THURSDAY, 23 NOVEMBER 2000
"ToThe Inhabitance of SPACE IN GENERAL
And H.C. IN PARTICULAR
This Work is Dedicated
By a Humble Native of Flatland
In the Hope that
Even as he was Initiated into the Mysteries
Of THREE DIMENSIONS
Having been previously conversant
With ONLY TWO
So the Citizens of that Celestial Region
May aspire yet higher and higher
To the Secrets of FOUR FIVE or EVEN SIX Dimensions
Thereby contributing
To the Enlargment of THE IMAGINATION
And the possible Development
Of that most and excellent Gift of MODESTY
Among the Superior Races
Of SOLID HUMANITY"
-- Edwin A. Abbott, in his dedication of Flatland
While we are most of the time comfortable with a world of three
dimensions, it may be intriguing to ask why exactly that is the
number. But perhaps it isn't. For roughly a century, scientists have
been introducing higher dimensions into their theories of nature. Often
it has merely been a mathematical convenience, for example the notion
of time as the fourth dimension among the three of space. However, the
practical results of Einstein's General Relativity suggest that space
behaves as if it were curved in a further dimension, the fourth
spatial dimension.
An analogue can be drawn with the most familiar of examples, the
surface of Earth. Only for a few thousands of years the Earth has been
known to be a sphere. Did we not know of the sun and the planets, most
everyday observations would show the world as flat. In the larger
scale, effects of curvature become apparent: two vertical poles of
equal height will cast shadows of slightly different length, if they
are separated by a sufficient distance. This is what made the ancient
Greeks conjecture the idea of a the world as a sphere.
The geometry of the surface of a sphere leads to the practical
conclusion that, although the surface has a definite area, it has no
borders. Traveling in any direction will eventually bring you to where
you started. If this is the case with the universe, being curved in a
higher spatial dimension, it might naively look like it would solve
the problem of the border of the universe, and what lies beyond, for
there would be no beyond.
On the contrary, there indeed is a border. On the Earth, we are aware
of the third dimension and hence the curvature of ''space'' is not
something mystical. Once we lift off the ground, we are outside the
two-dimensional space. The situation would be more interesting if we
actually perceived only the two dimensions of the surface. The kind of
world is described by Edwin A. Abbot in his fascinating classic of
science fiction, Flatland. If we were two-dimensional beings on the
surface of the Earth, we would have no direct way of observing the
curvature, like that used by the ancient Greeks. In fact, our brains
would be so accustomed to the two dimensions that it would take a
giant leap of faith even to consider the idea of the third dimension,
let alone perform any measurements. The important lesson the book
attempts to convey is that of a three-dimensional being who contacts
the main character, doing his best to explain what his world of higher
dimensionality is like. In fact, Abbott was a mathematics teacher and
wrote the novel in order to give a hint of a four-dimensional
existence to us in three-space.
If some of us (in the two-dimensional world) had the arrogance to
suggest a third dimension, there would, in fact, be a rather simple
method for determining the curvature. If you draw a circle on a
sphere, the circumference is generally less than the familiar pi times
the diameter - note that the diameter is measured along the surface,
if you think you only know of two dimensions. Alternatively, if the
surface resembles a saddle, the circumference will be greater than
expected.
So far, such an experiment has not shown any signs of the curvature of
our three-space in the fourth dimension. However, as predicted by
General Relativity, the curvature will be larger in the vicinity of
heavy objects such as the Sun. Indeed, the first direct evidence for
the theory was obtained during the solar eclipse of 1945 when the
position of a distant star was observed to differ from what it usually
was. The ray of light from the star had been slightly bent, because of
the curvature of space caused by the Sun. From the two-dimensional
analogy of a membrane curved in the third dimension, the universe
which we perceive is thought to exist on a ''brane'' contained in a
space of higher dimensions.
Given the experimental evidence, why is it that we only observe three
dimensions? More precisely, why does so much of physics work as if
there were only three of them? For example, the inverse square law of
gravitation is a direct consequence of three dimensions: the flux of
gravity from a massive object is evenly spread over an area
proportional to the square of the distance. Similarly, in an imaginary
four-dimensional world, the force of gravity would follow an inverse
cube law.
