Conventional wisdom says the universe is infinite. But

it could be finite, merely giving the illusion of infinity.

Upcoming measurements may finally answer this ancient question

by Jean-Pierre Luminet, Glenn D. Starkman and Jeffrey R. Weeks

...........

SUBTOPICS:

Comfort in the Finite

Eightfold

Cosmic Crystals

Circular Reasoning

ILLUSTRATIONS: Infinity Box

Local Geometry

Doughnut Space

Finite Hyperbolic Space

Distances Between Galaxy Clusters

Wrapped Around

Three Possible Universes

FURTHER READING

RELATED LINKS

Image: Bryan Christie

Looking up at the sky on a clear night, we feel we can see forever. There

seems to be no end to the stars and galaxies; even the darkness in between

them is filled with light if only we stare through a sensitive enough

telescope. In truth, of course, the volume of space we can observe is limited

by the age of the universe and the speed of light. But given enough time,

could we not peer ever farther, always encountering new galaxies and

phenomena?

Maybe not. Like a hall of mirrors, the apparently endless universe might be

deluding us. The cosmos could, in fact, be finite. The illusion of infinity

would come about as light wrapped all the way around space, perhaps more than

once--creating multiple images of each galaxy. Our own Milky Way galaxy would

be no exception; bizarrely, the skies might even contain facsimiles of the

earth at some earlier era. As time marched on, astronomers could watch the

galaxies develop and look for new mirages. But eventually no new space would

enter into their view. They would have seen it all.

The question of a finite or infinite universe is one of the oldest in

philosophy. A common misconception is that it has already been settled in

favor of the latter. The reasoning, often repeated in textbooks, draws an

unwarranted conclusion from Einstein's general theory of relativity.

According to relativity, space is a dynamic medium that can curve in one of

three ways, depending on the distribution of matter and energy within it.

Because we are embedded in space, we cannot see the flexure directly but

rather perceive it as gravitational attraction and geometric distortion of

images. To determine which of the three geometries our universe has,

astronomers have been measuring the density of matter and energy in the

cosmos. It now appears to be too low to force space to arch back on itself--a

"spherical" geometry. Therefore, space must have either the familiar

Euclidean geometry, like that of a plane, or a "hyperbolic" geometry, like

that of a saddle [see illustration]. At first glance, such a universe

stretches on forever.

One problem with this conclusion is that the universe could be spherical yet

so large that the observable part seems Euclidean, just as a small patch of

the earth's surface looks flat. A broader issue, however, is that relativity

is a purely local theory. It predicts the curvature of each small volume of

space--its geometry--based on the matter and energy it contains. Neither

relativity nor standard cosmological observations say anything about how

those volumes fit together to give the universe its overall shape--its

topology. The three plausible cosmic geometries are consistent with many

different topologies. For example, relativity would describe both a torus (a

doughnutlike shape) and a plane with the same equations, even though the

torus is finite and the plane is infinite. Determining the topology requires

some physical understanding beyond relativity.

The usual assumption is that the universe is, like a plane, "simply

connected," which means there is only one direct path for light to travel

from a source to an observer. A simply connected Euclidean or hyperbolic

universe would indeed be infinite. But the universe might instead be

"multiply connected," like a torus, in which case there are many different

such paths. An observer would see multiple images of each galaxy and could

easily misinterpret them as distinct galaxies in an endless space, much as a

visitor to a mirrored room has the illusion of seeing a huge crowd.

A multiply connected space is no mere mathematical whimsy; it is even

preferred by some schemes for unifying the fundamental forces of nature, and

it does not contradict any available evidence. Over the past few years,

research into cosmic topology has blossomed. New observations may soon reach

a definitive answer.

Comfort in the Finite

Many cosmologists expect the universe to be finite. Part of the reason may be

simple comfort: the human mind encompasses the finite more readily than the

infinite. But there are also two scientific lines of argument that favor

finitude. The first involves a thought experiment devised by Isaac Newton and

revisited by George Berkeley and Ernst Mach. Grappling with the causes of

inertia, Newton imagined two buckets partially filled with water. The first

bucket is left still, and the surface of the water is flat. The second bucket

is spun rapidly, and the surface of the water is concave. Why?

