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Does
Space Have More Than 3 Dimensions?
The intuitive
notion that the universe has three dimensions seems to be
an irrefutable fact. After all, we can only move up or down,
left or right, in or out. But are these three dimensions all
we need to describe nature? What if there are, more dimensions
? Would they necessarily affect us? And if they didn't, how
could we possibly know about them?
Some physicists
and mathematicians investigating the beginning of the universe
think they have some of the answers to these questions. The
universe, they argue, has far more than three, four, or five
dimensions. They believe it has eleven! But let's step back
a moment. How do we know that our universe consists of only
three spatial dimensions? Let's take a look at two of these
"proofs."
Proof
1: There
are five and only five regular polyhedra. A regular polyhedron
is defined as a solid figure whose faces are identical polygons
- triangles, squares, and pentagons - and which is constructed
so that only two faces meet at each edge. If you were to move
from one face to another, you would cross over only one edge.
Shortcuts through the inside of the polyhedron that could
get you from one face to another are forbidden. Long ago,
the mathematician Leonhard Euler demonstrated an important
relation between the number of faces (F), edges (E), and corners
(C) for every regular polyhedron: C - E + F = 2. For example,
a cube has 6 faces, 12 edges, and 8 corners while a dodecahedron
has 12 faces, 30 edges, and 20 corners. Run these numbers
through Euler's equation and the resulting answer is always
two, the same as with the remaining three polyhedra. Only
five solids satisfy this relationship - no more, no less.
Not content
to restrict themselves to only three dimensions, mathematicians
have generalized Euler's relationship to higher dimensional
spaces and, as you might expect, they've come up with some
interesting results. In a world with four spatial dimensions,
for example, we can construct only six regular solids. One
of them - the "hypercube" - is a solid figure in
4-D space bounded by eight cubes, just as a cube is bounded
by six square faces. What happens if we add yet another dimension
to space? Even the most ambitious geometer living in a 5-D
world would only be able to assemble thee regular solids.
This means that two of the regular solids we know of - the
icosahedron and the dodecahedron - have no partners in a 5-D
universe.
For those of you who successfully mastered visualizing a hypercube,
try imagining what an "ultracube" looks like. It's
the five- dimensional analog of the cube, but this time it
is bounded by one hypercube on each of its 10 faces! In the
end, if our familiar world were not three-dimensional, geometers
would not have found only five regular polyhedra after 2,500
years of searching. They would have found six (with four spatial
dimension,) or perhaps only three (if we lived in a 5-D universe).
Instead, we know of only five regular solids. And this suggests
that we live in a universe with, at most, three spatial dimensions.
All right, let's
suppose our universe actually consists of four spatial dimensions.
What happens? Since relativity tells us that we must also
consider time as a dimension, we now have a space-time consisting
of five dimensions. A consequence of 5-D space-time is that
gravity has freedom to act in ways we may not want it to.
Proof
2: To the
best available measurements, gravity follows an inverse square
law; that is, the gravitational attraction between two objects
rapidly diminishes with increasing distance. For example,
if we double the distance between two objects, the force of
gravity between them becomes 1/4 as strong; if we triple the
distance, the force becomes 1/9 as strong, and so on. A five-
dimensional theory of gravity introduces additional mathematical
terms to specify how gravity behaves. These terms can have
a variety of values, including zero. If they were zero, however,
this would be the same as saying that gravity requires only
three space dimensions and one time dimension to "give
it life." The fact that the Voyager space- craft could
cross billions of miles of space over several years and arrive
vithin a few seconds of their predicted times is a beautiful
demonstration that we do not need extra-spatial dimensions
to describe motions in the Sun's gravitational field.
From the above
geometric and physical arguments, we can conclude (not surprisingly)
that space is three-dimensional - on scales ranging from that
of everyday objects to at least that of the solar system.
If this were not the case, then geometers would have found
more than five regular polyhedra and gravity would function
very differently than it does - Voyager would not have arrived
on time. Okay, so we've determined that our physical laws
require no more than the three spatial dimensions to describe
how the universe works. Or do they? Is there perhaps some
other arena in the physical world where multidimensional space
would be an asset rather than a liability?
