# Is the universe a four-dimensional sphere

At the sight of the clear night sky, the infinity of space seems to open up: Stars and galaxies, no matter where and how far you look. No matter how big our telescope is - the universe always turns out to be filled with luminous objects, even where one could only perceive darkness with smaller instruments. In reality, the size of the observable space is limited by the age of the universe and the speed of light. This means that so far no light has been able to reach us from even more distant areas of the cosmos because it has not yet had enough time to penetrate the vastness of space.

But shouldn't we, if we only wait long enough, be able to discover more and more distant galaxies and other new phenomena? Not necessarily. The cosmos could well be finite and only simulate its infinity for us like in a mirror cabinet - for example, by the light winding around the room one or more times so that the image of each galaxy is simulated several times. Our Milky Way System would be no exception. Somewhere in the firmament there could even be several copies of the earth in earlier stages of development hidden. Over time, astronomers could see the galaxies evolve and discover new reflections. Eventually, after all, they would no longer find any new areas of space; they would have already seen it all.

The question of whether the universe is finite or infinite is one of the oldest in philosophy - and it is by no means already decided in favor of infinity, as some books lead us to believe, because they draw an inadmissible conclusion from the general theory of relativity. According to Albert Einstein's theory, space is a dynamic medium that can be curved in three different ways, depending on the distribution of mass and energy in it. Since we ourselves are embedded in the room, we cannot see this curvature directly, but perceive it as gravitational attraction and deflection of light radiation. To find out which of the three possible geometries our universe has, the astronomers measured the density of matter and energy in the cosmos. According to current knowledge, this size is too small for the space to be closed and thus to have a "spherical" geometry. Consequently the space must either have the usual "Euclidean" geometry like a plane, or a "hyperbolic" geometry like a saddle surface (picture on page 52). At first glance, such a universe appears to be infinitely extensive.

One problem with this conclusion is that the universe could be spherical, but so vast that the observable parts appeared Euclidean to us - just as a small piece of the surface of the earth appears flat. It is even more important that the theory of relativity is a purely local theory. It predicts the curvature of space - i.e. the geometry - of a small section of space based on the mass and energy contained therein. But neither the theory of relativity nor the usual cosmological observations give any indication of how these small pieces of space contribute to the entire shape of the universe - its topology. The three standard geometries are consistent with many different topologies. For example, relativity theory would describe a torus and a plane with the same equations, although a torus is finite and a plane is infinite. In order to determine the topology of the universe, one needs a physical understanding that goes beyond the theory of relativity.

**Satisfied with the finite**

It is commonly assumed that the universe is "simply connected" like a plane. This means that there is only one direct path from a light source to an observer. A simply connected Euclidean or hyperbolic universe would indeed be infinite. But the universe could also be "multi-connected" like a torus, and then there would be many different such paths. An observer would then see many images of one and the same galaxy and could easily mistake them for different star systems in an infinite space - just as the visitor to a cabinet of mirrors is led to believe the illusion of a large crowd.

A multiply connected space is more than a purely mathematical construct. It is even favored by some of the currently favored theories for unifying the fundamental forces of nature and does not contradict what has been observed. During the last few years, research has increasingly focused on the topology of the cosmos. New observations may soon provide an unequivocal answer.

Many cosmologists expect a finite universe. Part of this may simply be due to convenience: Humans are more likely to imagine a finite universe than an infinite one. There are, however, two scientific reasons for this. The first is based on a thought experiment devised by the English natural scientist Isaac Newton (1643-1727), which the Irish philosopher and theologian George Berkeley (1685-1753) and the Austrian physicist and philosopher Ernst Mach (1838-1916) later refined. Newton thought about the cause of the inertia of the masses and imagined two buckets, each half filled with water (see "Newton: A natural philosopher and the system of the worlds", Spectrum of Science, Biography 1/1999, p. 59 ). One bucket is at rest and the other is rotating quickly around its vertical axis. While the water surface is flat in the first bucket, it takes on a concave shape in the second. Why?