The naive answer is centrifugal force. But how does the second bucket know it

is spinning? In particular, what defines the inertial reference frame

relative to which the second bucket spins and the first does not? Berkeley

and Mach's answer was that all the matter in the universe collectively

provides the reference frame. The first bucket is at rest relative to distant

galaxies, so its surface remains flat. The second bucket spins relative to

those galaxies, so its surface is concave. If there were no distant galaxies,

there would be no reason to prefer one reference frame over the other. The

surface in both buckets would have to remain flat, and therefore the water

would require no centripetal force to keep it rotating. In short, it would

have no inertia. Mach inferred that the amount of inertia a body experiences

is proportional to the total amount of matter in the universe. An infinite

universe would cause infinite inertia. Nothing could ever move.

In addition to Mach's argument, there is preliminary work in quantum

cosmology, which attempts to describe how the universe emerged spontaneously

from the void. Some such theories predict that a low-volume universe is more

probable than a high-volume one. An infinite universe would have zero

probability of coming into existence [see "Quantum Cosmology and the Creation

of the Universe," by Jonathan J. Halliwell; Scientific American, December

1991]. Loosely speaking, its energy would be infinite, and no quantum

fluctuation could muster such a sum.

Historically, the idea of a finite universe ran into its own obstacle: the

apparent need for an edge. Aristotle argued that the universe is finite on

the grounds that a boundary was necessary to fix an absolute reference frame,

which was important to his worldview. But his critics wondered what happened

at the edge. Every edge has another side. So why not redefine the "universe"

to include that other side? German mathematician Georg F. B. Riemann solved

the riddle in the mid-19th century. As a model for the cosmos, he proposed

the hypersphere--the three-dimensional surface of a four-dimensional ball,

just as an ordinary sphere is the two-dimensional surface of a

three-dimensional ball. It was the first example of a space that is finite

yet has no problematic boundary.

One might still ask what is outside the universe. But this question supposes

that the ultimate physical reality must be a Euclidean space of some

dimension. That is, it presumes that if space is a hypersphere, then that

hypersphere must sit in a four-dimensional Euclidean space, allowing us to

view it from the outside. Nature, however, need not cling to this notion. It

would be perfectly acceptable for the universe to be a hypersphere and not be

embedded in any higher-dimensional space. Such an object may be difficult to

visualize, because we are used to viewing shapes from the outside. But there

need not be an "outside."

By the end of the 19th century, mathematicians had discovered a variety of

finite spaces without boundaries. German astronomer Karl Schwarzschild

brought this work to the attention of his colleagues in 1900. In a postscript

to an article in Vierteljahrschrift der Astronomischen Gesellschaft, he

challenged his readers:

"Imagine that as a result of enormously extended astronomical experience, the

entire universe consists of countless identical copies of our Milky Way, that

the infinite space can be partitioned into cubes each containing an exactly

identical copy of our Milky Way. Would we really cling on to the assumption

of infinitely many identical repetitions of the same world?. . . We would be

much happier with the view that these repetitions are illusory, that in

reality space has peculiar connection properties so that if we leave any one

cube through a side, then we immediately reenter it through the opposite

side."

Schwarzschild's example illustrates how one can mentally construct a torus

from Euclidean space. In two dimensions, begin with a square and identify

opposite sides as the same--as is done in many video games, such as the

venerable Asteroids, in which a spaceship going off the right side of the

screen reappears on the left side. Apart from the interconnections between

sides, the space is as it was before. Triangles span 180 degrees, parallel

laser beams never meet and so on--all the familiar rules of Euclidean

geometry hold. At first glance, the space looks infinite to those who live

within it, because there is no limit to how far they can see. Without

traveling around the universe and reencountering the same objects, the ship

could not tell that it is in a torus [see illustration]. In three dimensions,

one begins with a cubical block of space and glues together opposite faces to

produce a 3-torus.

The Euclidean 2-torus, apart from some sugar glazing, is topologically

equivalent to the surface of a doughnut. Unfortunately, the Euclidean torus

is food only for the mind. It cannot sit in our three-dimensional Euclidean

space. Doughnuts may do so because they have been bent into a spherical

geometry around the outside and a hyperbolic geometry around the hole.

Without this curvature, doughnuts could not be viewed from the outside.

When Albert Einstein published the first relativistic model of the universe

in 1917, he chose Riemann's hypersphere as the overall shape. At that time,

the topology of space was an active topic of discussion. Russian

mathematician Aleksander Friedmann soon generalized Einstein's model to

permit an expanding universe and a hyperbolic space. His equations are still

routinely used by cosmologists. He emphasized that the equations of his

hyperbolic model applied to finite universes as well as to the standard

infinite one--an observation all the more remarkable because, at the time, no

examples of finite hyperbolic spaces were known.

