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parallaxicality
2009-May-01, 07:21 PM
It seems that WMAP is telling us the universe is flat. But if it's flat, then it's infinite. But it has existed for only a finite amount of time, which means it must have expanded infinitely quickly. This seems wrong to me.

loglo
2009-May-01, 07:32 PM
But if it's flat, then it's infinite.

Why do you say this?

parallaxicality
2009-May-01, 07:34 PM
Because it would then have no boundary. It can only be finite if it was curved.

speedfreek
2009-May-01, 07:38 PM
But if it's flat, then it's infinite. But it has existed for only a finite amount of time, which means it must have expanded infinitely quickly. This seems wrong to me.

It may have been infinite to begin with.

parallaxicality
2009-May-01, 07:41 PM
But isn't that effectively the steady state theory? How could the universe be expanding, since infinity cannot be larger than itself?

PraedSt
2009-May-01, 07:50 PM
I think the balloon analogy comes in here. Flat does not have to mean infinite.

loglo
2009-May-01, 07:51 PM
Higher dimensional mathematics can provide for finite yet unbounded spaces as pointed out by Einstein as early as 1920. His description is here (http://www.bartleby.com/173/31.html), its fairly accessible, "flatlander" type of language.

speedfreek
2009-May-01, 08:07 PM
How could the universe be expanding, since infinity cannot be larger than itself?

Infinity is not a real number. You can add 1 to infinity, an infinite number of times. It does not describe a real distance or quantity. It has no end.

Our observable universe might always have been part of an infinite universe.

Or on the other hand, the observable universe may not be quite flat. The radius of the curvature might be so large as to be undetectable at our scale. The last I heard, the observable universe is thought to be within 2% of being flat.

Or there is also the possibility of a flat universe that is finite.

01101001
2009-May-01, 08:10 PM
It can only be finite if it was curved.

Does not follow.

ESA: Interview with Professor Joseph Silk (http://www.esa.int/esaSC/SEMR53T1VED_index_0_iv.html)

So you have two possibilities for a flat Universe: one infinite, like a plane, and one finite, like a torus, which is also flat.

publius
2009-May-01, 08:19 PM
This is one of those tricky questions that can be hard to understand. First of all, let's clarify exactly what is meant by the statement the "universe is flat". What is meant (and sometimes I wonder if the sources making those statements even understand what it means) is that, *in a certain coordinate system*, the so-called co-moving coordinates, the spatial hyperslices are flat. The spatial hyperslices are not invariant, and thus have no real meaning in the covariant sense of GR.

In terms of the invariants, which do have meaning, *space-time* has curvature, no question about it. The space-time manifold, a 4D psuedo-Riemannian geometric structure which models our universe, has Riemann curvature. It has Ricci curvature as well. It is not Minkowski (and Minkowski itself is a psuedo-Riemannian structure, flat is not Euclidean because the metric is not postive-definite). What does that mumbo jumbo mean? It means space-time doesn't work like you think it does. It's an abstract mathematical structure that models what we can observe.

Now, that curved 4D geometric structure admits coordinatizations where the spatial part is flat, indeed a nice postive definite flat space (at constant co-moving time).

Now, let's try to put that mumbo-jumbo in some terms we can understand. Consider an infinite Euclidean plane, a 2D flat structure. Now, let's coordinatize it.

First choice that comes to mind is a Cartesian x-y system. Those are flat, straight lines. We've coordinatized a flat space with flat coordinates. But we don't have to do that.

Let's coordinatize with polar coordinates, r and theta. r = constant is a family of circles. These circles are curved (positive curvature) and finite. That is, at r = constant, varying theta moves us around in a closed, curved, structure. (theta = constant are a family of straight lines, but they are not parallel. r = constant are non-intersecting).

Now, we have an infinite flat plane coordinatized by curved, closed "hyperslices".