The unbiased answer points to centrifugal force. But how does the second bucket know that it is rotating? How is the inertial frame of reference defined relative to which the second bucket rotates and the first remains at rest? Berkeley and Mach's answer was that all the masses in the universe together define the frame of reference. The first bucket is at rest with respect to the distant galaxies, so the water surface remains level. The second bucket rotates relative to these masses, so the surface of the water in it is concave. If there were no distant galaxies, there would be no reason to prefer one frame of reference to another. The surface in both buckets would then have to remain level. In short: there would be no inertial forces. Mach concluded from this that the inertia of a body must be proportional to the total mass in the universe. An infinite universe would result in infinite inertia. Movement would not be possible.

Modern works from quantum cosmology, which attempt to describe how the universe spontaneously emerged from nowhere, point in the same direction. Some of these theories predict that the smaller its total volume, the more likely a universe can arise from a quantum fluctuation. Accordingly, an infinite cosmos could not have formed at all; the probability of this would have been zero (see "Quantum Cosmology and the Origin of the Universe", by Jonathan J. Halliwell, Spectrum of Science, February 1992, p. 50). Put simply, this is because no quantum fluctuation could produce an infinite amount of energy that would be required for an infinite universe.

Historically, the idea of a finite universe has not remained undisputed. Aristotle (384-322 BC) considered the cosmos to be finite because a limit was necessary to define an absolute frame of reference that was important in his view of the world. But his critics wondered what was behind it, because every boundary separates two sides. So why not understand the universe as the totality of the before and the behind? The German mathematician Bernhard Riemann (1826-1866) finally proposed a hypersphere as a model for the cosmos: the three-dimensional surface of a four-dimensional sphere. Like the surface of an ordinary sphere, the hypersphere is finite, but yet limitless; it was the first example of a room with such properties.

One might be tempted to ask what is outside of the universe. With this one would assume that physical reality is ultimately a Euclidean space of some dimension. Applied to the image of the hypersphere, this meant that it had to be embedded in a four-dimensional Euclidean space so that it could be viewed from the outside.

By the end of the 19th century, mathematicians had discovered a whole series of finite spaces without boundaries. The German astronomer Karl Schwarzschild (1873-1916) drew the attention of his colleagues to it in 1900. In an addendum to the article "About the permissible degree of curvature of space" in the "Vierteljahrsschrift der Astronomische Gesellschaft", he challenged the readers:

"Imagine as the result of an enormously expanded astronomical experience that the whole world consists of innumerable identical repetitions of our Milky Way system, that infinite space can be split into nothing but cubes, each of which would contain a Milky Way system that is absolutely identical to ours. Then we would actually stop at the assumption of an infinite number of identical repetitions of the same whole of the world? ... We would much rather turn to the view that these repetitions are only apparent, that in reality the space has such peculiar relationships that we, by looking at the relevant cube leave on one side, come back in on the opposite side by walking straight out. "

Schwarzschild's example shows how one can mentally construct a torus from Euclidean space. In two dimensions you start with a square and equate the opposite edges. It's like with many video games: a spaceship that leaves the screen on the right side reappears on the left. Apart from this assignment of the edges, the space is unchanged: All the familiar laws of Euclidean geometry still apply: For example, the sum of the angles in a triangle is 180 degrees, and parallel rays of light never intersect. Residents of such a room would consider it to be infinitely extensive, because they could see as far as they wanted. They would only recognize the shape of their two-dimensional ring universe (called two-torus for short) when they hit the same objects again with a spaceship after crossing the imaginary connecting line (picture on the left). In an analogous way, a toric space can be obtained in three dimensions (a three-torus) by equating the opposite side faces of a cube and mentally gluing them together.

When Einstein presented the first relativistic model of the universe in 1917, he chose Riemann's hypersphere as its shape on a large scale. At that time there was a lot of discussion about the topology of space. The Russian mathematician Alexander Friedmann (1888-1925) generalized Einstein's equations so that an expanding universe could also be described with hyperbolic geometry. The Friedmann equations are still part of the tools of the trade of cosmologists today. Their author emphasized that they apply to both a finite and the usually considered infinite universe - this is all the more remarkable since no examples of finite hyperbolic spaces were known at the time.

