Center of Universe?
Victor Grauer
The farther out we look all around us at the universe, at the
deepest penetration our telescopes allow, the closer we come to the
origin point of the Big Bang. This is easy to grasp, since, at least
according to currently accepted theories due to Hubble, we are looking
not only at distant space but distant time.
Now, according to General Relativity, the universe has no center
in space (the famous balloon analogy). But the expanding universe theory
seems to imply that it does have an *origin* in time and the farther out
we look all around us, the farther back in time we go toward that origin
point.
This seems to lead to some fantastic paradoxes, e.g. the center of
the universe is located at the periphery (which would imply that our
universe is "inside-out"). I know that Hawking has argued that there was
no actual beginning point in time. Perhaps some extrapolation from that
argument would save us here.
Hannu Poropudas
By the way do you know where in our Universe is the center of
the space (I mean the mass center of all galaxies).?
---
Best Regards,
Hannu Poropudas.
"It's Not What You Know That Matters
... It's Knowing What You Don't."
Emory F. Bunn
Miguel Lerma <mle...@arthur.ma.utexas.edu> wrote:
>Hannu Poropudas (hannu.P...@ericsson.fi) wrote:
>
>: By the way do you know where in our Universe is the center of
>
>masses of the continents in the surface of the Earth.
That's right. As far as we can tell, the Universe is roughly
homogeneous on large scales. There doesn't seem to be any special
point that you'd want to call "the center." The analogy with the
surface of the Earth is a good one. Let me expand on it just a little
bit, in case anyone missed the point.
The Universe may be spatially finite and yet have no boundary. This
initially may sound impossible, but it's not. The surface of sphere
is a two-dimensional surface that has finite area and yet has no
boundary, and one can imagine analogous three-dimensional "surfaces"
(although such objects are hard to visualize).
If the Universe is of this form, then we can get some intuition about
what's going on by considering the analogy with the spherical
surface. It's crucial to remember that in this analogy the entire
three-dimensional volume of space is being compared to the
two-dimensional surface of the sphere. It may help to imagine a
species of little two-dimensional creatures crawling around on the
surface of the sphere. Since they live in only two dimensions, they
have no idea that their world is actually "curved" into "the third
dimension." As far as they're concerned, that two-dimensional surface
is all there is.
Now, these creatures could do experiments and discover that their
world is shaped like a sphere. They might find this result surprising
at first, but they'd get used to it eventually, and they'd realize
that although their world has finite area, it has no boundary. Nor
does it have a center, at least not one that they can point to. There
is no point on the surface of the sphere that you'd call the center:
all points on the surface look exactly the same. Similarly, if our
three-dimensional Universe is curved like a sphere, it could be finite
and yet have no center.
On the other hand, our Universe might be infinite. We have no way to
tell. If it's infinite, it still might not have a center. It might
just stretch on forever, with each point looking more or less like
every other point.
In fact, if you take our observations of the distribution of stuff in
the Universe, and extrapolate them in the simplest possible way, you
find that these two possibilities (curved like a sphere or extending
forever homogeneously) are the two simplest, most natural hypotheses.
That doesn't necessarily mean that either is right, of course. The
big problem is that we can only see a finite amount of the Universe:
since the Universe is only about 15 billion years old, we can't see
anything further away than about 15 billion light-years. That
distance is known as our "horizon." The Universe seems to be pretty
much homogeneous throughout the volume enclosed by our horizon, so
it's natural to guess that maybe things continue that way outside of
the horizon, but since we can't see out there, that's just a guess.
>You can try to define it as (1/M) SUM r_i m_i, where M is
>the total mass of the universe, m_i is the mass of the i-th
>particle of the universe, r_i is a position vector of that
>particle, and the sum goes throght all the particles in the
>universe (perhaps about 10^80, I am ignoring quantum efects).
>But in a curved space there is no such a thing as "position
>vector". If you substitute it by, say, the lenght of the
>geodesic from some "fix" point taken as origin, the result
>is going to depend on the point chosen.
Absolutely right. But things are even worse than that, because that
number 10^80 is the number of particles in the *observable* Universe
(i.e., within our horizon), not the number in the whole Universe. The
observable Universe is a sphere centered on us, since it's the set of
all points close enough to us for us to see them. So even if you
could get around the problems of spatial curvature, you wouldn't get
reliable results by computing the center of mass of all of the
observable particles. If you did compute such a thing, you'd find
that the center of the Universe was right here (or pretty close to
it), simply because you've artificially restricted your attention to a
sphere centered on us, instead of considering the whole Universe.
An alien in a distant galaxy could perform the same computation
using his own observable Universe, and he'd find that *he*
was at the center instead.
