Matti Pitkanen (matpitka@pcu.helsinki.fi)
Fri, 27 Aug 1999 10:27:24 +0300 (EET DST)
When this book is published?
By the way, 8-dimensional imbedding space is of maximal dimension in the
sense that only in this dimension two 4-surfaces in general intersect
in point just as two curves in plane intersect generalically
but miss each other in dimensions D>2.
This might have some deep implications. For instance, cognitive
spacetime sheets would generically intersect material
spacetime sheets in D=8 but not in D>9. D<9 maximimizes
geometric contact interactions in well defined sense.
Furthermore, two 3-surfaces in 7-dimensional lightcone
boundary delta M^4_+c xCP_3 or in a=constant hyperboloid can get
linked just as curves in 3-space get linked. Could the linking
of 3-surfaces can have any physical effects? It could if
the topological reaction destroying the linking requires
large enough energy: in this case linked 3-surface would be
confined.
Best,
MP
On Thu, 26 Aug 1999, Stephen P. King wrote:
>
>
> 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!
t!
> ake n = 7.
>
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