Registered: Mar 2004
Posts: 132

I know that there is already a simple definition for Planck length. The simple one that I've relied on can be found here: http://www.physlink.com/Education/AskExperts/ae644.cfm . What still bothers me about how this deriviation of a discrete unit for space (and I know that there is much more to this constant than it's quick description) is that it has always been found using some sort of equational method. I am curious if a this value can be explained as a fundamental phenomenon regarding computational irreducibility, the distribution of difficulty in computations and densities among them.

That's a mouthful, and I'm not sure if it's all needed. Some deep thought on NKS pages 770 and 771 led me to this curiosity, and I'd reccomend reading them and their preceeding pages. To my eye, this is a ball of twine that can be unwraveled. Could there be an alternate deriviation of a discrete unit for space and time coming from NKS?

(here are the pages)
http://www.wolframscience.com/nksonline/page-770
http://www.wolframscience.com/nksonline/page-771

Jesse Nochella

Last edited by Jesse Nochella on 10-18-2004 at 08:07 PM

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Registered: Feb 2004
Posts: 35

Jesse,

Space only has meaning when related to time, and both space and time are ultimately defined by the speed of light. So it is the "speed" of light that has the more fundamental meaning. However, the idea of light traveling at a conventional speed is somewhat misleading; light "travels" at the absolute maximum speed that anything could possibly travel at. So once a beam of light goes, it can never be caught, and any information contained in the beam can never be removed from the universe.

If you were to travel with a beam of light from point A to point B, you would arrive at B without experiencing any passage of time. If you were to return back to A, with the light, you would again experience no passage of time. However, the local time at A would have increased during your journey, and you can consider that the increase in "time" at A represents the same amount of distance traveled in "space" from point A to B and back to A again: Therefore, any displacement in space also represents an equivalent displacement in time, and is regardless of how fast the displacement occurs (up to the speed of light, at which time would effectively stop to preserve causality).

So it is far more meaningful to think about the speed of light as the fundamental relationship between space and time, and then consider what this relationship actually represents. One way to think of the speed of light is as a "rate of causality". This is quite simply one "unit" of "space", divided by one "unit" of "time". If you take these units to be the Planck Units then all the mathematics of physics reduces to a very simple form indeed.

I presented a poster at the NKS 2004 conference in April related to this, which you can find here: http://www.wolframscience.com/confe...l/ahewitt-1.pdf , and here: http://www.wolframscience.com/confe...l/ahewitt-2.pdf

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Registered: Mar 2004
Posts: 132

I think that an essentially NKS derived discrete unit for observed space has yet to be found, could indeed be found, and when done so can give light as to why we encounter discreteness on small scales.

This is quoted from NKS pages 770-771. I'm not sure if the original thought behind what was written was indeed related to fundamental units of space. It just seems like that to me:

"Just as with the Turing machines of pages 761 and 763 there will be a certain density of cases where the problem is fairly easy to solve. But it seems likely that as one increases t, no ordinary Turing machine or cellular automaton will ever be able to guarantee to solve the problem in a number of steps that grows only like some power of t.
Yet even so, there could still in principle exist in nature some other kind of system that would be able to do this. And for example one might imagine that this would be possible if one were able to use exponentially small components. But almost all the evidence we have suggests that in our actual universe there are limits on the sizes and densities of components that we can ever expect to manipulate."

Where is observed discrete length in the computational universe? From what I've read in NKS I don't see any reason why it would be an actual constant. Instead, it would vary much in the way time does and correspond exactly to the computational difficulty of finding various solutions.

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Registered: Jun 2004
Posts: 78

Did anyone played with the idea that everything
moves with a speed of light?
Suppose in addition to 3+1 dimensions we have one
compact (small) dimension, and each line is not
a line, but a spiral trajectory on the surface
of a pipe. Suppose, next, that observed time
is proportional to the number of turns this
spiral makes. Is this idea crazy enough to be true?

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Registered: Mar 2004
Posts: 132

I've come to a conclusion about the matter on pages 770-771 and their relating to fundamental limits.

The physical limits of our universe at small and large scales are both caused by the phenomenon of computational irreducibility, and are exactly analagous to the kinds of limits we observe from within our own perception and analysis.

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Registered: Feb 2004
Posts: 4

The fundamental physical constants have all a relationship with the golden section, see
http://www.volkmar-weiss.de/chaos.html
and the publications of El Naschie and his school.

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