Of all aspects of cosmic topology, perhaps the hardest thing to understand is how a hyperbolic space can be finite. For the sake of simplicity, let's first consider a two-dimensional universe. Imitate the construction of a two-torus, but start with a hyperbolic surface. Cut out a regular octagon and set opposite edges equal, so that every path that leaves the octagon over one edge re-enters the opposite one. Alternatively, one could also think of an octagonal screen for a corresponding computer game (picture on page 54). You get a multi-connected universe, topologically equivalent to a pretzel with two holes. An observer in the center of the octagon would see the next pictures of himself in eight different directions. This gave the illusion of infinite hyperbolic space when in reality this universe would be finite. Something similar - albeit much more difficult to illustrate - can be constructed in three dimensions. To do this, a "solid" polyhedron is cut out of a three-dimensional hyperbolic space and opposing side surfaces are glued in such a way that every path that runs out through one side surface re-enters through the corresponding point on the associated other side surface.

The angles of the octagon deserve careful attention. With a flat surface, the angles of a regular polygon do not depend on its size. All interior angles of a regular octagon of any size always have a value of 135 degrees. On a curved surface, however, the angles change with the size of the figure: they grow on a spherical surface, but shrink on a hyperbolic surface. The above construction requires an octagon of a certain size so that its angles are 45 degrees and its edges fit together when abstractly glued. The eight corners then meet at one point, and the sum of the angles at this point is 360 degrees. This subtlety explains why such a construction is not possible with a flat octagon. In Euclidean geometry, eight angles of 135 degrees each cannot meet in one point. The two-dimensional universe, which is created by equating opposite edges of an octagon, must therefore be hyperbolic. The topology thus defines the geometry.

The size of the polygon or polyhedron is measured relative to the only geometrically meaningful length scale for a space: the radius of curvature. A sphere, for example, can be of any size, but its surface will always be exactly 4PI times the square of the sphere's radius, i.e. 4PI square radians. In the same way, the size of a hyperbolic topology can be determined, for which a radius of curvature can also be defined.

The most compact hyperbolic topology that one of us (Weeks) discovered in 1985 comes about when you equate the surfaces of an 18-surface element in pairs; it has a volume of about 0.94 cubic radians. Other topologies can be constructed with larger polyhedra. The universe can also be measured in terms of radians. Various astronomical observations indicate that the density of matter in space is only about a third of what would be required for Euclidean space. Either the difference is due to a cosmological constant (see "New Buoyancy for an Accelerated Universe" by Lawrence M. Krauss, Spektrum der Wissenschaft, March 1999, p. 46), or the universe has a hyperbolic geometry with a radius of curvature of 18 billion Light years. In the latter case, the observable universe has a volume of 180 cubic radians - enough space for almost 200 weeks polyhedra. In other words, when the universe has the Weeks topology, its volume is only 0.5 percent of what it appears to be. With increasing expansion of the space, the proportions do not change, so neither does the topology.

In fact, almost all topologies require hyperbolic geometries. In two dimensions, a finite Euclidean space must either have the topology of a two-torus or that of a Klein bottle. In three dimensions there are only ten Euclidean possibilities: namely the three-torus and nine simple variations of the same (which are obtained if opposing surfaces are only glued together after a quarter turn or a reflection). In contrast, there are infinitely many possible topologies for a finite hyperbolic three-dimensional universe. Their rich mathematical structures are the subject of intensive research (see "The Mathematics of Three-Dimensional Manifolds" by William P. Thurston and Jeffrey R. Weeks, Spectrum of Science, September 1984, p. 110).

Because no method was available in the 1920s to directly determine the topology of the universe, cosmologists gradually lost interest in the subject. In the decades that followed, the authors of astronomical textbooks mostly quoted each other, claiming that the universe must either be a hypersphere, an infinite Euclidean space, or an infinite hyperbolic space. Other topologies were largely forgotten. Interest has only recently flared up again: in the last three years almost as many papers on cosmic topology have been published as in the previous 80 years. The most exciting thing about it is that the cosmologists could soon be able to determine the topology of the universe on the basis of observations.