-Ted
Miguel Lerma
: By the way do you know where in our Universe is the center of
: the space (I mean the mass center of all galaxies).?
Probably it is indefined. It is like asking about a center of
masses of the continents in the surface of the Earth.
You can try to define it as (1/M) SUM r_i m_i, where M is
the total mass of the universe, m_i is the mass of the i-th
particle of the universe, r_i is a position vector of that
particle, and the sum goes throght all the particles in the
universe (perhaps about 10^80, I am ignoring quantum efects).
But in a curved space there is no such a thing as "position
vector". If you substitute it by, say, the lenght of the
geodesic from some "fix" point taken as origin, the result
is going to depend on the point chosen.
Miguel A. Lerma
Andrew Cooke
everything you say up to the last paragraph is more-or-less
correct (in my humble opinion!). the `fantastic paradoxes'
i think you are noticing are:
- spacetime is curved. if the universe is smaller at earlier times
then it must have a smaller volume. so volume is not increasing
as we go further and further away (back in time). one effect of
this is that the angular size of galaxies stops getting smaller
as they get further away(!). the details depend on which model
of the universe you use, but if you want more info check out
`the angular diameter distance relation' in astronomy textbooks.
- even at the big bang, for a flat or open universe, the spatial
extent of the universe is infinite. this gives me a headache if
i think about it for too long, but you might consider it
`paradoxical'....
don't forget that as we approach the moment of the big-bang
the physics becomes less and less well known as the energies get
higher and higher. the `big bang' is only a model and it becomes
more and more uncertain as we approach the initial moment
(incidentally, if anyone out there knows, to what extent are
singularity theorems immune to changing physics?)
andrew
--
work phone/fax: 0131 668 8356, office: 0131 668 8357
institute for astronomy, royal observatory, blackford hill, edinburgh
http://www.roe.ac.uk/ajcwww
Andrew Cooke
Andrew Cooke <A.C...@roe.ac.uk> wrote:
> - spacetime is curved. if the universe is smaller at earlier times
> then it must have a smaller volume. so volume is not increasing
...as fast as you would expect...
> as we go further and further away (back in time). one effect of
> this is that the angular size of galaxies stops getting smaller
> as they get further away(!).
andrew
(just thought i better correct that!)
Edward Green
>This is a question that has bothered me for years --
> The farther out we look all around us at the universe, at the
>deepest penetration our telescopes allow, the closer we come to the
>origin point of the Big Bang. This is easy to grasp, since, at least
>according to currently accepted theories due to Hubble, we are looking
>not only at distant space but distant time.
at distant time" just follows from the finite speed of light.
> Now, according to General Relativity, the universe has no center
>in space (the famous balloon analogy).
matter. What you describe may be a feature of a particular *solution* to
the field equations, but so far as I know is not a necessary feature of
all solutions.
>But the expanding universe theory
>seems to imply that it does have an *origin* in time and the farther out
>we look all around us, the farther back in time we go toward that origin
>point.
> This seems to lead to some fantastic paradoxes, e.g. the center of
>the universe is located at the periphery (which would imply that our
>universe is "inside-out").
out in all directions, we are really be looking at the same distant region
of space. We cannot look far enough out to see times right after the Big
Bang, so we cannot see all the way around the balloon to a point, but
merely see the perimeter of some "hidden region" (spooky).
Now, even assuming a universe with this topology, the distant region we
see in all directions is by no means special... it's more like our
antipode on the balloon. We happen to see it an earlier epoch, but it's
not "closer" to the "center". Heck, you just got through telling us that
the universe has no center :-) The Big Bang "happened" "everywhere" at
once, and the residual radiation that is supposed to be its signature is
permeating space.
In a nutshell, your "center is at the periphery" paradox mixes time and
space. Even relativity preserves *some* distinction. You can forget about
the balloon and see the situation clearly in a one-dimensional "rubber
band" universe, with signals of finite speed propagating around the band.
The Big Bang is the extrapolated time when the infinitesimal band first
popped into being. :-)
Suggestions for further handwaving:
Since we can't see 'round and 'round our universe, does this mean the
expansion was at some time faster than light? Or that now distant regions
of space have never communicated since the dawn of time? Strange. Then
again, everything in cosmology is strange. All cosmologies are
inconceivable, and yet one simply *is*.
Disclaimer:
This reply has started in certainty and ended in speculation. An
increasing density of "so far as I know" is to be assumed to the extent
necessary to vacate false assertions. :-)
>...Hawking has argued that there was
>argument would save us here.
