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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
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Image: Bryan Christie
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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
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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 São Paulo
have looked for and found no repeating images among galaxies within one
billion light-years of the earth.
Image: Bryan Christie
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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 Lachièze-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.
Further Reading:
La Biblioteca De Babel (The Library of Babel).
Jorge Luis Borges in Ficciones. Emecé 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 Lachièze-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.
Related Links:
D.A.R.C. Homepage
Science Tour: Learning to
See the Infinite Universe
The Authors:
JEAN-PIERRE
LUMINET, GLENN D. STARKMAN and JEFFREY R. WEEKS say they relish
participating in the boom years of cosmic topology, as researchers come
together across disciplinary boundaries and no question is considered
stupid. Luminet, who studies black holes at Paris
Observatory, has written several books of science and of poetry and
collaborated with composer Gérard Grisey on the musical performance Le
Noir de l'Etoile. Starkman was institutionalized for six years--at the
Institute for Advanced Study in Princeton, N.J., and then at the Canadian
Institute for Theoretical Astrophysics in Toronto. He has been released
into the custody of Case Western Reserve University in Cleveland. Weeks,
the mathematician of the trio, resigned his position at Ithaca College to
care for his newborn son and now receives funding from the National
Science Foundation to develop research software.
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