The easiest way would be to study the arrangement of the galaxies. If they were to lie in a rectangular grid, with the images of the same galaxy repeated in equivalent grid points, then the universe would be a three-torus. Other patterns indicated more complicated topologies.The recognition of regularities would be made more difficult, however, by the fact that the images of one and the same galaxy would show them in different stages of development. The astronomers would have to find out despite changes in shape and spatial displacement to neighboring star systems that it is the same galaxy. Over the past 25 years, researchers such as Dmitri Sokoloff from Moscow State University, Viktor Shvartsman from the Soviet Academy of Sciences, J. Richard Gott III. from Princeton University and Helio V. Fagundes from the Institute for Theoretical Physics in São Paulo, but found no repeating images among the galaxies up to a billion light years from Earth.

Others - such as Boudewijn F. Roukema from the Inter-University Center for Astronomy and Astrophysics in Pune (India) - have tried the same with quasars, which can be observed from great distances because of their enormous brightness. The researchers identified all groupings of four or more quasars and checked whether any of them could be seen again elsewhere in the sky, just from a different perspective. In two cases this seems to be entirely possible, but this result is not statistically significant.

Roland Lehoucq and Marc Lachièze-Rey from the Service d'Astrophysique in Saclay (France) and one of us (Luminet) approached the problem differently. We have developed a cosmic crystallography, so to speak: With statistical methods we can determine regularities in a Euclidean universe without having to recognize certain galaxies as images of others. In the case of a periodic repetition, a histogram, in which all distances between galaxies are plotted, should have peaks at certain values, which indicate the true size of the universe. So far we haven't found any patterns (graph above), but that could be due to the lack of data on galaxies more than two billion light years away from us. An American-Japanese project to survey the sky in strips, the so-called Sloan Digital Sky Survey, will provide larger amounts of data for our investigation.

Finally, several other research groups are planning to determine the topology of the universe using the cosmic background radiation. This microwave radiation - a remnant from the early days of the universe, when the hot plasma condensed into hydrogen and helium atoms - is remarkably homogeneous: its temperature and intensity are the same in all parts of the sky except for 1 in 100,000. However, the small deviations from the mean, which the satellite COBE determined in 1991, indicate density fluctuations in the early universe - seeds from which galaxies and galaxy clusters later developed.

These fluctuations are key to solving a variety of cosmological questions, one of which is the topology of the universe. The photons of the background radiation arriving at a certain moment began their journeys at approximately the same time and at the same distance to the earth. Your starting points thus form a spherical surface with the earth in the center. If this sphere were larger than the universe, it would penetrate itself - like a large paper disk overlaps when you wrap it around a cylindrical broomstick (picture on page 55). The intersection of a spherical surface with itself is simply a circle in space. From Earth, astronomers would see two circles in the sky showing the same pattern of temperature fluctuations. In reality, these two circles would only be one that could be seen from two directions.

Two of us (Starkman and Weeks) work with Princeton’s David N. Spergel and Neil J. Cornish to discover such circle pairs. The beauty of this method is that it is independent of the uncertainties of contemporary cosmology - it relies on the observation that space has constant curvature, but makes no assumptions about the density of matter, the geometry of space, or the existence of a cosmological constant. The main problem is to identify the two circles despite the expected distortions. The merging of galaxies, for example, changes the gravitational influence on the background radiation, so that the energy of the photons shifts on their way to earth.

Unfortunately, COBE was unable to resolve structures with an angular diameter smaller than 10 degrees, or to clearly identify individual areas of higher or lower temperature. It is only certain that, from a statistical point of view, some of the fluctuations measured represent real structures and are not instrumental artifacts. Progress is expected from new instruments with higher resolution and lower noise. Some of them are already observing from the ground or from balloons, but they do not cover the entire sky. The decisive observations are to be carried out by the Microwave Anisotropy Probe (MAP), a probe that the American aerospace authority NASA plans to launch at the end of next year, and the Planck satellite of the European Space Agency ESA, which is to be launched in 2007.