--
Ed Green
egr...@nyc.pipeline.com
Andrew Cooke
Edward Green <egr...@nyc.pipeline.com> wrote:
>'t...@physics2.berkeley.edu (Emory F. Bunn)' wrote:
>
>>The
>>big problem is that we can only see a finite amount of the Universe:
>>since the Universe is only about 15 billion years old, we can't see
>>anything further away than about 15 billion light-years.
>
>big bang.
just after the `big bang' everything was so hot that photons were
continually scattering off particles (mainly electrons).
as things expanded and cooled this became less important, and
the effective change was actually quite sudden - so we can see
back to a certain point, but then no further because it is
`misty' due to all the scattering.
this `misty' early universe is what people are looking at when
they look at fluctuations in the microwave background. since
the universe was still expanding very rapidly at that time it
is at a very high redshift (about 1000 if i remember correctly).
that's why people were so interested in the fluctuations - they
were the very beginnings of the structures that are now galaxies.
and they are only just big enough (if they hadn't been seen people
would have had big problems because it would have been too smooth
to form the universe we have today).
andrew
Richard A. Schumacher
>Earth have a center, but it is outside of Earth's surface.
Correct! The center of Earth's surface is not on the surface.
And the same is true of the universe: the center of the universe
is not in the universe. It lies outside the universe, 15 billion
or so years in the past. The center is not "near the center of
Virgo Super Cluster" or anywhere else accessible to us.
Paul J. Kossick
of the universe in 3-dimensional space? To answer this would require
knowledge of the actual 'shape' of the universe in that space, and I
don't believe anyone actually agrees on that.
--
*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
Paul J. Kossick Standing on a hill in my mountain of dreams
kos...@crl.com Telling myself it's not as hard, hard, hard as it seems
*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
Hannu Poropudas
In article <4chedk$i...@geraldo.cc.utexas.edu>
which is dated 4 Jan 1996 20:47:48 GMT
mle...@arthur.ma.utexas.edu (Miguel Lerma) wrote:
>Hannu Poropudas (hannu.P...@ericsson.fi) wrote:
>
>: By the way do you know where in our Universe is the center of
>: the space (I mean the mass center of all galaxies).?
>
>Probably it is indefined. It is like asking about a center of
>masses of the continents in the surface of the Earth.
>
Earth have a center, but it is outside of Earth's surface.
Perhaps center of the space (center of all galaxies) is
outside of visible Universe, perhaps in the side of
contracting part of it (please take a look of README-articles
in my directory mentioned below).?
Perhaps it could be somewhere near center of Virgo Super Cluster.
Perhaps in invisible center of M87.?
>You can try to define it as (1/M) SUM r_i m_i, where M is
>the total mass of the universe, m_i is the mass of the i-th
>particle of the universe, r_i is a position vector of that
>particle, and the sum goes throght all the particles in the
>universe (perhaps about 10^80, I am ignoring quantum efects).
>But in a curved space there is no such a thing as "position
>vector". If you substitute it by, say, the lenght of the
>geodesic from some "fix" point taken as origin, the result
>is going to depend on the point chosen.
>
>
>Miguel A. Lerma
Sure there is no such a thing as "position vector" in a curved
space. You have to generalize vector concept to be as "directed
geodesic line" in a curved space. You have to define also how
to add, subtract and multiply by scalar these "directed geodesic
lines". For example on two dimensional sphere surface you can
define multiplication and division of these "directed geodesic
lines". One problem remain and that is how to define these
four operations for these objects which starts from different
points on the sphere. Also this "algebra" has two characteristic
features, namely non-associativity and non-distributivity.
Please take a look at my trial in WWW.FUNET.FI (or FTP.FUNET.FI)
and directory pub/doc/misc/HannuPoropudas article is PostScript
file called surface_algebras.Z.
David Jacob
On 4 Jan 1996, Hannu Poropudas wrote:
>
> By the way do you know where in our Universe is the center of
> the space (I mean the mass center of all galaxies).?
>
>
> ---
>
>
> Best Regards,
>
> Hannu Poropudas.
In the beginning, the Universe issued from a single point, and was
composed of a at least 11 dimensions. Then the universe became unstable,
and all but 3 of the dimensions collapsed, releasing tremendous energies,
and causing the center of the universe to actualy detach and become a
free wandering anomolie. The Center of the Universe has been wandering
around ever since, and has been sought after by the various civilizations
and species of the galaxy since the dawn of life. It is said that
whomsoever possesses the true center of the universe possesses infinite
power to reshape reality and bend others to his will...
Dave
ps--if any hollywood types wish to use the above for the
plot of a movie or something, dont forget my check...
Miguel Lerma
[...]
: Earth have a center, but it is outside of Earth's surface.
I guess the closest to this idea is what Richard A. Schumacher says
in his post: the center is 15 million years in the past (Big Bang).