Image: Bryan Christie

Eightfold

Of all the issues in cosmic topology, perhaps the most difficult to grasp is

how a hyperbolic space can be finite. For simplicity, first consider a

two-dimensional universe. Mimic the construction of a 2-torus but begin with

a hyperbolic surface instead. Cut out a regular octagon and identify opposite

pairs of edges, so that anything leaving the octagon across one edge returns

at the opposite edge. Alternatively, one could devise an octagonal Asteroids

screen [see illustration]. This is a multiply connected universe,

topologically equivalent to a two-holed pretzel. An observer at the center of

the octagon sees the nearest images of himself or herself in eight different

directions. The illusion is that of an infinite hyperbolic space, even though

this universe is really finite. Similar constructions are possible in three

dimensions, although they are harder to visualize. One cuts a solid

polyhedron out of a hyperbolic three-dimensional space and glues pairs of

faces so that any object leaving from one face returns at the corresponding

point on the matching face.

The angles of the octagon merit careful consideration. On a flat surface, a

polygon's angles do not depend on its size. A large regular octagon and a

small regular octagon both have inside angles of 135 degrees. On a curved

surface, however, the angles do vary with size. On a sphere the angles

increase as the polygon grows, whereas on a hyperbolic surface the angles

decrease. The above construction requires an octagon that is just the right

size to have 45-degree angles, so that when the opposite sides are

identified, the eight corners will meet at a single point and the total angle

will be 360 degrees. This subtlety explains why the construction would not

work with a flat octagon; in Euclidean geometry, eight 135-degree corners

cannot meet at a single point. The two-dimensional universe obtained by

identifying opposite sides of an octagon must be hyperbolic. The topology

dictates 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 have any physical size (in meters, say), but its

surface area will always be exactly 4p times the square of its radius--that

is, 4p square radians. The same principle applies to the size of a hyperbolic

topology, for which a radius of curvature can also be defined. The most

compact hyperbolic topology, discovered by one of us (Weeks) in 1985, may be

constructed by identifying pairs of faces of an 18-sided polyhedron. It has a

volume of approximately 0.94 cubic radian. Other topologies are built from

larger polyhedra.

The universe, too, can be measured in units of radians. Diverse astronomical

observations agree that the density of matter in the cosmos is only a third

of that needed for space to be Euclidean. Either a cosmological constant

makes up the difference [see "Cosmological Antigravity," by Lawrence M.

Krauss; Scientific American, January], 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 room

for nearly 200 of the Weeks polyhedra. In other words, if the universe has

the Weeks topology, its volume is only 0.5 percent of what it appears to be.

As space expands uniformly, its proportions do not change, so the topology

remains constant.

In fact, almost all topologies require hyperbolic geometries. In two

dimensions, a finite Euclidean space must have the topology of either a

2-torus or a Klein bottle; in three dimensions, there are only 10 Euclidean

possibilities--namely, the 3-torus and nine simple variations on it, such as

gluing together opposite faces with a quarter turn or with a reflection,

instead of straight across. By comparison, there are infinitely many possible

topologies for a finite hyperbolic three-dimensional universe. Their rich

structure is still the subject of intense research [see "The Mathematics of

Three-Dimensional Manifolds," by William P. Thurston and Jeffrey R. Weeks;

Scientific American, July 1984].

Cosmic Crystals

Despite the plethora of possibilities, the cosmologists of the 1920s had no

way to measure the topology of the universe directly, and so they eventually

lost interest in the issue. The decades from 1930 to 1990 were the dark ages

of the subject. Most astronomy textbooks, quoting one another for support,

stated that the universe must be either a hypersphere, an infinite Euclidean

space or an infinite hyperbolic space. Other topologies were largely

forgotten. But the 1990s have seen the rebirth of the subject. Roughly as

many papers have been published on cosmic topology in the past three years as

in the preceding 80. Most exciting of all, cosmologists are finally poised to

determine the topology observationally.

The simplest test of topology is to look at the arrangement of galaxies. If

they lie in a rectangular lattice, with images of the same galaxy repeating

at equivalent lattice points, the universe is a 3-torus. Other patterns

reveal more complicated topologies. Unfortunately, looking for such patterns

is difficult, because the images of a galaxy would depict different points in

its history. Astronomers would need to recognize the same galaxy despite

changes in appearance or shifts in position relative to neighboring galaxies.