The relative location of the two circles - if they really exist - will reveal the specific topology of the universe. If the spherical surface from which the photons of the background radiation originate is just big enough that it winds once around the universe, it will only intersect its closest ghost image. If it is bigger, it will reach further and also cut the next pictures. With the appropriate size, we can even expect several hundred or thousand pairs of circles (picture above).

The data would be highly redundant. The largest circles would completely determine the topology of the room and also the position and orientation of all smaller pairs of circles. This internal consistency of the pattern would not only confirm the topological results, but also the interpretation of the cosmic background radiation.

Other working groups have different plans. John D. Barrow and Janna J. Levin from the University of Sussex in Brighton (England), Emory F. Bunn from Bates-Collegein Lewiston (Maine) and Evan Scannapieco and Joseph I. Silk from the University of California at Berkeley want the areas to be elevated and the lower temperature in the background radiation. This team has already constructed sample cards on which the microwave background is simulated for various topologies. They multiplied the temperatures in each direction by the temperatures in every other direction, creating a huge four-dimensional map called the 2-point correlation function. Such maps can be used to quantitatively compare topologies.

**The topology of the universe**

J. Richard Bond, Dmitry Pogosyan, and Tarun Souradeep of the Canadian Institute for Theoretical Astrophysics are applying similar new methods to existing COBE data that may prove sufficiently accurate to identify the smallest hyperbolic spaces.

In addition to intellectual satisfaction, determining the topology of space would have far-reaching consequences for physics. In contrast to the theory of relativity, newer, more comprehensive theories currently being worked on should predict the topology of the universe, or at least give probabilities for various possibilities. These "theories about everything" are used to explain gravity in the first moments after the Big Bang, when quantum mechanical effects played a role (see "Quantum Theory of Gravitation" by Bryce S. DeWitt, Spectrum of Science, February 1984, p. 30).

The tentative steps towards a unification of physics have already brought about the branch of quantum cosmology. Three basic hypotheses for the origin of the universe are defended, namely by Andrej Linde from Stanford University (California), Alexander Vilenkin from Tufts University in Medford (Masachesetts) and Stephen W. Hawking from Cambridge University (England). A particular controversial issue is whether the volume of a newborn universe is very large (Linde and Vilenkin) or very small (Hawking). Topological data could provide the decision.

If observations should show that the universe is finite, the question of why it is on the whole homogeneous could also be solved. To explain this uniformity, the big bang model was expanded to include the theory of inflation, but it ultimately ran into difficulties because the standard form reveals that the geometry of the universe must be Euclidean - in contradiction to the observed density of matter. This is why some theorists have postulated hidden forms of energy and others modified the inflation theory (see "What happened before the Big Bang" by Martin A. Bucher and David N. Spergel, Spektrum der Wissenschaft, March 1999, p. 54).

However, the universe could also be smaller than it looks. Then inflation might have stopped prematurely - before it forced Euclidean geometry - and still made the universe homogeneous. Igor Y. Sokolov of the University of Toronto (Canada) and others have used COBE's data to rule out this possibility when the universe is a three-torus. But it is still an option if the space is hyperbolic.

Since ancient times all civilizations have wondered how the universe began and whether it is finite or infinite. Through mathematical insight and careful observation, science in this century has partially answered the first question. Perhaps she can begin the next century with an answer to the second question. n

**Bibliography**

Cosmic Topology. By Marc Lachièze-Rey and Jean-Pierre Luminet in: Physics Reports, Vol. 254, No. 3, pp. 135-214 (March 1995).

Circles in the Sky: Finding Topology with the Microwave Background Radiation. By Neil J. Cornish, David N. Spergel, and Glenn D. Starkman in: Classical and Quantum Gravity, Vol. 15, No. 9, pp. 2657-2670 (1998).

Reconstructing the Global Topology of the Universe from the Cosmic Microwave Background. By Jeffrey R. Weeks in: Classical and Quantum Gravity, Vol. 15, No. 9, pp. 2599-2604 (1998)

From: Spektrum der Wissenschaft 7/1999, page 50

© Spektrum der Wissenschaft Verlagsgesellschaft mbH

#### This article is included in Spectrum of Science 7/1999

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