But I think this assumes that only the 3-space is curved, and that
the whole 4-dimensional manifold made up with the whole past, present
and future history of the universe is Euclidean. It is still possible
that the universe is even more complicated than that. I think Hawking
suggested that the whole 4-dimensional manifold is curved, and that
the Big Bang and the Big Cruch are just points like any other
else except for the fact that in a certain system of cordinates
all coordinates except one (i.e.: the three spacial coordinates
but not time) converge there (something similar to what happens
in the surface of the Earth with meridians and paralels). If that
is so, then the "center" of the universe can be conceived only
with help of a fith dimension, so than the universe is a 4-sphere
inmersed in a 5-dimencional Euclidean space.
Miguel A Lerma
Emory F. Bunn
>I think the question might be better stated as: Where would be the center
>of the universe in 3-dimensional space? To answer this would require
>knowledge of the actual 'shape' of the universe in that space, and I
>don't believe anyone actually agrees on that.
I'm not sure I understand what you mean here. Space, as far as we can
tell, looks like a three-dimensional manifold. It fills all of
three-dimensional space, more or less by definition, and so it doesn't
have a "shape in 3-dimensional space." It seems to me that one only
talks about the "shape" of something if the something is embedded in
some larger volume. For example, we might talk about the shape of a
spherical volume that's embedded in a larger three-dimensional space.
But since the Universe doesn't seem to be embedded in a larger
three-dimensional space, I don't know what meaning one would attach to
a phrase like "the shape of the Universe in three-dimensional space."
If you like, you can choose to imagine that the Universe is embedded
in some space of a larger number of dimensions. This is a useful
thing to do when trying to get some intuition about curved spacetime,
although we have no particular reason to believe that space really is
embedded in such a way. In this context, it would make sense to talk
about the shape of the Universe within this larger (four or more
dimensional) space.
-Ted
Edward Green
>big problem is that we can only see a finite amount of the Universe:
>since the Universe is only about 15 billion years old, we can't see
>anything further away than about 15 billion light-years.
big bang.
Ed Green
egr...@nyc.pipeline.com
Andrew Cooke
Edward Green <egr...@nyc.pipeline.com> wrote:
>'a...@reaxp01.roe.ac.uk (Andrew Cooke)' wrote:
>> as things expanded and cooled this became less important, and
>> the effective change was actually quite sudden - so we can see
>> back to a certain point, but then no further because it is
>> `misty' due to all the scattering.
>years? A thousand? A nanosecond?
i can't remember. certainly not long compared to the age of
the universe. it would also depend on the details of the model
(density, inflation, etc).
i *should know* (and so should my office mate!) - i'll check it
when i'm next in the library.
>> this `misty' early universe is what people are looking at when
>> they look at fluctuations in the microwave background. since
>> the universe was still expanding very rapidly at that time it
>> is at a very high redshift (about 1000 if i remember correctly).
>
>I though I remembered the explanation that this radiation had "cooled".
>Now I suppose cooling can be distinguished form a straight red-shift by
>spectral signature. This is the "4K" background radition I take it.
>Comment?
yes, it was cooling and expanding and yes, it's the 4(3?)K
background - at a redshift of 1000 it would be at a temperature
of 4 x 10^12 K - a lot hotter!
when i say `redshift of 1000' i mean that if there was something
at that distance then, if we could observe a recognisable
spectrum, it would have a redshift of 1000 (ish). in practice
you are not going to able to do that because there wasn't any
significant line emission (the universe was in almost perfect
thermal equilibrium?). so all i am doing is giving the redshift
from a mathematical model, not an observed measurement. the
most distant things we *observe* (apart from the structure in
the microwave background) have a redshift of about 5.
for a `black body' (which is the spectrum of the background) you
can't measure a redshift - at different redshifts you just have
different temperature black bodies!
hope that makes sense,
MARSHALL DUDLEY
-> By the way do you know where in our Universe is the center of
-> the space (I mean the mass center of all galaxies).?
This is an interesting question. I believe some of the hyperspace equations
show that if the universe is closed then there is no center, or alternative
every point can be viewed as the center. Of course to see this requires
viewing in more than 3 dimensions, but an analogy is to view the earth's 2
dimensions from 3 dimensions, and then it is obvious that no point on the
surface of the earth is the center, and that every point can also be viewed as
the center, since all other points on the surface are symetrically surrounding
it.
Marshall
Edward Green
> just after the `big bang' everything was so hot that photons
were
> continually scattering off particles (mainly electrons).
>
> the effective change was actually quite sudden - so we can see
> back to a certain point, but then no further because it is
> `misty' due to all the scattering.
