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THE TOPOLOGY OF THE UNIVERSE
Is the Universe crumpled?
Is the Universe spatially closed or open? Has it holes or handles, is it connected or not ?
Often neglected by cosmologists, the study of the topological classification of threedimensional manifolds is likely to bring original answers to the question of the space
extension. In the so-called " crumpled " universe models, the sky is the arena of a gigantic
optical illusion, due a topological lens effect.
by Jean-Pierre Luminet
Research Director at CNRS
Astrophysicist at the Paris-Meudon Observatory.
Relativistic cosmos
General relativity upsets the concepts of time and space. The universe does not have a structure
of immutable Euclidean space woven by an independent time; it is described as a space-time
distorted by the presence of matter and energy. As a manifestation of the curvature of spacetime, gravitation dictates the trajectories of particles and light rays, compelled to marry contours
of a non-Euclidean four-dimensional geometry.
The basic equations of relativity describe the way the material contents of the universe
determines the background geometry of space-time. In this way, the theory makes it possible to
describe the universe as a whole according to plausible cosmological models. Among the
solutions, only a few correctly describe the universe, providing a theoretically consistent
explanation of astronomical observations.
In 1917 Einstein built the first universe model based on his theory of relativity. Its great
breakthrough was to propose a new approach of the question of space. Indeed, the non-Euclidean
geometry makes it possible to describe a space which is both finite and unbounded: the
hypersphere. Einstein thus offered, for the first time in the history of cosmology, a model of a
finite universe free from any " edge " paradox.
A finite space without edge
The advocates of a finite world butted a long time against a fundamental difficulty. It seemed
essential to imagine a center and a border to the world, but Archytas of Tarente, a pythagorician
from Vth century, raised a paradox aiming at showing the nonsense of the idea of a physical
edge of the world. Its argument knew a considerable fortune in all the debates on space : " If I
am at the end of the sky, can I lengthen the hand or a stick? It is absurd to think that I cannot ;
and if I can , what is beyond is either a body, or space. We can thus go beyond that still, and so
on. And if there is always a new space towards which one can tighten the stick, that implies an
extension without limits clearly ".
If" what is beyond the world" always formed part of the world, the world cannot logically be
limited without there being paradox! It was necessary to await the development of the nonEuclidean geometries in XIXth century to solve the controversy. These geometries make it
possible to conceive a space as finite but without edge (just like, in two dimensions, the surface
of a sphere). This design is not so natural and confusion is still found today in a number of
minds; when, for example, a lecturer describes to a popular audience the expansion of the
universe, it is often seen raising the question: in what the universe does inflate? The question
itself is a semantic non-sense, since there is no space apart from itself ! But to really understand
this, it should be adopted a non-Euclidean mental framework.
Besides the conceptual revolution resulting from relativity, observational progress led Hubble to
announce, in 1929, that the other galaxies move away systematically from ours, with velocities
proportional to their distance. The Einstein model thus had to be abandoned because it described
a static universe, and replaced by dynamical universe models, independently discovered by the
Russian Alexandre Friedmann and the Belgian George Lemaître.
The question of the extension of space is perfectly well put within the framework of the
Friedmann-Lemaître models, called more commonly "big-bang models". These ones assume that
the universe has the same properties everywhere (space is known as " homogeneous and
isotropic "). The geometrical properties of space are of two kinds only: the curvature, constant in
space when matter is uniformly distributed, and the topology. Regarding the curvature, three
families of spaces can be considered: Euclidean space (zero curvature), spherical space (positive
curvature) and hyperbolic space (negative curvature). Spherical space is, in all the cases, finite (it
is one of the reasons for which Einstein, in the spirit of Parmenides, choose it initially). For
spaces belonging to the two other families, the finite or infinite character depends on topology.
In the simplest versions however (simply-connected topologies), they are infinite.
The cosmologists generally neglect the topological aspect to consider only the curvature
properties. This simplification is dramatic as for the problem of infinite space since, in such a
case, the dilemma finite/infinite is brought back to know the sign of the space curvature only.
General relativity indicates how to calculate this curvature. Its value depends on the average
density of matter-energy it contains, as well as a parameter Lambda, called the cosmological
constant. Generally, a second simplification is introduced, that to suppose a vanishing Lambda.
Then, the finite/infinite character of space does not depend any more but on the average matterenergy density: according to whether it is higher or lower than a certain "critical value ", 10^(29) g/cm3, the curvature is positive or negative, and space is finite or infinite. What are the
observational data? They indicate an average density approximately ten times lower than the
critical value. Apparently, if the topological complications and the cosmological constant are
neglected, space would be thus infinite. In fact, the actual value is only a lower limit. It would be
non-sense to believe that we see all the matter in the universe. Various reasons suggest that, in
addition to visible matter, great quantities of dark matter exist, sufficiently perhaps so that the
true density of the universe reaches exactly the critical value. In this case, the universe would
marginally remain open in space and time. This is the Euclidean model, first proposed by
Einstein and de Sitter in 1931, and which keeps still today the favours of many cosmologists,
without decisive argument to justify it (if not... an aesthetic feeling)
Is the Universe closed or open ?
