Universe topology

What does it mean when people speak of the universe's topology? Does this
topology imply any type of large scale variation in the universe's
fundamental physical constants?

-- Michael S.

In article , "Michael S."
writes:

What does it mean when people speak of the universe's topology? Does this
topology imply any type of large scale variation in the universe's
fundamental physical constants?

In general, it refers to the "shape" of the universe. A two-dimensional
analogue would be saying that a torus has a different topology than the
surface of a sphere etc. If you think of differences between surfaces
which are preserved if you are allowed to stretch but not tear the
surface, you get the idea.

The sum of the densities of matter and the cosmological constant
determine the curvatu if the sum is 1, the simplest case is the
3-dimensional analogue of a plane, if it is greater than 1, the
3-dimensional analogue of the surface of a sphere, if it is less than 1,
the 3-dimensional analogue of a saddle (roughly speaking). These are
the "simplest topologies". However, the surface of the cylinder has the
same curvature as a plane (you can wrap a sheet of paper around a
cylinder without tearing it) but has a different topology. For the
cases of positive and negative curvature, it is also possible that the
topology is more complex than the simplest case. It could be the
3-dimensional analogue of a torus, say, or (as is now in the news) the
Poincare' dodecahedron.

[Mod. note: accented character fixed up -- mjh]

In article , "Michael S."
writes:

What does it mean when people speak of the universe's topology? Does this
topology imply any type of large scale variation in the universe's
fundamental physical constants?

In general, it refers to the "shape" of the universe. A two-dimensional
analogue would be saying that a torus has a different topology than the
surface of a sphere etc. If you think of differences between surfaces
which are preserved if you are allowed to stretch but not tear the
surface, you get the idea.

The sum of the densities of matter and the cosmological constant
determine the curvatu if the sum is 1, the simplest case is the
3-dimensional analogue of a plane, if it is greater than 1, the
3-dimensional analogue of the surface of a sphere, if it is less than 1,
the 3-dimensional analogue of a saddle (roughly speaking). These are
the "simplest topologies". However, the surface of the cylinder has the
same curvature as a plane (you can wrap a sheet of paper around a
cylinder without tearing it) but has a different topology. For the
cases of positive and negative curvature, it is also possible that the
topology is more complex than the simplest case. It could be the
3-dimensional analogue of a torus, say, or (as is now in the news) the
Poincare' dodecahedron.

[Mod. note: accented character fixed up -- mjh]

"Michael S." writes:

What does it mean when people speak of the universe's topology?

They are speaking of the possiblity that distant points within the universe
may be "indentified" in some regular fashion. For example, the universe
might have the topology of a "3-torus," wherein points on opposite faces
of a hexahedral "unit cell" would be considered "identical." This would be
somewhat analogous to the situation you would be in if attempting to leave
your house by the front door simply brought you right back in through the
back door, trying to climb out onto the roof from the attic simply braought
you back in via the basement, etc. Alternatively, one might identify two of
the faces of the "unit cell" mapped via a 180 degree twist; this would imply
that the Universe is a three-dimensional analog of a moebius strip or Klein
bottle. Other possible "unit cells" and identifications of faces would lead
to still different topologies.

In the most general case, one might imagine identifying points with each
other in an essentially arbitrary (but smooth) fashion, in which case the
Universe would be riddled with "wormholes" --- a so-called "spacetime foam."

It is still an open question whether it is possible for the topology of the
Universe to change with time, or whether its topology was "chiseled in stone"
at the beginning of Time.


Does this topology imply any type of large scale variation in the
universe's fundamental physical constants?

No. That is an entirely unrelated (and possibly undefinable) question.
(Undefinable, because it makes no sense to talk about dimensionful
quantities "varying" --- it only makes sense to talk about variations
of dimensionless ratios.) Whether or not the "physical constants" might
vary with position or time has nothing whatsoever to do with questions
of what sort of nontrivial topology the Universe might or might not have.