The I assume this era was relatively brief by current models. A million
years? A thousand? A nanosecond?
> they look at fluctuations in the microwave background. since
> the universe was still expanding very rapidly at that time it
> is at a very high redshift (about 1000 if i remember correctly).
I though I remembered the explanation that this radiation had "cooled".
Now I suppose cooling can be distinguished form a straight red-shift by
spectral signature. This is the "4K" background radition I take it.
Comment?
> were the very beginnings of the structures that are now galaxies.
> and they are only just big enough (if they hadn't been seen people
> would have had big problems because it would have been too smooth
> to form the universe we have today).
Ed Green
egr...@nyc.pipeline.com
Bruce Scott TOK
: 'a...@reaxp01.roe.ac.uk (Andrew Cooke)' wrote:
:
: > just after the `big bang' everything was so hot that photons
: were
: > continually scattering off particles (mainly electrons).
: >
: > as things expanded and cooled this became less important, and
: > the effective change was actually quite sudden - so we can see
: > back to a certain point, but then no further because it is
: > `misty' due to all the scattering.
:
: The I assume this era was relatively brief by current models. A million
: years? A thousand? A nanosecond?
A few hundred thousand years. Pretty brief.
:
: > this `misty' early universe is what people are looking at when
: > they look at fluctuations in the microwave background. since
: > the universe was still expanding very rapidly at that time it
: > is at a very high redshift (about 1000 if i remember correctly).
:
: I though I remembered the explanation that this radiation had "cooled".
: Now I suppose cooling can be distinguished form a straight red-shift by
: spectral signature. This is the "4K" background radition I take it.
: Comment?
Yes. The temperature at recombination was some 3000 to 4000 K (it helps
to look things up; I almost told you the time at recombination was 3000
years, but that was the temperature I had remembered :-).
The cooling is different from a red shift. A red shift tells you a
relative velocity, while the temperature refers to a _random_ velocity
(in this case, momentum, since we're speaking of photons after
recombination).
BTW... recombination refers to the capturing of the free electrons in
the 'cosmic soup' by protons. Before this time, you had several species
of particle -- which mix is a function of the temperature and density --
all interacting via scattering. This keeps all the constituent
temperatures the same. So right before recombination everything had the
same temperature -- about 3000 K -- and right afterward, the radiation
bath and hydrogen bath still had the same temperature. But afterward
they evolved independently; the photon bath cools as the 'scale factor'
R expands (T is inversely proportional to R).
It is called "re"-combination because the physical process involved is
seen in the lab after you ionise a gas. A bit of historical baggage.
source:
J N Islam, An Introduction to Mathematical Cosmology (Cambridge, 1992)
--
Mach's gut!
Bruce Scott The deadliest bullshit is
Max-Planck-Institut fuer Plasmaphysik odorless and transparent
b...@ipp-garching.mpg.de -- W Gibson
Victor Grauer
red shifted photons which are relics of a time very close to the time of
the big bang, then, to me, this clearly means that the center of the
universe is at the periphery.
Bruce Scott TOK
: If, at the farthest reaches of space in all directions we see extremely
: the big bang, then, to me, this clearly means that the center of the
: universe is at the periphery.
Don't forget you are looking back in time. As someone else said, this
means the center of the universe is its beginning.
[At t=0 in the standard model, the curvature is infinite for all values
of the other three coordinates.]
Emory F. Bunn
Victor Grauer <gra...@oberon.pps.pgh.pa.us> wrote:
>If, at the farthest reaches of space in all directions we see extremely
>red shifted photons which are relics of a time very close to the time of
>the big bang, then, to me, this clearly means that the center of the
>universe is at the periphery.
That conclusion doesn't follow. When we look at the farthest reaches
of space, we are looking far into the past, since light from those
points has taken a long time to reach us. In fact, the limit on the
furthest points we can see is set by the age of the Universe: we can't
see objects further than about 15 billion light-years, since light
from more distant objects hasn't had time to reach us.
So when we look at the furthest objects we can see, we're necessarily
going to be seeing them as they were when the Universe was very
young. That's why points near the edge of the observable Universe
look like they're "close to the big bang". They're not *spatially*
any closer to the center than we are. The light we see from those
points did originate from *times* close to the big bang, but that's
just because we happen to be looking at those points from very far
away; it's not anything special about that region of space.
Here's another way to put it. At this very moment there could be a
race of creatures in a galaxy at the edge of our observable Universe.
If they look our way, they will see radiation that left our patch of
space shortly after the moment of the big bang. They might conclude
that our patch of space is "close to the big bang", just as we might
conclude the same thing from our observations of their patch of
space. But we'd both be wrong.