In the Friedmann-Lemaître cosmological models with vanishing cosmological
constant, the curvature is directly related to the average energy density: the
curvature is positive (spherical space) when the density is higher than the critical
value, zero (Euclidean space) if the density is equal to the critical value, and
negative (hyperbolic space) if density is lower. The curvature thus dictates only the
time evolution: the universe is (temporally) closed in the spherical case,
(temporally) open in the Euclidean and hyperbolic cases. The Einstein-de Sitter
flat model (that some cosmologists estimate favoured by inflationary models of the
early universe) corresponds to the diagram of the middle. If moreover the simplest
topology is assumed, the curvature dictates also the finite or infinite character of
space: finite in the spherical case, infinite in the Euclidean and hyperbolic cases.
With these two (unjustified) simplifications, there is strict equivalence between time
finiteness /infiniteness and space finiteness / infiniteness. In the FriedmannLemaître models with non zero cosmological constant, the curvature is related to
the matter density and the cosmological constant. There is no more direct link
between the curvature and the cosmic dynamics : the universe can be spherical but
temporally open. If, moreover, topology is not the simplest one, there is no more
correspondence between time finiteness / infiniteness and space finiteness /
infiniteness.
The darkness of night
If the paradox of the edge made obstacle to the concept of finite space, the " dark night paradox
" made obstacle to space infinity. The darkness of the night indeed hides a mystery involving the
cosmos as a whole, its extension and its history. It is stated like follows: if space is infinite and
uniformly filled with eternal stars, in any direction which one looks at one must end up finding a
star on the line of sight. In other words, the sky background should be a radiant tapestry,
continuously made up of stars, not leaving any place to the dark. Why isn't it thus? The question,
put as of XVIIth century by Kepler (and later by Olbers), raised tens of explanations and models.
The American writer Edgar Poe provided the first satisfactory answer. In a premonitory text
entitled Eureka, Poe explained why the darkness of the night rested on the finitude of cosmic
time. Indeed, as he pointed out, the light can propagate only at finite speed. However, in a noneternal universe, the stars did not always exist. We can thus receive their light only if this one
had time to reach us, i.e. if the stars which emitted it were sufficiently close. Thus, the sky is not
uniformly brilliant because the stars (not necessarily the entire universe) have existed only for a
finite time. By understanding how the night darkness privided to us a deep teaching about the
time finiteness of the world, Poe anticipated by several decades the big-bang relativistic models.
The cosmic microwave background
radiation
Since the universe has not existed
(at least in a state allowing the
existence of stars) for more than a
few billion years, the sky
background is hardly brilliant. It
emits a weak gleam, unperceivable
to our eyes, but that radiotelescopes
can collect. Discovered in 1965, it is
the vestige of dazzling primitive fire
cooled by fifteen billion years of
time travel.
Whatever space is infinite or not,
only a finite volume is accessible to
the observations. The cosmic
microwave background radiation
marks a horizon, an ultimate wall
against which will stop any
observation forever in the
electromagnetic spectrum. Because,
in its early phase during about one
million years, the universe did not
give anything to see: the
electromagnetic radiation could not
propagate, the stars and the galaxies
were not still formed!
The topology of the universe
The questions related to the global shape of space and, in particular, its finite or infinite
extension, cannot be fully answered by general relativity (a local physical theory), but by
topology (a global mathematical theory).
Nothing obliges space to have the simplest topology (known as " simply-connected ") because
general relativity does not impose any constraint on the global properties of space-time. Many
topological "alternatives" of three-dimensional spaces can thus be used to build relevant universe
models, i.e. both compatible with relativity and observations.
Thanks to " multi-connected " topologies, it becomes possible to consider universe models where
space is finite whatever its curvature, even if the matter density and the cosmological constant
are very low.
Historically, W. de Sitter pointed out in 1917 to Einstein that his static and spherical universe
model could put up with a different topology, namely that of projective space. The difference
was not very large because these two alternatives are finite. The outstanding article by
Friedmann, in 1922, makes mention of a finite Euclidean space form (normally infinite). Einstein
remained unaware of that since, in 1931, he published with de Sitter an article where they
selected the infinite Euclidean universe model. Only in 1958, Lemaître mentioned the existence
of compact hyperbolic spaces, also suitable for application to big-bang models. In spite of that,
the subject of cosmic topology always remained confidential and widely ignored by the
community of cosmologists.