Since the 1920s,
physicists have tried numerous approaches to unifying the
principal natural interactions: gravity, electromagnetism,
and the strong and weak forces in atomic nuclei. Unfortunately,
physicists soon realized that general relativity in a four-dimensional
space-time does not have enough mathematical "handles"
on which to hang the frameworks for the other three forces.
Between 1921 and 1927, Theodor Kaluza and Oskar Klein developed
the first promising theory combining gravity and electromagnetism.
They did this by extending general relativity to five dimensions.
For most of us, general relativity is mysterious enough in
ordinary four-dimensional space-time. What wonders could lie
in store for us with this extended universe?
General relativity
in five dimensions gave theoreticians five additional quantities
to manipulate beyond the 10 needed to adequately define the
gravitational field. Kaluza and Klein noticed that four of
the five extra quantities could be identified with the four
components needed to define the electromagnetic field. In
fact, to the delight of Kaluza and Klein, these four quantities
obeyed the same types of equations as those derived by Maxwell
in the late 1800s for electromagnetic radiationl Although
this was a promising start, the approach never really caught
on and was soon buried by the onrush of theoretical work on
the quantum theory of electromagnetic force. It was not until
work on supergravity theory began in 1975 that Kaluza and
Klein's method drew renewed interest. Its time had finally
come.
What do theoreticians
hope to gain by stretching general relativity beyond the normal
four dimensions of space-time? Perhaps by studying general
relativity in a higher-dimensional formulation, we can explain
some of the constants needed to describe the natural forces.
For instance, why is the proton 1836 times more massive than
the electron? Why are there only six types of quarks and leptons?
Why are neutrinos massless? Maybe such a theory can give us
new rules for calculating the masses of fundamental particles
and the ways in which they affect one another. These higher-dimensional
relativity theories may also tell us something about the numbers
and properties of a mysterious new family of particles - the
Higgs bosons - whose existence is predicted by various cosmic
unification schemes. (See "The Decay of the False Vacuum,"
ASTRONOMY, November 1983.)
These expectations
are not just the pipedreams of physicists - they actually
seem to develop as natural consequences of certain types of
theories studied over the last few years. In 1979, John Taylor
at Kings College in London found that some higher- dimensional
formalisms can give predictions for the maximum mass of the
Higgs bosons (around 76 times that of the proton.) As they
now stand, unification theories can do no more than predict
the existence of these particles - they cannot provide specific
details about their physical characteristics. But theoreticians
may be able to pin down some of these details by using extended
theories of general relativity.
Experimentally,
we know of six leptons: the electron, the muon, the tauon,
and their three associated neutrinos. The most remarkable
prediction of these extended relativity schemes, however,
holds that the number of leptons able to exist in a universe
is related to the number of dimensions of space-time. In a
6-D space-time, for example, only one lepton - presumably
the electron - can exist. In a 10-D space-time, four leptons
can exist - still not enough to accommodate the six we observe.
In a 12-D space- time, we can account for all six known leptons
- but we also acquire two additional leptons that have not
yet been detected. Clearly, we would gain much on a fundamental
level if we could increase the number of dimensions in our
theories just a little bit.
How many additional
dimensions do we need to consider in order to account for
the elementary particles and forces that we know of today?
Apparently we require at least one additional spatial dimension
for every distinct "charge" that characterizes how
each force couples to matter. For the electromagnetic force,
we need two electric charges: positive and negative. For the
strong force that binds quarks together to form, among other
things, protons and neutrons, we need three "color"
charges - red, blue, and green. Finally, we need two "weak"
charges to account for the weak nuclear force. if we add a
spatial dimension for each of these charges, we end up with
a total of seven extra dimensions. The properly extended theory
of general relativity we seek is one with an 11 -dimensional
space-time, at the very least. Think of it - space alone must
have at least 10 dimensions to accomodate all the fields known
today.
Of course, these
additional dimensions don't have to be anything like those
we already know about. In the context of modern unified field
theory, these extra dimensions are, in a sense, internal to
the particles themselves - a "private secret," shared
only by particles and the fields that act on them! These dimensions
are not physically observable in the same sense as the three
spatial dimensions we experience; they'stand in relation to
the normal three dimensions of space much like space stands
in relation to time.
With today's
veritable renaissance in finding unity among the forces and
particles that compose the cosmos, some by methods other than
those we have discussed, these new approaches lead us to remarkably
similar conclusions. It appears that a four-dimensional space-time
is simply not complex enough for physics to operate as it
does.
