Stephen P. King (stephenk1@home.com)
Thu, 26 Aug 1999 15:20:50 -0400
Looking at a paper by I.J. Good: "Winding Space" in "The Scientist
Speculates" I.J. Good ed.
"A problem that naturally fascinates philosophers of science and
theoretical physicists is why space has three dimensions, or whether it
does. [1] In this paper I shall first briefly discuss this question [2],
and then speculate on the possibility that ordinary three-dimensional
space in embedded in space of higher dimensionality. The suggestion I
shall make is that ordinary space is of infinite extent, but winds
around in space of say seven dimensions, without intersecting itself.
If you take six rods of equal lengths you can fit them together as a
tetrahedron, but, try as you may, you cannot add four more rods as you
should be able to do in four spatial dimensions. So it is clear that in
some sense space has just three dimensions when we are concerned with
ordinary lengths. But we should remember, with A. N. Whitehead [3], that
different numbers of dimensions may be appropriate for different kinds
of phenomena.
We may just happen to be the right size to think that space has just
three dimensions. Perhaps if our lengths were 10^24k meters (k = -3, -2,
-1, 0, 1, 2, ...) we would think space as having k + 3 dimensions. At
certain intermediate sizes we may have physical significance.
Courant [5] has stated that only in three dimensions is high fidelity
communication possible. We may speculate that there are really an
infinite number of dimensions, but that we can appreciate only three of
them because we ignore low-fidelity signals. I have not thought of any
reasonable rigorous formulation of this idea.
We may try to run away from the question by saying that 3 is a small
enough not to need an explanation. An explanation would have been more
in demand if the dimensionality had been 32650494425.
Finally, it is possible that God selected just three dimensions in
order that communication systems, such as ourselves, should be possible.
In the theory of relativity, space is assumed to be of three
dimensions, not embedded in more than four dimensions of space-time. In
most of what follows, I shall ignore the time dimension. In the Einstein
universe, the mean radius of curvature, R, of space is related to the
total mass, M, by the formula
4GM = pi
c^2 R,
where G is the gravitational constant and c is the velocity of light.[6]
In this model, space is assumed to be a 3-sphere, the hypersurface of a
hyperball. The curvature is an 'intrinsic' property of space, defined in
terms of the metric of space. For example, the curvature can be deduced
by measuring the angles of a very large triangle, and seeing by how much
the sum exceeds 180 degrees. Thus, the curvature can be defined without
assuming the real existence of the hyperball, but the hyperball is
valuable in providing a proof of the self-consistency of the model, if
the self-consistency of more elementary mathematical ideas is taken for
granted. Without the idea of the hyperball, the idea that space is
finite but unbounded would be intuitively difficult to accept.
It is interesting to consider some of the implications of the above
formula if we assume insisted that the universe is infinite. Then the
natural interpretation of M is the mass of the observable universe, i.e.
of those parts that are not receding from the observer faster than
light. Owing to the random distribution of matter in space we should
expect M to vary slightly when we switch from one observer to some other
very remote observer. Since it is thought that there are about 10^78
particles in any one observable universe, we may assume, on the basis of
binomial (heads-and-tails) random variation that the proportional
variation will be in the order of 10^-39. Since the radius of curvature
is about 10^28 cm, the variations will be in the order of 10^-11 cm, or
very roughly, h/(2 pi m c). Where h is Planck's constant, and m is the
mass of the electron.
The above argument should be compared with Eddington's method of
arriving at his 'Uncertainty constant', which measures the 'uncertainty
of the reference frame.'[7]
If, after smoothing out local irregularities, such as clusters of
galaxies, R varies, by any amount however small, then space would not
necessarily close on itself. It could, for example, bear much the same
relationship to a 3-sphere as a helix bears to a circle. The local
properties. The local properties of a helix of very small torsion
would be indistinguishable from those of a circle, and the likewise the
local properties of space might be indistinguishable from those of a
3-sphere.[8] But since space apparently has 'handedness,' [9] one's
belief in its isotropy is somewhat undermined, and the 3-sphere seems a
little less probable than it was before.
If the variations of R are somewhat random, as suggested above, then
space would be more analogous to a badly wound reel of cotton than to a
helix. We are thus led to a theory of 'winding space', but I have yet
determined whether the assumptions are mathematically self-consistent.
Suppose that optical or radio telescopes are one day powerful enough to
see up to a distance of 2piR (along a geodesic), that is enough to see
'right around the universe'. [10] If the universe is Einstein's, then,
if we looked in various directions we would see the same part of space
from various aspects, namely 'ourselves' 2piR/c seconds ago. But on the
present hypothesis of winding space we would see two different parts of
space when we looked in two opposite directions. Thus, there is some
hope of an experimental test. But even if no test could distinguish
Einstein's universe from 'winding space,' it would still be of
cosmological interest that space could be infinite and yet all the
physical consequences of Einstein's universe be valid. It would
exemplify the well-know hazards of extrapolation from our limited
knowledge to the universe as a whole.
Other consequences may follow from an elaboration of the hypothesis.
Suppose for example that out 3-space is embedded in an n-dimensional (n
> 3) having physical meaning, just as the hyperball could have. Imagine a trip taken along a geodesic with infinite speed. After going round 'full sphere', a point would be displaced by a small distance in a direction perpendicular to ordinary space. If we imagine the process repeated indefinitely, the displaced point would perform a random walk in space of n - 3 dimensions. (This is only an approximate description since there are three independent routes around the universe.) Now there is a theorem of Poyla's [11] to the effect that if a random walk is performed on a p-dimensional lattice, with unit steps, then return to the origin infinitely often is certain if p < 3. It is true that the conditions of this theorem are not quite applicable here, but it does suggest that we need n >/= 6 in order that the hypersheets should not be packed indefinitely densely. In fact, owing to the 'independent routes around the universe', which were mentioned parenthetically above, we need to take n = 7.
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