In addition to the interest of "compactifying" spaces, the multi-connected models cause many
surprises by creating an "illusion of the infinity". Let us see why. To build multi-connected
spaces, mathematics teach us that one can start from one of the three types of " ordinary "
(simply connected) spaces. Then, identification between some points change the shape of space
and makes it multi-connected. From this one can build universe models where space is finite
(although the curvature can be negative or zero) and of a really small volume. They are called
"small universes". The simplest example is when our space would be a hypertorus having a
radius lower than five billion light-years. In this case, the light rays would have had time to turn
three times "around" the universe. That would imply that each cosmic object (each galaxy for
example) should appear according to as many "ghost" images, observable in various areas of the
sky. The observed universe thus appears made up of the repetition of a same set of galaxies,
although viewed at different look-back times.
It is not easy to check if we live or not in a small universe. The ghost images of each " real "
galaxy would appear in different directions, with different luminosities, under different
orientations, and at different evolutionary times. It would be practically impossible to recognize
them like such! The universe could appear vast to us, " unfolded ", filled of billion galaxies,
while it would actually be much smaller, " folded up " but containing only a small number of
authentic objects. A gigantic cosmic optical illusion! Of course, the current observational data
make it possible to eliminate the possibility of a too small universe... If not we would have
already recognized, close to us, the multiple images of our own Galaxy! Various arguments of
this kind, applied to some well-known cosmic objects (e.g. the closest galaxy clusters), make it
possible to exclude a universe whose dimensions would be lower than a few hundreds of million
light-years. However statistical studies on the distribution of galaxy clusters may reveal in the
future the "crumpled" nature of space over a scale of a few billion light-years.
We see a sky filled with galaxies, but its aspect does not make it possible to decide if
the farthest galaxies are not ghost images of closer galaxies. The assumption of a
multi-connected Universe cannot be discarded: the Universe could appear vast to
us, " unfolded ", while it would be actually much smaller and "folded up".
The basics of topology
Topology is the branch of geometry which classifies spaces according to their global shape. By
definition, spaces belong to a same topological class if they can deduced from each other by
continuous deformation, i.e. without cutting nor tearing. In the case of two dimensional spaces,
i.e. surfaces, the sphere, for example, has the same topology as any ovoid closed surface. But the
plane has a different topology, since no continuous deformation will give it the shape of a sphere.
For better visualizing what is topology, start from the ordinary Euclidean plane. It is an infinite
2-dimensional layer (that one generally imagines as embedded in ordinary 3 dimensional space).
Let us cut out a tape with infinite " length " but finite width; then let us identify (i.e. restick) the
two edges of this tape: one gets a cylinder, i.e. a surface whose topology differs from that of the
initial plane. Let us take another infinite sheet and, this time, cut out it in rectangle. Let us
identify two by two the parallel edges. We obtain a closed (finite) surface. It is a flat torus. From
a simple paper sheet we could thus define 3 surfaces with different topologies, pertaining to the
same family of locally flat surfaces.
During the XXth century, mathematicians stuck to the classification of three-dimensional spaces.
Like surfaces, 3-spaces can be arranged, according to the sign of their curvature, into spherical,
Euclidean or hyperbolic types. Then one counts the topological forms inside each one of these
families. There are for example 18 kinds of three-dimensional spaces with zero curvature.
Simplest is " ordinary " infinite Euclidean space, the properties of which are teached at school,
but others space forms are closed and finite. It is for example the case of the hypertorus, which
generalizes in three dimensions the case of the torus. A hypertorus can be regarded as the interior
of an ordinary cube, whose opposite faces are identified two by two: while leaving by one, one
returns immediately by the opposite. Such a space is finite.
In addition, there are a countable infinity of spaces forms with positive curvature, all of them
closed, and an infinite number of spaces with negative curve, some closed (finite), some open
(infinite).
To visualize them, one represents them by the interior of a polyhedron of which some faces are
identified two by two.
The five regular polyhedrons, already called upon by Plato for geometrizing the " elements "
Earth, Water, Air, Fire, Quintessence, are used today to represent certain multi-connected
spaces, on the condition of considering that the faces are identified by pairs according to
specific geometrical transformations.
A compact hyperbolic space.
The interior of a regular dodecahedron, whose pentagonal faces are identified
("stuck") by pairs, is a closed space of negative curve. Seen from inside, such a
space would give the impression we live in a cellular space, paved ad infinitum by
dodecahedrons deformed by optical illusions.
Copyright 1990 by The Geometry Center, University of Minnesota.
