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 radiation .
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.
Source: http://www.astronomycafe.net/cosm/dimens.html
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