Surprisingly, recent theories (originally by Arkani-Hamed, Dimopoulos
and Dvali in 1998) suggest that, of all the fundamental forces, only
gravity might actually be aware of higher dimensions. This would
require, however, that space be ''curled up'' in the higher dimensions
on a very small, subatomic scale, leaving only three dimensions
accessible to anything practically observable. According to the
theory, gravitational flux from a mass would expand into all
dimensions (by an inverse nth power law, if there are n+1 dimensions)
in the small scale, but revert to the usual inverse square behaviour
in all measurements in practical scales. This would explain why
gravity is so much weaker than any other fundamental force, for
instance the electromagnetic. Moreover, the gravitational law has only
been tested down to the millimetre scale, while the scale of the
curled up dimensions is supposed to be significantly smaller than that
of fundamental particles.
Fundamental as electrons and quarks may seem, there are currently
known to be twenty-four different ''elementary'' constituents of matter,
so it is no surprise that scientists are working on a more elegant and
unified theory. A bold idea about as old as the higher dimensions, the
string theories are based on a single kind of entity which would
appear as different particles when in different working modes. These
would be open or closed loops of ''string'', and the kinds of vibration
of the string, for instance the frequency, would give the different
particle modes. There are several slightly different string theories,
all of which require the existence of ten or more dimensions to
work. As is the case with the above theory of gravitation, the extra
dimensions would be curled up in scales of the so-called Planck length
- about 1019 times smaller than the atomic nucleus. In
order to explore such details, we would need particle accelerators
with about 1016 times more energy than the present
facilitites, so there is currently no hope of directly testing string
theories.
Zooming up to the cosmological scale, there are yet more mysteries to
be solved. One of the currently most important is the notion of the
dark matter. The observed amount of matter in the universe appears to
be notably less, than what is required by General Relativity, to
account for recent experiments of the expansion of the universe. These
have shown that the universe has a critical mass density - i.e. if it
were any higher, gravity would eventually draw everything into a Big
Crunch. This result, by the way, also proves that in the large scale,
the space is flat instead of being curved. To compensate fof the mass
deficiency ''dark matter'' has been proposed. It is thought to be very
different from ordinary matter, for instance in that it does not
gravitate on itself to form clusters. This may be because the dark
matter resides on a different brane - which, from our point of view,
is truly a parallel universe. As expected by the new theory of
gravitation, the flux of gravity may spread between two branes,
although nothing else (e.g. light) can pass through. It may just be
ordinary matter, but on this brane the only effect we see is that of
gravity, which is why it appears so distorted in relation to the rest
of matter.
Probably the most famous known anomalies created by gravity are black
holes. In the context of branes, these are places of extreme curvature
created by ultra-dense masses. In the two-dimensional analogy, they
are points very sharply stretched away from the plane of the
membrane. It may happen that this stretch hits an adjacent brane and
we get a ''wormhole'' into the parallel world. Alternatively, the
wormhole may form a shortcut into another place on our brane. The term
was originally coined from the two-dimensional example of the surface
of an apple. A worm, living in this two-space, find a shortcut to the
opposite side by digging through in the third dimension. Some day,
wormholes may be viable method of transport, but so far the problem
with the extreme gravitational fields exerted on the passengers has
not been solved.
A further consideration of black holes, suggested by Lee Smolin and
others, is that in the ''inside'' of a black hole, a sub-universe might
start expanding into something not unlike our known brane of
universe. In the two-space, the end of the stretch would expand into a
sphere, while the connection to the ''mother brane'' would stay
infinitely narrow. The latter is what makes the holes black to us, for
no information can flow out into this world. Smolin believes this is a
widely occurring mechamism for the evolution of universes, and the
origin of our world. The Big Bang would correspond to the creation
of the black hole in the first place, but the origin of the first
universe remains a question, probably in the shadowy realms of quantum
theories.
With the brave new theories of branes in higher levels of reality, it
can be said that physics has once again peeked at a whole new world of
phenomena that could hardly have been predicted, until some
inquisitive minds had the arrogance to question the obvious. While the
work on higher dimensions appears like an intellectual game, it is
likely that we see real-world applications some day. Like quantum
mechanics, which was certainly perceived as a mathematical amusement
by some people at the time it was invented, has brought us electronics
and computing. Maybe some day when this planet gets too crowded, we
can just take the tube to the next brane...
Links:
Risto A. Paju is an Undergraduate in Physics at Queens'