-Ted
Edward Green
'a...@reaxp01.roe.ac.uk (Andrew Cooke)' wrote:
>
> for a `black body' (which is the spectrum of the background) you
> can't measure a redshift - at different redshifts you just have
> different temperature black bodies!
distinguished from an equivalent change in temperature. My thanks to Emory
Bunn also for mentioning this.
'b...@ipp-garching.mpg.de (Bruce Scott TOK )' was kind enough to look up
some numbers and also wrote:
>The cooling is different from a red shift. A red shift tells you a
>relative velocity, while the temperature refers to a _random_ velocity
>(in this case, momentum, since we're speaking of photons after
>recombination).
distinguish red shift from cooling, is that trying to tell us something?
Or this only an incredible coincidence brought on by lack of mathematical
imagination on the part of the universe?
(Not a rhetorical question, but I don't necessarily expect an answer)
Now, to shift gears, I've been trying my hand as a pro-aether crank
recently :-) , and when
't...@physics2.berkeley.edu (Emory F. Bunn)' writes:
>All that matters is that
>the expansion of the Universe stretched the wavelengths of all of
>those photons by a factor of 1100.
why that just gives me verbal fodder!
From a cosmological point of view it's apparent we regard the idea that
space is a *stuff* and matter in energy are embedded in that *stuff* as the
mildest milque-toast; indeed, isn't that the meta-model of GR, which
Einstein is supposed to have regarded as evidence for an aether?
This successful 'reification' of space seems to stand in spendid isolation,
a problem we sometimes describe as requiring the "quantization of
gravity", but may equally well require the "general relativization of the
remaining forces" . (Let me put myself out on a limb; when this happens,
I think virtual particles will be seen as a calculational method only)
Someday I hope to tell you what I think a string is, but not too soon.
I just can't model it mathematically at the moment. :-( :-(
--
Ed Green
egr...@nyc.pipeline.com
Miguel Lerma
> In article <4ck30f$b...@geraldo.cc.utexas.edu>,
> Miguel Lerma <mle...@arthur.ma.utexas.edu> wrote:
> >I guess the closest to this idea is what Richard A. Schumacher says
> >in his post: the center is 15 million years in the past (Big Bang).
> You mean "billion" rather than "million." Specifically, I should
> point out for the benefit of non-U.S. readers that I mean a
> U.S. billion, 1000000000, not a U.K. billion, which is 1000 times
> bigger. (I think even Nature uses "billion" in the U.S. sense
> now, by the way.)
Right, I meant (US) "billion". Sorry.
> >But I think this assumes that only the 3-space is curved, and that
> >the whole 4-dimensional manifold made up with the whole past, present
> >and future history of the universe is Euclidean.
> I can't speak for Richard Shumacher, but I can tell you that this isn't
> how I interpreted what he wrote. Standard theories of cosmology
> are based on the theory of general relativity. In general relativity,
> spacetime is modeled as a four-dimensional manifold, but definitely
> not a Euclidean one. Spacetime has curvature in all of these models.
[...]
Your remarks are perfectly sound. I was just playing around with
"naive" models of the universe, just to see in what extent they
could provide some meaning to the original question. In particular,
the idea that places the "center" of the universe in the Big Bang
comes from a model in which the universe is like a balloon growing
from an initial point, and the radial coordinate would be the time.
The center would correspond to t=0. Of course, if we want to
deal with "state of the art" models of the universe, we need to
look at relativistic cosmology.
By the way, I have always found intriguing the relation between
the local structure of the universe (as a differenciable manifold)
and its global topology. I think most of the time cosmologists make
implicit assumptions about how they are related. In particular they
estimate the size of the universe from its local curvature. However,
it seems to me that they are different problems. I can imagine, for
instance, manifolds of zero curvature and finite size, e.g. a plane
torus. Also I can conceive, say, infinite 2-manifolds with constant
positive curvature. A sofisticated but interesting example is the
following: let H be the the open upper half complex plane, Q the set of
rational numbers, H-hat = H union Q union {infinity}, j: H-hat -> P^1(C)
the j-invariant (P^1(C) is the Riemann sphere, j appears in the theory
of modular forms), and s: P^1(C) -> S^2 a stereographic projection. S^2
(2-sphere) is assumed to have its usual constant curvature differential
structure. Recall that j is invariant by the modular group, and that H-hat
can be partitioned into infinitely many fundamental domains. The interesting
thing is that the H-hat can be endowed with a contant positive curvature
differential structure via the map z -> s(j(z)), and each fundamental
domain maps bijectively to S^2. It is like having infinitely 2-spheres
glued together in a single 2-manifold.
In short, I do not think that the local structure of the universe
allows us to get conclusions about its global structure without
additional assumptions.
