The following text is a contribution from Cliff Pickover.
Physicists can give no good reason why space has three dimensions.
Perhaps the dimensionality of space in our universe was "accidentally"
determined during the Big Bang, billions of years ago. It does
seem that life would be more challenging in other dimensions.
As we discussed, it would be difficult for digestive tracts to
run through a creature in two dimensions because the tract would
cut the creature into two pieces. Richard Morris in _Cosmic Questions_
suggests that if the dimensionality of space were four or greater,
then stable planetary orbits would not be possible. Morris suggests
that if a planet did manage to form, it would follow a path that
caused it to spiral into the sun. This line of thinking is extended
in Max Tegmark's wonderful recent article "On the Dimensionality
of Spacetime" appearing in the journal _Classical and Quantum
Consider a universe with _m_ spatial dimensions and _n_ time
dimensions. These universes are classified as (_n_+_m_)-universes.
For example, our universe could be a (3+1)-universe with three
spatial dimensions and one dimension of time. Max Tegmark of
the Institute for Advanced Study in Princeton, New Jersey, suggests
that all universes -- except for a (3+1)-dimensional universe
-- may be "dead universes" in the sense they are devoid
of observers. He believes that higher dimensional spaces cannot
contain traditional atoms nor perhaps any stable structures.
In a space with less than three dimensions, there may be no gravitational
force. In universes with more or less than one time dimension,
living creatures could not make predictions. These ideas are
so fascinating that I would like to explain them just a bit further.
Some kinds of universes are more likely to contain observers
than others. Here is some background. As far back as 1917, P.
Ehrenfest suggested that neither classical atoms nor planetary
orbits can be stable in a space with _n_ > 3. In the 1960s,
F. Tangherlini further suggested that traditional quantum atoms
cannot be stable in higher dimensional universes (see "For
Further Reading"). For physicist readers, these properties
are related to the fact that the fundamental Green's functions
of the Poisson equation .eq del sup 2 phi = rho -- which gives
the electrostatic/gravitational potential of a point particle
-- is .eq r sup <2-n> for _n_ > 2. As Tegmark points
out, this means the inverse-square law of electrostatics and
gravity become an inverse-cube law if _n_=4, etc. When _n_ >
3, the two-body problem no longer has any stable orbits as solutions
(see I. Freeman's 1969 paper).
In simple English, this implies that if you were in a four-dimensional
universe and launched planets toward a sun, the planets would
either fly away to infinity or they would spiral into the sun.
(This is in contrast to a (3+1)-universe which can, for example,
have stable orbits of moons around planets.) A similar problem
occurs in quantum mechanics, where a study of the Schr&oe.dinger
equation shows that the hydrogen atom has no bound states for
_n_ > 3. This seems to suggest that it is difficult for higher
universes to be stable over time and contain creatures that can
make observations about the universe.
Lower dimensional worlds (such as 1- and 2-dimensional worlds)
may not be able to have gravitational forces, as discussed in
_Gravitation_ by John Wheeler and colleagues, and in a paper
by S. Deser.
So far we have been talking about spatial dimensions, but
we may also postulate the existence of different time dimensions.
Tegmark believes that a universe will only be able to have observers
if there is just one time dimension (i.e., _m_=1). What would
it be like to live in a universe with more than one timelike
dimension? Would we have difficulty going through our daily routines
of life, job, and the search for an ideal mate? Even with two
or more time dimensions, you might _perceive_ time as being one-dimensional,
thereby having a pattern of thoughts in a linear succession that
characterizes perception of reality. You may travel along an
essentially one-dimensional (timelike) world line through the
(_m_+_n_)-universe. Your wrist watch would work. However, the
world would be odd. If two people moving in different time directions
happen to meet on the street, they would inevitably drift apart
in separate time directions again, unable to stay together! Also,
as discussed by J. Dorling, particles like protons, electrons
and photons are unstable and may decay if there is more than
one dimension of time.
All sorts of causal paradoxes can arise with more than one
dimension of time. However, I do not think this precludes life,
even if the behavior or the universe would be quite disturbing
to us. Also, electrons, protons, and photons could still be stable
if their energies were sufficiently low -- creatures could still
exit in _cold_ regions of universes with greater than one time
dimension. However, without well-defined cause and effect in
these universes, it might be difficult for brains (or even computers)
to evolve and function.
None of these arguments rule-out the possibility of life in
the fourth spatial dimension (i.e. a (4+1)-universe). For example,
stable structures may be possible if they are based on short
distance quantum corrections to the .eq 1/r sup 2 potential or
on string-like rather than point-like particles.
Recently, scientists and mathematicians have researched the
theoretical melting properties of ice in higher dimensions. In
particular, mathematicians Nassif Ghoussoub and Changfeng Gui,
from the University of British Columbia, have developed mathematical
models for how ice turns from solid into liquid in the seventh
dimension and have proven that if such ice exists, it likely
exhibits a different melting behavior than ice in lower dimensions.
This dependence on dimension, although not very intuitive, often
arises in the field of partial differential equations and minimal
surfaces -- recent results suggest that geometry depends on the
underlying dimension in ways that were not suspected in the past.
Other research suggests that there is something about eight-dimensional
spaces that makes physical phase transitions inherently different
from seven-dimensional spaces. If you want to read more about
what happens when you lick a seven-dimensional popsicle, see:
Ekeland, I. (1998) How to melt if you must. _Nature_. April 16,
1998, 392(6677): 654-655.
A. Pickover's Home Page
and the Mind, Computer Art, and More
his book Surfing Through Hyperspace for more information on the
Uncle Taz Library
Site search Web search
powered by FreeFind