Now, rather than circles, do it with a family of hyperbolas (or parabolas). One coordinate selects an hyperbola, and the other chooses a point on that hyperbola. You've now coordinatized the plane by curves with negative curvature, infinite and open.

The invariants here deal with the structure of the plane itself, not the coordinates used to map it.

When one splits space-time into space and time, one is doing that, choosing coordinates. Space has no meaning other than the being the hypersurfaces orthogonal (and in the psuedo-Riemannian way) to some family of time-like word lines through the space-time.

It has no more meaning than that. Our universe has invariant Riemann curvature. What you say about "space" is up to you by whatever coordinates you choose.

-Richard

WaxRubiks
2009-May-01, 08:45 PM
well maybe it's about topography; that is more than about what co-ordinates you use. If something is cylindrical, then it must be curved; or spherical.

publius
2009-May-01, 08:51 PM
well maybe it's about topography; that is more than about what co-ordinates you use. If something is cylindrical, then it must be curved; or spherical.

That's exactly what I'm saying. The topology of the universe is about the space-time structure, which is an invariant. But the structure of space-time is very different from the structure of arbitrary spatial coordinate structures you can map onto the space-time manifold. Questions of what "space" is doing are about coordinates. They have nothing to do with invariants.

The "something" you imagine is a non-positive definite 4D dimensional space-time structure, not some notion of positive definite "space" you can imagine. Getting your mind wrapped around this, how different space-time is from "space" is difficult.

My point is, when 99% of people think and imagine "the shape of the universe" they are thinking of stuff that has no meaning.

-Richard

DrRocket
2009-May-02, 02:10 PM
It seems that WMAP is telling us the universe is flat. But if it's flat, then it's infinite. But it has existed for only a finite amount of time, which means it must have expanded infinitely quickly. This seems wrong to me.

There is in mathematics the notion of a deformation retract, which is, in crude language a map that can reflect either contraction or expansion, depending on how you look at it. It is quite possible to contract Euclidean space to a single point, continuously. Like this

For t in [0,1] and X a vector in Euclidean space (any dimension will do) let
f(t,X) = tX. Thenf(0,X)=0 for any X and f(1,X) = X for any X. f then describes a model for "expanding" a point to all of Euclidean space, simply by scaling. Moreover, for fixed t>0 the image of f(t,.) is the entire Euclidean space. Whether you consider that an "infinitely quick" expansion or not is a matter of semantics, but the map is continuous.

DrRocket
2009-May-02, 02:47 PM
This is one of those tricky questions that can be hard to understand. First of all, let's clarify exactly what is meant by the statement the "universe is flat". What is meant (and sometimes I wonder if the sources making those statements even understand what it means) is that, *in a certain coordinate system*, the so-called co-moving coordinates, the spatial hyperslices are flat. The spatial hyperslices are not invariant, and thus have no real meaning in the covariant sense of GR.

In terms of the invariants, which do have meaning, *space-time* has curvature, no question about it. The space-time manifold, a 4D psuedo-Riemannian geometric structure which models our universe, has Riemann curvature. It has Ricci curvature as well. It is not Minkowski (and Minkowski itself is a psuedo-Riemannian structure, flat is not Euclidean because the metric is not postive-definite). What does that mumbo jumbo mean? It means space-time doesn't work like you think it does. It's an abstract mathematical structure that models what we can observe.

Now, that curved 4D geometric structure admits coordinatizations where the spatial part is flat, indeed a nice postive definite flat space (at constant co-moving time).

Now, let's try to put that mumbo-jumbo in some terms we can understand. Consider an infinite Euclidean plane, a 2D flat structure. Now, let's coordinatize it.

First choice that comes to mind is a Cartesian x-y system. Those are flat, straight lines. We've coordinatized a flat space with flat coordinates. But we don't have to do that.

Let's coordinatize with polar coordinates, r and theta. r = constant is a family of circles. These circles are curved (positive curvature) and finite. That is, at r = constant, varying theta moves us around in a closed, curved, structure. (theta = constant are a family of straight lines, but they are not parallel. r = constant are non-intersecting).