Miguel A. Lerma
Emory F. Bunn
Miguel Lerma <mle...@arthur.ma.utexas.edu> wrote:
>By the way, I have always found intriguing the relation between
>the local structure of the universe (as a differenciable manifold)
>and its global topology. I think most of the time cosmologists make
>implicit assumptions about how they are related. In particular they
>estimate the size of the universe from its local curvature. However,
>it seems to me that they are different problems. I can imagine, for
>instance, manifolds of zero curvature and finite size, e.g. a plane
>torus.
You're absolutely right. The "standard model" of cosmology
involves a couple of assumptions that people sometimes don't bother
to state explicitly:
1. The density is roughly uniform over very large scales
2. The Universe is simply connected.
There's pretty good evidence for 1, at least over the scales that we
can observe. On the other hand, if the Universe is much larger than
our horizon, it's quite plausible that the density might vary
dramatically over scales of, say, a trillion light-years, and we'd
never know it. So even if someone found an incredibly clever way to
measure the density within our horizon and thereby established that
space was negatively curved around here, that wouldn't prove that the
Universe was truly open, since openness is a global property that
depends on what things look like at arbitrarily large distances.
The second assumption (the one about topology) is the one you were
talking about. Cosmologists usually say that if the Universe has zero
or negative curvature, then it goes on forever. That conclusion
depends on assumption 2. For both flat and negatively curved models,
the only simply connected topology is the one that goes on forever,
but in both cases you can change the topology to get a compact space.
(In the flat case, the simplest way is to make space a 3-torus, as you
say. If there's uniform negative curvature then you have to go to
more complicated topologies.)
There is a relatively small literature of attempts to place
constraints on these alternative topologies. As far as I know, only
the flat toroidal case has been considered, since the open case is
much more complicated. As you'd expect, if the size of the torus is
much larger than our horizon, there's no way you can tell you're in a
torus rather than an infinite space. If the size is comparable to the
horizon or smaller, then there are observational tests you can
perform. If anyone is really interested, I can look up the limits
that have been placed on the torus size in these models. (I think the
length of the torus is constrained to be larger than something like
0.2 to 0.5 horizon sizes.)
>Also I can conceive, say, infinite 2-manifolds with constant
>positive curvature.
That's interesting. I didn't know that such things existed. I'm
afraid I didn't really understand your construction of such a thing on
first reading; I'll try to look at it more carefully later.
I have the impression that you can't make a noncompact 3-manifold with
constant positive curvature. Do you know if that's true or not? (I'm
a couple of miles from my book on Riemannian geometry at the moment;
I'll try to look it up later if I remember.) If that's true, then the
conventional wisdom that locally positive curvature implies a finite
Universe depends only on assumption 1 above, not on assumption 2.
(Note that we cosmologists think we're doing pretty well if our
conclusions depend on only *one* wholly unverifiable assumption! :-)
>In short, I do not think that the local structure of the universe
>allows us to get conclusions about its global structure without
>additional assumptions.
Agreed. For what it's worth, most working cosmologists know
this, although we're frequently to careless to say so.
-Ted
Miguel Lerma
> >Also I can conceive, say, infinite 2-manifolds with constant
> >positive curvature.
> That's interesting. I didn't know that such things existed. I'm
> afraid I didn't really understand your construction of such a thing on
> first reading; I'll try to look at it more carefully later.
I mentioned that example because it is related to a problem I have
been studying recently, but there are simpler examples. However
one should be cautious, because that kind of surface is not
completely homogeneous, it contains some exceptional points
(perhaps I abused the language by calling it "manifold"). For instance,
consider the Riemann surface for f(z) = sqrt(z). It can be seen as two
Riemann spheres glued along the negative real axis. If you identify each
of those Riemann spheres with S^2 and look at its differential structure,
you see an object of constant positive curvature with the size of two
spheres of the same curvature. But it contains two branching points:
0 and infinity. Almost everywhere that manifold looks like a sphere,
but at z=0 (and at z=infinity) little circles surounding that point
have length close to 4*pi*r instead of 2*pi*r.
In my example, the surface can be seen as infinitely many
spheres glued in a certain way along the negative real axis
and the interval [0,1728]. At almost every point that manifold
looks like a piece of sphere, but the point 0 is exceptional.
If you are close to z=0 in one of the spheres and go around it
in a small circle, you will go through six diferent spheres
making an arc of 180 degrees in each one, so you need to turn
180*6 = 1080 degrees arround that point to return to the
starting point. At the point z=1728, a path arround it goes
through two different spheres 360 degrees each, 720 in total.
At z=infinity the circles go through infinitely many spheres
and have infite length.