Over the past 25 years researchers such as Dmitri Sokoloff of Moscow State

University, Viktor Shvartsman of the Soviet Academy of Sciences, J. Richard

Gott III of Princeton University and Helio V. Fagundes of the Institute for

Theoretical Physics in S�o Paulo have looked for and found no repeating

images among galaxies within one billion light-years of the earth.

Image: Bryan Christie

Others--such as Boudewijn F. Roukema of the Inter-University Center for

Astronomy and Astrophysics in Pune, India--have sought patterns among

quasars. Because these objects, thought to be powered by black holes at the

cores of galaxies, are bright, any patterns among them can be seen from large

distances. The observers identified all groupings of four or more quasars. By

examining the spatial relations within each group, they checked whether any

pair of groups could in fact be the same group seen from two different

directions. Roukema identified two possibilities, but they may not be

statistically significant.

Roland Lehoucq and Marc Lachi�ze-Rey of the Center for Astrophysical Studies

in Saclay, France, together with one of us (Luminet), have tried to

circumvent the problems of galaxy recognition in another way. We have

developed the method of cosmic crystallography, which in a Euclidean universe

can make out a pattern statistically without needing to recognize specific

galaxies as images of one another. If galaxy images repeat periodically, a

histogram of all galaxy-to-galaxy distances should show peaks at certain

distances, which represent the true size of the universe. So far we have seen

no patterns [see illustration],but this may be because of the paucity of data

on galaxies farther away than two billion light-years. The Sloan Digital Sky

Survey--an ongoing American-Japanese collaboration to prepare a

three-dimensional map of much of the universe--will produce a larger data set

for these studies.

Finally, several other research groups plan to ascertain the topology of the

universe using the cosmic microwave background, the faint glow remaining from

the time when the primordial plasma of the big bang condensed to hydrogen and

helium gas. The radiation is remarkably homogeneous: its temperature and

intensity are the same in all parts of the sky to nearly one part in 100,000.

But there are slight undulations discovered in 1991 by the Cosmic Background

Explorer (COBE) satellite. Roughly speaking, the microwave background depicts

density variations in the early universe, which ultimately seeded the growth

of stars and galaxies [see "The Evolution of the Universe," by P. James E.

Peebles, David N. Schramm, Edwin L. Turner and Richard G. Kron; Scientific

American, October 1994].

Circular Reasoning

These fluctuations are the key to resolving a variety of cosmological issues,

and topology is one of them. Microwave photons arriving at any given moment

began their journeys at approximately the same time and distance from the

earth. So their starting points form a sphere, called the last scattering

surface, with the earth at the center. Just as a sufficiently large paper

disk overlaps itself when wrapped around a broom handle, the last scattering

surface will intersect itself if it is big enough to wrap all the way around

the universe. The intersection of a sphere with itself is simply a circle of

points in space.

Looking at this circle from the earth, astronomers would see two circles in

the sky that share the same pattern of temperature variations. Those two

circles are really the same circle in space seen from two perspectives [see

illustration]. They are analogous to the multiple images of a candle in a

mirrored room, each of which shows the candle from a different angle.

Two of us (Starkman and Weeks), working with David N. Spergel and Neil J.

Cornish of Princeton, hope to detect such circle pairs. The beauty of this

method is that it is unaffected by 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 presence of a cosmological constant. The main problem is to identify the

circles despite the forces that tend to distort their images. For example, as

galaxies coalesce, they exert a varying gravitational pull on the radiation

as it travels toward the earth, shifting its energy.

Unfortunately, COBE was incapable of resolving structures on an angular scale

of less than 10 degrees. Moreover, it did not identify individual hot or cold

spots; all one could say for sure is that statistically some of the

fluctuations were real features rather than instrumental artifacts.

Higher-resolution and lower-noise instruments have since been developed. Some

are already making observations from ground-based or balloon-borne

observatories, but they do not cover the whole sky. The crucial observations

will be made by the National Aeronautics and Space Administration's Microwave

Anisotropy Probe (MAP), due for launch late next year, and the European Space

Agency's Planck satellite, scheduled for 2007.

The relative positions of the matching circles, if any, will reveal the

specific topology of the universe. If the last scattering surface is just

barely big enough to wrap around the universe, it will intersect only its

nearest ghost images. If it is larger, it will reach farther and intersect

the next nearest images. If the last scattering surface is large enough, we

expect hundreds or even thousands of circle pairs. The data will be highly

redundant. The largest circles will completely determine the topology of

space as well as the position and orientation of all smaller circle pairs.

