Wednesday, February 09, 2000

2



Heh.  So, I've been playing with a physics problem (among others) for the last few months,
and came to an answer just today.  The answer, you see, is...

2

Heh.  This is the type of thing that physicists find outrageously funny.  OK, the question,
that's going to take some time, but I'll give it a go.  It has to do with hyperspace -- the
theory of other dimensions besides our four of space and time.
The game I'm playing is to try to come up with a geometric theory of everything.  My
strategy is to begin with Einstein's Geometrodynamics -- the exquisitely beautiful theory
of spacetime curvature that so well describes gravity -- and extend it, in substance but
not in form, to account for all other physical laws; in particular, the laws and particle
zoo of the phenomenally successful Standard Model and Quantum Field Theory.
So, I've got a lot of the pieces worked out so far.  But for now I'll describe the one
with the 2.  It has to do with the five fundamental forces: gravity, electromagnetism, the
weak force, and the strong force.  Each of these forces is described by a set of vector
fields -- each of which you can visualize as an infinite set of arrows, one at each point
of spacetime.  Now, to measure anything in curved spacetime, you need a standard by which
to measure these arrows.  So, for a space with N dimensions, one defines a collection of N
arrows at each point to be the N unit length arrows all perpendicular to each other.  This
construction is called the "vielbein," which means "many legs" in German (according to
Hop).  These fields of N arrows, orthogonal and perpendicular, completely describe the
intrinsic geometry of the space.  So, for example, in four dimensional spacetime, the way
these arrows stretch and squeeze together around a massive body completely describes the
gravitational
field, and thus how a freely falling body will move in the nearby spacetime.  And
Einstein's equations tell you what set of arrows are allowed, i.e. what partial
differential equations the components of these arrows must satisfy.
Now, really neat stuff happens when you consider spacetimes with more than four
dimensions.  Lets say, five.  We assume that the fifth dimension is wrapped up on itself,
so we don't see it without looking down smaller than we are able -- just like a garden hose
looks like a one dimensional object, until we get up close and see that the surface is
really more of a cylinder.  So, to describe the five dimensional geometry, with one wrapped
up, we need a fifth standard arrow.  And, quite astoundingly, when we work out what
Einstein's equations are for the four components of the fifth arrow that point along four
dimensional spacetime, we recognize them as identical to Maxwell's equations for
electromagnetism. :)  So that's the magic of hyperspace -- we get the existence and
equations of motion for all the existing force fields by assuming Einstein's equations and
the existence of several extra wrapped up dimensions.
Ah, but what exactly is the shape of these wrapped up dimensions, such that they
produce not only electromagnetism, but the funny weak and strong forces as well?  Well,
since we're sticking by Al, they're going to have to satisfy his equations -- which takes
work to check.  But, it turns out, that all the forces appear if one assumes the wrapped up
surface to be the product of a five dimensional sphere and a three dimensional sphere, with
a circle divided out of it.  Here "product" is meant as in "the product of a line and a
circle is a cylinder."  And the tricky part was figuring out just how to frickin divide by
the circle.  Now, this surface, S5xS3/S1 will satisfy Einstein's equations if and only if
the radius of the five dimensional sphere divided by the radius of the three dimensional
sphere is...

2


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