Four images of the same quasar, bent around a central galaxy: a common astronomical
mirage.
A very simple two-dimensional universe illustrates how an observer in galaxy A (red)
can see multiple images of galaxy B (yellow). This model of the universe, called
a tore (a doughnut shape), is constructed from a square stuck together on two
sides. Instead of one line linking the galaxies, light from galaxy B can now reach
galaxy A by various routes; as a result, the observer in galaxy A sees images of
galaxy B from various directions. Though the tore is finite, the observer is under
the illusion that space is much larger than it really it–his or her image
of space is similar to a grid of repeated cells. |
Could
we be living deep in a cosmic mirage, where rays of light multiply and distort our
perceptions of space? Instead of being flat and infinite, might space not in fact
be folded up–and our sense of the universe’s vastness just an illusion?
It is extremely difficult
for most of us to give a shape to something as intangible as space. For the physicist,
the question only has meaning if couched in the language of geometry. So then the
question becomes: how can geometry best represent physical space–the space we inhabit?
The problem is much more complex than it first appears. There is no doubt that the
space we live in and which we are familiar with is correctly described by what is
known as Euclidean geometry, a set of laws devised by the Greek geometer Euclid in
the third century BC.1 But space at the tiny microscopic level, according to quantum
mechanics, is a chaotic, fluctuating domain somewhat like the foam on the surface
of an ocean, while at the very large cosmological level we learn that space is in
truth “curved.” When we ask what shape or form space has, therefore, modern physics
actually gives us a variety of answers depending on the level at which we are looking.
Yet what exactly do we mean when we say that space is “curved”? The modern science
of cosmology stems largely from the equations of general relativity formulated by
Albert Einstein in the first two decades of the 20th century. According to these
equations, all space is deformed, or rather curved, by the distribution of matter
such as galaxies inside it. This curvature manifests itself as one of the universe’s
most fundamental forces: gravity.
Now if we turn to the universe as a whole–by which we mean huge scales of over 1025
metres–it appears that the virtually uniform distribution of galaxies throughout
the cosmos must curve space in a likewise uniform fashion. Aside from this constant
curvature, the universe should also have an underlying dynamic: in other words, it
can either be expanding or contracting.
On the basis of Einstein’s equations, Alexander Friedmann and Georges Lemaître
discovered in the 1920s a set of models for such curved space. The most simple version
points to so-called positive curvature, resembling a simple sphere that dilates from
the big bang onwards to reach a maximum size before contracting back into a final
“big crunch.” Space could also have no curvature or a negative curvature (forming
a “hyperbolic” shape that resembles a saddle). In both of these cases, the universe
expands forever, though the rate of expansion slows over time.
Recent observations have suggested that cosmic space is in fact close to the “no
curvature” model–in other words it is flat and almost Euclidean, like our normal
understanding of space. But the data also indicates that the universe is expanding
at an increasing rate, suggesting that some kind of “cosmological constant” is accelerating
the expansion rate. This constant may be understood as the energy of empty space.
But important questions remain. Can general relativity really give us a satisfactory
description of the shape of space at the largest scales? It is, after all, still
unclear whether space is finite or infinite: a spherical universe would certainly
be finite, but a Euclidean or negatively curved universe could be either finite or
infinite.
At this stage we need a new approach: that of topology, a branch of geometry devoted
to exploring the properties that define particular spatial objects. Where the science
of curvature falls shorts, topology can step in to tell us about the overall structure
of space.
Euclidean space can certainly be a lot more complicated than imagined. A surface
with no curvature, for example, is not necessarily flat. All one has to do to prove
this is take a rectangular piece of paper and stick it together at the edges to make
a cylinder. Just as with the original flat piece of paper, the surface of the cylinder
is still Euclidean. The cylinder’s surface thus still has no curvature, but unlike
the flat paper, it is finite in one direction. This is the sort of property that
topology, not curvature, can reveal. Cutting and sticking the flat piece of paper
into various new shapes does not alter the curvature of the paper itself, but radically
changes it overall shape–its topology.
The
dawn of ruffled space
In space which is flat, any two points can be joined only by a single line, or geodesic–meaning
simply a line in curved space. In what we call multi-connected space, on the other
hand, an infinite number of geodesics join two points together, as can be seen in
the diagram. It is this very property which gives such spaces extraordinary cosmological
relevance.
When we look at a faraway galaxy, we normally think that we are seeing just one unique
object, in one particular direction and at one distance. But if cosmic space is multi-connected,
rays of lights will replicate to produce multiple images of the observed galaxy.
Since our entire perception of space comes from analysis of the trajectories of light
rays, such a multi-connected space would mean we are plunged in a massive optical
illusion that makes the universe appear much larger than it really is. Distant galaxies
that we believe to be “original” would in fact be repeated images of a single galaxy.
A “ruffled” space would be just such a cosmos: a multi-connected domain of finite
volume, whose size would be smaller than the observed universe (whose radius is around
15 billion light-years). This space would create a topological mirage that constantly
repeated the images from shining sources–akin to standing in a hall of mirrors.
Similar optical illusions are already well known by astronomers, and go under the
name gravitational mirages. These occur when light from a distant object such as
a quasar is bent by the effect of a massive body situated on its sightline. As a
result, light rays from the star follow the curvature of space and are scattered
(see photo). What the observer sees is a group of phantom images surrounding the
intervening heavy object (known by astronomers as a “lentil”). This sort of mirage
can be directly ascribed to the local curvature of space around the lentil.
In the event of a topological mirage, however, there is no single heavy body like
a galaxy deforming space. Instead, space itself plays the role of a lentil: phantom
images are distributed in all directions and for all their points in history across
the entirety of space. Such a global mirage allows us to examine objects not only
from every possible angle, but also from every possible phase in their evolution.
For space to have this ruffled shape, it would have to be highly subtle and constructed
over very large scales. If not, we would have already identified phantom images of
our own galaxy and other well-known structures–a feat we have not yet achieved.
So how can we then establish the true topology of the universe? One method, cosmic
crystallography, attempts to pinpoint repetitions in the distribution of distant
objects. Studies of fluctuations in the cosmic background radiation–a fossil from
the big bang–might also indicate that space is ruffled by uncovering specific recurrences.
Experimental projects in both fields are underway, though neither the depth nor the
resolution of observations are yet good enough to draw conclusions about space’s
overall topology. The next few years, however, promise great things: deep surveys
of distant galaxies and quasars, along with new satellites probing background radiation.
Perhaps we will soon be able to give a form to space.
1. Based on five
axioms (including his most famous one, namely that parallel lines do not meet), Euclid’s
13-volume text Elements includes commonsense geometric theorems such as that the
angles in a triangle add to 180º. Non-Euclidean geometry was created in the
19th century. |
