaps because the easiest way to look at it is through the "widely ignored" lens of topology.]

            Riemann's hypersphere was the groundbreaking formal description of finite but unbounded space in the middle of the nineteenth century, and Einstein used Riemann's idea to derive the first "modern" model of the universe: a "static" - that is, non-expanding - hypersphere in the early part of the twentieth century. In fact, this model reigned supreme until Edwin Hubble announced that the universe was expanding in 1929. Then it fell by the wayside in the 1930s, when the conventional cosmological wisdom presumed that one of Friedmann's three variations was the best "fit" for an expanding universe.

            But although Riemann's hypersphere was considered a realistic model of the universe roughly a century ago, the following list of questions remains surprisingly unanswered to this day:

You'd think that these questions would have been answered a long time ago and that the answers would appear frequently in FAQ files (Frequently Asked Questions) or in textbooks.

            Unfortunately you'd be wrong; this is not the case.

            The answers are not especially complicated, but they involve two disciplines: cosmology and ("widely ignored") topology. Since I'm assuming you're somewhat familiar with the big bang model but don't know that much about hyperspheres, Essay #1 describes the 4D (four-dimensional) hypersphere by showing how this structure is related to its lower-dimensional Euclidean cousins: the 2D (two-dimensional) circle and the 3D (three-dimensional) spherical surface. This essay also shows an easy way to visualize the four-space hypersphere by using a subtle trick in 3D Euclidean geometry.

            Next Essay #2 discusses how a very interesting position called an "antipode" will affect our observations of discrete, luminous objects near it. If we were at the "north pole" of a spherical surface and looked out far enough, eventually we would encounter a "south pole" in virtually every direction at the same distance. Similarly, an "antipode" is on the opposite side of one's position on the hypersphere. If an unsuspected antipode were actually within our cosmic viewing distance, it would cause very odd observations in what we ordinarily assume is Euclidean space.

            Without stretching credulity too far, several recent observations in deep space actually hint that an antipode might be close enough to affect what we see. Although a density induced antipode could exist within our Hubble horizon in the closed Friedmann model (provided the density is much higher than we think it is), I believe the more likely explanation for an antipode close enough to affect what we see is due to the existence of that long suspected, but ever elusive, "large" fourth spatial dimension in the FRW metric. So we have a topologically induced antipode (which means it's there because of an assumption we can make about the singularity).

            After providing a quick summary of the big bang model and providing a few links which provide the current estimates of the most critical parameters in that model, Essay #3 answers three questions:

  1. Given what we know about the expanding universe, what assumption is necessary to get the "closed cosmic hypersphere" with a (large) fourth spatial dimension?

  2. Is that assumption realistic?

  3. Since we'll encounter the antipode at least once in this model, at what range can we predict it's most likely to occur?

            Strictly from a topological viewpoint, I believe there is a persuasive argument given in Essay #3 favoring a universe with an "ultra-uncompactified" fourth spatial dimension, that is, a fourth spatial dimension that exists in a way that causes the three obvious spatial dimensions to lose their apparent orthogonal relationship on a large scale. The basic principle is expressed via a very succinct - and as far as I know original - extra dimension theorem.

[The key question is whether or not the space in which our geodesics - "light paths" - occur is the same space in which the global expansion of the universe occurs. There is a reasonable argument in topology which shows how the two spaces can be different, which is a major dent in the armor of Friedmann space (because it assumes that both types of space are the same). While the argument is presented in Essay #3, two "stand alone" short essays are available.]

            The next five essays - although Essay #4 has two parts - explore the possibility that several unusual phenomena in deep space might be suggesting a non-conventional (hyperspheric) model.

            Assuming you're a disciple of convention, I can't blame you for being skeptical because a concept like this is usually relegated to a minor footnote or to some obscure appendix in a typical textbook on cosmology. But if you look at the anomalies and statistics fairly and start to notice the ever longer litany of head scratching about a variety of oddities in deep space, you'll realize that this "alternate cosmology" goes much further than you think it does.

            And do not be surprised if you ask the same pair of questions I've asked myself many times: Are the numbers trying to tell us something? Or are we trying to read something into the numbers that isn't there?

            Finally, there are three cautions:

            The original 4D Riemannian hypersphere is now over a century and a half old. Whatever the current cosmological fad, fancy, or gospel happens to be, this particular version of a "finite but unbounded" universe has never disappeared entirely. (Like your typical mother-in-law, it keeps hanging around!) Nevertheless, it doesn't look like anyone has examined this model recently to see whether or not the latest data are consistent with it.

            So these essays will try to improve your understanding of this vaguely suggested and often hypothesized model, and then take a survey of the latest observations and see how much sense they make in this one (compared to the popular flat Friedmann model).

            Thanks for your interest, good luck, and take one of the following links to continue... 

 
 
 

Table of Contents, Links to Other Sites, Condensed Arguments 

 
 
 

(E-mail me at any time with comments or questions about this or any of the other essays.)