-- Gordon D. Pusch

perl -e '$_ = \n"; s/NO\.//; s/SPAM\.//; print;'

"Michael S." writes:

What does it mean when people speak of the universe's topology?

They are speaking of the possiblity that distant points within the universe
may be "indentified" in some regular fashion. For example, the universe
might have the topology of a "3-torus," wherein points on opposite faces
of a hexahedral "unit cell" would be considered "identical." This would be
somewhat analogous to the situation you would be in if attempting to leave
your house by the front door simply brought you right back in through the
back door, trying to climb out onto the roof from the attic simply braought
you back in via the basement, etc. Alternatively, one might identify two of
the faces of the "unit cell" mapped via a 180 degree twist; this would imply
that the Universe is a three-dimensional analog of a moebius strip or Klein
bottle. Other possible "unit cells" and identifications of faces would lead
to still different topologies.

In the most general case, one might imagine identifying points with each
other in an essentially arbitrary (but smooth) fashion, in which case the
Universe would be riddled with "wormholes" --- a so-called "spacetime foam."

It is still an open question whether it is possible for the topology of the
Universe to change with time, or whether its topology was "chiseled in stone"
at the beginning of Time.


Does this topology imply any type of large scale variation in the
universe's fundamental physical constants?

No. That is an entirely unrelated (and possibly undefinable) question.
(Undefinable, because it makes no sense to talk about dimensionful
quantities "varying" --- it only makes sense to talk about variations
of dimensionless ratios.) Whether or not the "physical constants" might
vary with position or time has nothing whatsoever to do with questions
of what sort of nontrivial topology the Universe might or might not have.


-- Gordon D. Pusch

perl -e '$_ = \n"; s/NO\.//; s/SPAM\.//; print;'

Michael S. wrote:

What does it mean when people speak of the universe's topology? Does this
topology imply any type of large scale variation in the universe's
fundamental physical constants?

-- Michael S.


No, it does not assume any variations in the fundamental constants.
It is essentially a way of describing in how many ways you draw a
straight line from point A that eventually hits point A again. Let
me give you a few examples. In the most trivial form of a flat
universe, the universe will behave like an infinitely large flat
sheet of paper. You let the line start at point A, and it will never
come back to point A, but if you take your sheet of paper, and roll
it into a cylinder there is exactly one straight line starting at A
that will hit A from the other side after having gone one turn
around the cylinder. Now you can take the cylinder and glue its
ends together to get a torus, and then there are two different
kinds of lines that will close at A, and you can also find
straight lines that are combinations of these two kinds of lines.
On a sphere things are a bit different again, and there is only
one kind of "straight lines", great circles, but on the other
hand all the lines that start at A will return to A after one
round around the circles. These are just some of the simplest
examples of the possible topologies of the universe. There
is a good article on this in Scientific American, April 1999,
I believe.

Ulf Torkelsson

Michael S. wrote:

What does it mean when people speak of the universe's topology? Does this
topology imply any type of large scale variation in the universe's
fundamental physical constants?

-- Michael S.


No, it does not assume any variations in the fundamental constants.
It is essentially a way of describing in how many ways you draw a
straight line from point A that eventually hits point A again. Let
me give you a few examples. In the most trivial form of a flat
universe, the universe will behave like an infinitely large flat
sheet of paper. You let the line start at point A, and it will never
come back to point A, but if you take your sheet of paper, and roll
it into a cylinder there is exactly one straight line starting at A
that will hit A from the other side after having gone one turn
around the cylinder. Now you can take the cylinder and glue its
ends together to get a torus, and then there are two different
kinds of lines that will close at A, and you can also find
straight lines that are combinations of these two kinds of lines.
On a sphere things are a bit different again, and there is only
one kind of "straight lines", great circles, but on the other
hand all the lines that start at A will return to A after one
round around the circles. These are just some of the simplest
examples of the possible topologies of the universe. There
is a good article on this in Scientific American, April 1999,
I believe.

Ulf Torkelsson