From: "JACK L. URETSKY (C) 1996; HEP DIV., ARGONNE NATIONAL LAB, ARGONNE, IL 60439" <JLU@hep.anl.gov>
Date: Sat, 29 Jun 1996 12:37:41 -0500 (CDT)
Hi Dwight-
You ask:
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I was wondering if someone could help me out? I was discussing with a
friend about dimensions. I was explaining that something in one
dimension is a point. Two dimensions is something with length and width,
but not heighth. Three dimensions has length, width, and heighth. If I
remember correctly, the fourth dimension is time.
I was wondering if first, am I correct about the fourth dimension being
time and why? My next question is there anything beyond the fourth (or
3rd) dimension? If so, what are they?
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My unequivocal answer is, "Yes and no".
Your descriptions of dimensions 1-3 show that you are
envisioning a special kind of space - one in which the distance is
given by x^2+y^2+z^2 (the "pythagorean" theorem in the example of
3 dimensions). But if you want to make time the "4th dimension", then
distance squared is given by x^2+y^2+z^2-t^2. Obviously, then, time as a
dimension means something quite different than x,y,z. Thus, if by
"dimension" you are talking about something that we "sense" then time
as a "dimension" is just a mathematical fictions.
I think that this is the strict answer to your question.
Now here is probably more than you want to know. In the
4 dimensional example, there were 3 + signs and 1 - sign. In the
3 dimensional example there were no minus signs. A space where distance
is given with no minus signs is called "Euclidean". There are many
useful multi-dimensional Euclidean spaces. The most notable are
the phase-spaces for many particles. A "phase-space" is a space where
the coordinates are the respective positions (3 coordinates each) and
momenta (3 coordinates each) of each particle. Phase space for 3 particles
is 18-dimensional. One can study hydrodynamics in thes multi-dimensional
spaces to arrive at interesting conservation theorems about many-particle
systems.
Now go back to the 4-dimensional example. Note that there 2
possible non-euclidean spaces, one with 1 minus sign and one with two
minus signs (it doesn't matter what "names" we give to the different
coordinates). In three dimensions there is one possible non-euclidean
space. In both the 4- and 3- dimensional cases I didn't care about
the overall sign. In 2-dimensions there appear to be 2 possible cases,
one Euclidean and one non-Euclidean (spaces with even dimensions always
seem to behave very differently from spaces with odd-dimensions). But
in some deep sense the two different 2-spaces are the "same". That
sameness is the basis for the "duality workshop" that my colleague
Zachos is currently sponsoring here at Argonne.
What do I mean by "the same"? I mean that any analytic function
or its complex conjugate goes over into a right-moving solution of the
wave equation or its left moving counterpart as the space goes from
Euclidean to non-Euclidean. This observation forms the basis of early
"string theory".
For more on two dimensional spaces (surfaces) embedded in higher
dimensional Euclidean spaces you could spend the summer working through
the classic text by Eisenhart, "An Introduction to Differential Geometry"
(Princeton 1947) which is still in print, easy, and fun.
Regards,
Jack
"What did Barrow's lectures contain? Bourbaki writes with some
scorn that in his book in a hundred pages of the text there are about
180 drawings (citation). (Concerning Bourbaki's books it can be said
that in a thousand pages there is not one drawing, and it is not at all
clear which is worse)." Arnold, "Huygens & Barrow, Newton & Hooke", p 40.