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Written by Sten Odenwald
Copyright (C) 1984 Kalmbach Publishing. Reprinted by permission ----------------------------------------------------------------------------
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 aree, 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 some "proofs."
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. 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. 