TWO ADDENDA TO MY PAPER "On the shape of the universe"
***1***
Famous Topologist Dennis Sullivan played the role in this paper of
preventing me from being too much of a topology ignoramus. He refused
to be a coauthor though.
Anyhow, the major issue which I wanted Sullivan to settle, but he did
not, was this.
W.P.Thurston's Geometrization Conjecture, now proved by G.Perelman
(and there is a book on it by John W. Morgan & Gang Tian),
classifies compact orientable 3-manifolds.
QUESTION: which of them have the property that
they contain at most ONE isotopy class of nonseparating surfaces?
Classify them into those having 0, 1, or more than 1.
(This is open problem #2 at the end of my paper.)
Arguments about charge quantization and orientability
by me, strongly suggest the universe (if a compact 3manifold)
must be in this subclass. (There also are arguments by me and others
suggesting the universe ought to be finite and 3D...)
Re that, Igor Rivin pointed out that a partial answer is in this paper:
NM Dunfield & WP Thurston: The virtual Haken Conjecture, experiments
and examples, Geometry & Topology 7 (June 2003) 399-441.
It appears (conjecturally) that "almost all" irreducible
3-manifolds are "Haken" and are NOT "rational homology spheres."
If this conjecture is correct then almost every possible irreducible
universe would feature quantized charge without difficulty.
In some sense that would be good, but in another not good
- namely charge quantization would not rule out many topologies.
However, any universe featuring TWO OR MORE
isotopy class of nonseparating surfaces is generically inconsistent
with quantized charge, and I conjecture "almost every" 3-manifold
is in this (physically excluded) class.
***2***
There is an additional instance of deducing something about the
topology of the universe from local physics. I did not mention it in
the paper, but probably I should have, because it is very famous and also because
the technique and its limitations both happen to coincide quite well
with the sort of technique I used.
It is this. After Einstein invented General Reativity in 1915 using
Riemannian Geometry, Hermann Weyl in 1918 invented a new kind
of geometry generalizing Riemann's and leading to a new physics theory
generalizing GR.
In Weyl's geometry, when you parallel-translate a vector around a
closed path P, you do not get back the original vector necessarily.
It may have moved. If the path P is a circle that is
infinitesimal in size (say area=epsilon) then this movement is
proportional to epsilon and the vector-valued
proportionality constant is proportional to the area of the
circle and is given by the "Riemann curvature tensor."
So far, that is normal Riemannian geometry. But with Riemann
the vector's length stays constant. With Weyl, no such restriction is
imposed.
Einstein immediately objected to Weyl's geometry and claimed it could
not be the geometry of our universe. Why?
Because we could translate a standard meter stick round a big circle
and the length of the meter would change. The meter is defined as
"The metre is the length of the path travelled by light in vacuum
during a time interval of 1/299,792,458 of a second."
The second is the duration of 9,192,631,770 cycles of radiation
corresponding to the transition between two energy levels of the
ground state of the caesium-133 atom.
So anyhow, with Weyl geometry, there would be no standard metre or
standard second. All Caesium atoms would not be the same. But
experimentally they all ARE the same.
So, Einstein concluded this could not be the geometry of our universe.
But let us think a bit more. Suppose to change the Caesium atom
enough to alter the standard second by 1 part in 10^12, we need to move
it around a path the size of the galaxy. Perhaps then we might live in
Weyl geometry and did not notice it.
I.e. it might be the numbers are small enough that Weyl might have
been right. Einstein's argument does not really rule it out.
Now you might be able to REALLY rule it out by arguing that somehow,
even the LOGICAL POSSIBILITY of a Caesium atom making a large journey -
even if it never actually happened - would create some sort of
contradiction preventing some kind of solution of a physics equation
from globally existing. Or you might argue that in the early
"pre-inflationary" universe, such large journeys did happen often
enough that since we see no evidence of them today, we cannot have Weyl geometry. If you
could do either, then Einstein's argument would again become valid.
Nobody actually HAS done either, but still there is nobody disputing
Einstein's argument (including Weyl did not dispute it either).
This episode later became famous because Weyl's mathematics later turned out
(BUT with a few crucial changes) to underly "gauge theories of particle physics."
Einstein's argument is quite similar to my own reasoning in my paper.
I too deduce, from observations about local physics,
that certain conceivable topologies for the universe,
cannot be the universe we live in. Like Einstein's
(and for the same reasons),
some of my arguments are not really fully valid, but nevertheless (also like
Einstein's) I do not see how they can seriously be questioned.
So as a result, we get, at the end of my paper, a large chart. You
choose the assumptions about the behavior of local physics you want to have,
and then look up in the chart, the conclusions about the topology of
the universe, that then seem forced.
--Warren D. Smith 5/13/2007