Cosmic sets of mirrors
Who wasn't fascinated by sets of mirrors? That it is about the Galerie des Glaces at the Chateau
de Versailles or most modest Palais of the Ices of open attractions, each one is filled with wonder
at the illusion generated by the phantom images. The mirrors conceal certain secrecies of
infinity.
Everyone noted that to put mirrors on the walls of a room gives the illusion of a larger room.
Let us take a room filled with mirrors on its six walls (floor and ceiling included). If you
penetrate in the room and light some candles, by the play of the multiple reflexions on the walls
you have immediately the impression to see the infinity, as if you were suspended on the node of
a bottomless well, ready to be swallowed in a direction or another with the least movement.
It could well be thus of cosmic space!
It may be that the topology of the universe is
multiconnected, i.e. that space resembles
inside a room papered with complicated
mirrors. This multiconnexity would create
additional paths for the light rays which reach
us from the remote galaxies. It would result a
great number of ghost images of these
galaxies. The diagrams on the left result from
numerical simulations of "crumpled"
universes, carried out with my collaborators.
top diagram : space is a hypertorus,
represented by the interior of a cube of 5
billion light-years size, whose opposite faces
are identical. 50 galaxies are randomly
distributed in space.
middle diagram : positions, on a celestial
planisphere, of the 50 " original " galaxies.
bottom diagram : appearance of the sky taking
account of the multiple paths of light rays.
Each " real " galaxy generates about fifty
"ghost" images. It is impossible to distinguish
the "real" images from the ghost images. If
one points out the resemblance of this diagram
to the appearance of the large scale structure
in the universe, one deduces that it is quite
possible that we live in a cosmic optical
illusion, giving the impression that space is
immense, whereas real space is small and
"crumpled".
The quantum universe
It is clear that the concept of a small crumpled universe concerns Parmenidian aesthetics. This
one took besides the step at the majority of modern physicists, who seek to eliminate infinite
quantities from their theories. Space infinity is not the only infinity occurring in relativistic
cosmology. The theory predicts configurations indeed where certain geometrical (e.g. curvature)
and physical (e.g. energy density, temperature) quantities become infinite: gravitational
singularities. Most known are the initial big-bang singularity, and the final singularity hidden at
the bottom of a black hole. The physicists doubt that a theory predicting singularities can be
correct. The fact is that general relativity is incomplete, since it does not take account of the
principles of quantum mechanics. This last governs the evolution of microscopic world, in
particular the field of elementary particles. Its essential characteristic is to give a " fuzzy "
description of the phenomena, insofar as the events can be calculated only in terms of
probabilities. However, the occurrence of singularities brings into play the structure of spacetime at very small scale. There is a length (called Planck length, equal to 10^(-33) centimetre)
representing the smallest dimension to which space-time can still be regarded as smooth. Below,
even the texture of space-time would not be continuous any more but, just like the matter and the
energy, formed of small grains. The gravitational infinities would be replaced by quantum
fluctuations of space-time.
With " quantum cosmology ", a theory hardly outlined and promised to attractive developments,
are profiled multiple, simultaneous and not-interacting bubble universes, differing from each
other by their geometry, their topology, their fundamental constants of physics.
All these universes would be like the foam of a single Universe, a kind of infinite and eternal
bubbling ocean, in perpetual transformation, called by the physicists " quantum vacuum ". With
such a conception, Heraclitus' sons did not say their last word...
The Foam of Vacuum
Quantum cosmology makes it possible to
consider multiple universes, without
interaction between them. Our observable
universe would occupy a " bubble " born of
the spontaneous fluctuations of the quantum
vacuum, like many other bubbles.
Copyright : Manchu/Ciel et Espace
The black hole infinities
A hybrid creature given birth to by non-Euclidean geometry and relativistic gravitation, the black
hole offers two pretty problems of infinity: a false one and a true one. A black hole results from
the gravitational collapse of a mass below some critical volume. Like the edge of a bottomless
well dug in the elastic fabric of space-time, its surface - called the event horizon - marks the
geometrical border of a no return zone. For an external observer, the beats of a clock placed
close to the black hole slow down as the clock is closer to event horizon, until "freezing" when
the clock reaches the surface. All occurs then as if time were indefinitely delayed. Consequently,
the black hole by itself is inobservable, because it belongs to the infinitely remote future of any
observer. This infinite time is only apparent because it can be made finite in a correct
representation (proper time).
The situation is quite different with the interior of the black hole. The general relativity theory
predicts the existence of an inescapable singularity inside the black hole, where the curvature of
space and the density of matter become infinite.
A traveller exploring the surroundings of a black hole would be plunged in optical illusions.
Misled by the infinite forgery related to the surface of the hole, it would never see the interior,
unless plunging in person to discover there with its costs the infinite truth of the singularity!
PS : I apologize for my clumsy English. The article is much better written in French, as you can
check here