We know that
particles called bosons mediate the natural forces. We also
know that particles called fermions are affected by these
forces. Members of the fermion family go by the familiar names
of electron, muon, neutrino, and quark; bosons are the less
well known graviton, photon, gluon, and intermediate vector
bosons. Grand unification theories developed since 1975 now
show these particles to be "flavors" of a more abstract
family of superparticies - just as the muon is another type
of electron. This is an expression of a new kind of cosmic
symmetry - dubbed supersymmetry, because it is all-encompassing.
Not only does it include the force-carrying bosons, but it
also includes the particles on which these forces act. There
also exists a corresponding force to help nature maintain
supersymmetry during the various interactions. It's called
supergravity. Supersymmetry theory introduces two new types
of fundamental particles - gravitinos and photinos. The gravitino
has the remarkable property of mathematically moderating the
strength, of various kinds of interactions involving the exchange
of gravitons. The photino, cousin of the photon, may help
account for the "missing mass" in the universe.
Supersymmetry
theory is actually a complex of eight different theories,
stacked atop one another like the rungs of a ladder. The higher
the rung, the larger is its complement of allowed fermion
and boson particle states. The "roomiest" theory
of all seems to be SO(8), (pronounced ess-oh-eight), which
can hold 99 different kinds of bosons and 64 different kinds
of fermions. But SO(8) outdoes its subordinate, SO(7), by
only one extra dimension and one additional particle state.
Since SO(8) is identical to SO(7) in all its essential features,
we'll discuss SO(7) instead. However, we know of far more
than the 162 types of particles that SO(7) can accommodate,
and many of the predicted types have never been observed (like
the massless gravitino). SO(7) requires seven internal dimensions
in addition to the four we recognize - time and the three
"every day" spatial dimensions. If SO(7) at all
mirrors reality, then our universe must have at least 11 dimensions!
Unfortunately, it has been demonstrated by W. Nahm at the
European Center for Nuclear Research in Geneva, Switzerland
that supersymmetry theories for space-times with more than
11 dimensions are theoretically impossible. SO(7) evidently
has the largest number of spatial dimensions possible, but
it still doesn't have enough room to accommodate all known
types of particles.
It is unclear
where these various avenues of research lead. Perhaps nowhere.
There is certainly ample historical precedent for ideas that
were later abandoned because they turned out to be conceptual
dead-ends. Yet what if they turn out to be correct at some
level? Did our universe begin its life as some kind of 11-dimensional
"object" which then crystallized into our four-
dimensional cosmos?
Although these
internal dimensions may not have much to do with the real
world at the present time, this may not always have been the
case. E. Cremmer and J. Scherk of I'Ecole Normale Superieure
in Paris have shown that just as the universe went through
phase transitions in its early history when the forces of
nature became distinguishable, the universe may also have
gone through a phase transition when mensionality changed.
Presumably matter has something like four external dimensions
(the ones we encounter every day) and something like seven
internal dimensions. Fortunately for us, these seven extra
dimensions don't reach out into the larger 4-D realm where
we live. If they did, a simple walk through the park might
become a veritable obstacle course, littered with wormholes
in space and who knows what else!
Alan Chocos
and Steven Detweiler of Yale University have considered the
evolution of a universe that starts out being five- dimensional.
They discovered that while the universe eventually does evolve
to a state where three of the four spatial dimensions expand
to become our world at large, the extra fourth spatial dimension
shrinks to a size of 10^-31 centimeter by the present time.
The fifth dimension to the universe has all but vanished and
is 20 powers of 10 - 100 billion billion times - smaller than
the size of a proton. Although the universe appears four-
dimensional in space-time, this perception is accidental due
to our large size compared to the scale of the other dimensions.
Most of us think of a dimension as extending all the way to
infinity, but this isn't the full story. For example, if our
universe is really destined to re-collapse in the distant
future, the three- dimensional space we know today is actually
limited itself - it will eventually possess a maximum, finite
size. It just so happens that the physical size of human beings
forces us to view these three spatial dimensions as infinitely
large.
It is not too
hard to reconcile ourselves to the notion that the fifth (or
sixth, or eleventh) dimension could be smaller than an atomic
nucleus - indeed, we can probably be thankful that this is
the case.
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