Thus, the internal consistency of the patterns will verify not only the

correctness of the topological findings but also the correctness of the

microwave background data.

Other teams have different plans for the data. John D. Barrow and Janna J.

Levin of the University of Sussex, Emory F. Bunn of Bates College, and Evan

Scannapieco and Joseph I. Silk of the University of California at Berkeley

intend to examine the pattern of hot and cold spots directly. The group has

already constructed sample maps simulating the microwave background for

particular topologies. They have multiplied the temperature in each direction

by the temperature in every other direction, generating a huge

four-dimensional map of what is usually called the two-point correlation

function. The maps provide a quantitative way of comparing topologies. J.

Richard Bond, Dmitry Pogosyan and Tarun Souradeep of the Canadian Institute

for Theoretical Astrophysics are applying related new techniques to the

existing COBE data, which could prove sufficiently accurate to identify the

smallest hyperbolic spaces.

Beyond the immediate intellectual satisfaction, discovering the topology of

space would have profound implications for physics. Although relativity says

nothing about the universe's topology, newer and more comprehensive theories

that are under development should predict the topology or at least assign

probabilities to the various possibilities. These theories are needed to

explain gravity in the earliest moments of the big bang, when

quantum-mechanical effects were important [see "Quantum Gravity," by Bryce S.

DeWitt; Scientific American, December 1983]. The theories of everything, such

as string theory, are in their infancy and do not yet have testable

consequences. But eventually the candidate theories will make predictions

about the topology of the universe on large scales.

The tentative steps toward the unification of physics have already spawned

the subfield of quantum cosmology. There are three basic hypotheses for the

birth of the universe, which are advocated, respectively, by Andrei Linde of

Stanford University, Alexander Vilenkin of Tufts University and Stephen W.

Hawking of the University of Cambridge. One salient point of difference is

whether the expected volume of a newborn universe is very large (Linde's and

Vilenkin's proposals) or very small (Hawking's). Topological data may be able

to distinguish among these models.

If observations do find the universe to be finite, it might help to resolve a

major puzzle in cosmology: the universe's large-scale homogeneity. The need

to explain this uniformity led to the theory of inflation, but inflation has

run into difficulty of late, because in its standard form it would have made

the cosmic geometry Euclidean--in apparent contradiction with the observed

matter density. This conundrum has driven theorists to postulate hidden forms

of energy and modifications to inflation [see "Inflation in a Low-Density

Universe," by Martin A. Bucher and David N. Spergel; Scientific American,

January]. An alternative is that the universe is smaller than it looks. If

so, inflation could have stopped prematurely--before imparting a Euclidean

geometry--and still have made the universe homogeneous. Igor Y. Sokolov of

the University of Toronto and others have used COBE data to rule out this

explanation if space is a 3-torus. But it remains viable if space is

hyperbolic.

Since ancient times, cultures around the world have asked how the universe

began and whether it is finite or infinite. Through a combination of

mathematical insight and careful observation, science in this century has

partially answered the first question. It might begin the next century with

an answer to the second as well.

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Further Reading:

La Biblioteca De Babel (The Library of Babel). Jorge Luis Borges in

Ficciones. Emec� Editores, 1956. Text available on the World Wide Web at

muni2000.com/babel/biblbabe.htm (in Spanish) and at

jubal.westnet.com/hyperdiscordia/library_of_babel.html (in English).

Cosmic Topology. Marc Lachi�ze-Rey and Jean-Pierre Luminet in Physics

Reports, Vol. 254, No. 3, pages 135-214; March 1995. Preprint available at

www.lanl.gov/abs/gr-qc/9605010 on the World Wide Web.

Poetry of the Universe. Robert Osserman. Anchor Books, 1995.

Circles in the Sky: Finding Topology with the Microwave Background Radiation.

Neil J. Cornish, David N. Spergel and Glenn D. Starkman in Classical and

Quantum Gravity, Vol. 15, No. 9, pages 2657-2670; September 1998. Preprint

available at www.lanl.gov/abs/astro-ph/9801212 on the World Wide Web.

Reconstructing the Global Topology of the Universe from the Cosmic Microwave

Background. Jeffrey R. Weeks in Classical and Quantum Gravity, Vol. 15, No.

9, pages 2599-2604; September 1998. Preprint available at www.lanl.gov/abs/

astro-ph/9802012 on the World Wide Web. Free software for exploring topology

is available at www.geom.umn.edu/software/download and www.northnet.org/weeks

on the World Wide Web.