> I have the impression that you can't make a noncompact 3-manifold with
> constant positive curvature. Do you know if that's true or not? (I'm
[...]
I guess that can also be done in a similar way as above, but if
exceptional branching points are not allowed, you might be right.
Miguel A. Lerma
Ben Weiner
>By the way, I have always found intriguing the relation between
>the local structure of the universe (as a differenciable manifold)
>and its global topology. I think most of the time cosmologists make
>implicit assumptions about how they are related. In particular they
>estimate the size of the universe from its local curvature. However,
>it seems to me that they are different problems. I can imagine, for
>instance, manifolds of zero curvature and finite size, e.g. a plane
>torus.
Interestingly, universes with torus-like geometry (that is, periodic
in one or more spatial dimensions, often called "small universes")
can be ruled out by observations. The spatial periodicity introduces
a long-wavelength cutoff and this distorts the spectrum of the cosmic
microwave background radiation. I believe the limits from observations
are actually good enough now to rule out these small-universe models.
>Also I can conceive, say, infinite 2-manifolds with constant
>positive curvature. A sofisticated but interesting example ... [deleted]
>In short, I do not think that the local structure of the universe
>allows us to get conclusions about its global structure without
>additional assumptions.
Well, quite probably, but the limits on alternative (non-Friedmann)
universes may be quite strict. If one can cook up a model which
fits all the observations and makes no predictions differently from
a Friedmann model, that is of theoretical interest but may not be
of any practical interest to astronomers, especially observers.
Them's the breaks.
--
NO STEP
Robert Israel
Ben Weiner <bwe...@electron.rutgers.edu> wrote:
> Outside the horizon, the universe could do just
>about anything it wants and we'd have no way of knowing. There could
>be a domain wall sweeping everything into oblivion heading for us at
>the speed of light, ready to come through the horizon tomorrow, and we
>wouldn't know - nor should we care much. It wouldn't get here for
>approximately a bazillion years anyway.
??? If it's heading directly for us at the speed of light, we wouldn't
know about it until it was already here.
--
Robert Israel isr...@math.ubc.ca
Department of Mathematics (604) 822-3629
University of British Columbia fax 822-6074
Vancouver, BC, Canada V6T 1Y4
Ben Weiner
>Interestingly, universes with torus-like geometry (that is, periodic
>in one or more spatial dimensions, often called "small universes")
>can be ruled out by observations. The spatial periodicity introduces
>a long-wavelength cutoff and this distorts the spectrum of the cosmic
>microwave background radiation. I believe the limits from observations
>are actually good enough now to rule out these small-universe models.
That is, to rule out models in which the periodicity is significantly
smaller than the horizon size (the size of the observable universe),
as Ted Bunn said. Outside the horizon, the universe could do just
about anything it wants and we'd have no way of knowing. There could
be a domain wall sweeping everything into oblivion heading for us at
the speed of light, ready to come through the horizon tomorrow, and we
wouldn't know - nor should we care much. It wouldn't get here for
approximately a bazillion years anyway.
This is what I alluded to when I said that nonstandard topologies
may be theoretically amusing but not of much practical interest.
I believe de Oliveira-Costa & Smoot (1995, Ap.J. 448, 477 - "Constraints
on the Topology of the Universe from the 2 Year COBE Data") discusses
the present limits. Oh, I found another abstract:
Stevens, Scott & Silk, 1993, Phys Rev Lett 71, 20 -
"Microwave background anisotropy in a toroidal universe."
Abstract: Large-scale cosmic microwave background temperature
fluctuations are calculated for a universe with the topology of a
3-torus. In such a universe only perturbations which are harmonics of
the fundamental mode are permitted. By comparison with data from the
Cosmic Background Explorer satellite, we find that the minimum
(comoving) scale of a cubic toroidal universe is 2400/h Mpc for an n =
1 inflationary model. This is approximately an order of magnitude
greater than previous limits and 80 percent of the horizon scale,
implying that a topologically 'small' universe is no longer an
interesting cosmological model.
--
"If current World Wide Web usage trends continue, as with prior Internet growth,
we project that the US economy will collapse on June 10, 1998, as the rate of
white collar workers going pointy-clicky pointy-clicky all day goes asymptotic."
--- President's Council of Economic Advisors report, 12/1/95 (classified)
Edward Green
>Now, to shift gears, I've been trying my hand as a pro-aether crank
100 times: I will *not* write aether. I will *not* write aether...
I'd still like to know if the red-shift/temperature-shift equivalence in
black-body spectrums is some incredible cosmic coincidence, or the source
of deep insight. Does anybody have an opinion on that?
--
Ed Green
egr...@nyc.pipeline.com