**Matti Pitkanen** (*matpitka@pcu.helsinki.fi*)

*Fri, 27 Aug 1999 10:27:24 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Stephen P. King: "[time 625] Re: [time 623] Re: [time 620] [Fwd: Quantum General Relativity (was: Does afundamental time exist in GR and QM?)]"**Previous message:**Matti Pitkanen: "[time 623] Re: [time 620] [Fwd: Quantum General Relativity (was: Does a fundamental time exist in GR and QM?)]"**In reply to:**Stephen P. King: "[time 620] [Fwd: Quantum General Relativity (was: Does a fundamental time exist in GR and QM?)]"**Next in thread:**Stephen P. King: "[time 626] Re: [time 624] Re: [time 621] Winding Space. Space-time sheets"

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:

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

**Next message:**Stephen P. King: "[time 625] Re: [time 623] Re: [time 620] [Fwd: Quantum General Relativity (was: Does afundamental time exist in GR and QM?)]"**Previous message:**Matti Pitkanen: "[time 623] Re: [time 620] [Fwd: Quantum General Relativity (was: Does a fundamental time exist in GR and QM?)]"**In reply to:**Stephen P. King: "[time 620] [Fwd: Quantum General Relativity (was: Does a fundamental time exist in GR and QM?)]"**Next in thread:**Stephen P. King: "[time 626] Re: [time 624] Re: [time 621] Winding Space. Space-time sheets"

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