Now, we have an infinite flat plane coordinatized by curved, closed "hyperslices".

Now, rather than circles, do it with a family of hyperbolas (or parabolas). One coordinate selects an hyperbola, and the other chooses a point on that hyperbola. You've now coordinatized the plane by curves with negative curvature, infinite and open.

The invariants here deal with the structure of the plane itself, not the coordinates used to map it.

When one splits space-time into space and time, one is doing that, choosing coordinates. Space has no meaning other than the being the hypersurfaces orthogonal (and in the psuedo-Riemannian way) to some family of time-like word lines through the space-time.

It has no more meaning than that. Our universe has invariant Riemann curvature. What you say about "space" is up to you by whatever coordinates you choose.

-Richard

I think one needs be even a bit more careful. This is the explanation that I have found in Wald's General Relativity.

First one assumes that the universe is homogeneous and isotropic. Then one has to define what that means. The definition involves the existence of a foliation of space-time into a one-parameter family of space-like hypersurfaces. On those hypersurfaces the metric inherited from the Lorentzian structure is actually Riemannian (positive-definite). The parameter plays the role of "time" but it is not really time in the Newtonian sense. Then one shows that the family of space-like hypersurfaces is of constant (sectional) curvature, and appeals to a classification scheme from general differential geometry to say that there only a limited number of possibilities for such spaces. Wald in fact throws out some constant curvature spaces obtained as quotient spaces on physical grounds -- and I am not totally comfortable with that. Manifolds such as flat tori are for instance not allowed in his scheme. That leaves the possibilities of Euclidean 3-space (the flat case), the 3-sphere (positive curvature) and hyperbolic manifolds (negative curvature). There are some fairly exotic hyperbolic three-manifolds (see the work of Bill Thurston), and I am not sure whether or not some of them ought to be considered as candidates, but I am not an expert on Thurston's work.

I have been told elsewhere that the foliation is related to some solutions of the Einstein field equations that are called "globally hyperbolic", which again relies on the assumption of homogeneity and isotropy. I think that this implies that the topology of the space-like slices is not dependent on any specific coordinate system, although a concrete realization of them may be. In other words the foliation itself is not unique, but there are properties of it that are common to all such foliations.

It is quite clear that this can only be a large-scale approximation. Space-time is clearly neither homogeneous nor isotropic on small scales. It does not foliate cleanly in any global way, and space-like slices at local levels are not of constant curvature. There is no global "time" parameter in the real universe, only in this "toy" universe.

In any case, it is my impression that in this toy universe there are things that can be said about the space-like slices that can be stated in an invariant way, or at least in a way that does not depend on the specific foliation chosen.

Jeff Root
2009-May-02, 06:34 PM
Richard, DrRocket,

You guys may be smart, knowledgeable, and maybe even right, but I
don't appreciate "answers" that cannot be understood by anyone besides
you two and Ken G. I doubt that the original poster can get anything
useful from your verbiage except keywords to search on. I can't.

-- Jeff, in Minneapolis

Jeff Root
2009-May-02, 07:02 PM
ESA: Interview with Professor Joseph Silk (http://www.esa.int/esaSC/SEMR53T1VED_index_0_iv.html)
I have read Silk's 1989 revision of his book, 'The Big Bang'. It does
not mention a toroidal geometry. I can't imagine what it is supposed
to mean. He says "Flat is just a two-dimensional analogy." Does he
also mean that a torus is just an analogy? He says:

imagine the geometry of the Universe in two dimensions as a plane.
It is flat, and a plane is normally infinite. But you can take a sheet of
paper [an 'infinite' sheet of paper] and you can roll it up and make a
cylinder, and you can roll the cylinder again and make a torus [like
the shape of a doughnut]. The surface of the torus is also spatially
flat, but it is finite. So you have two possibilities for a flat Universe:
one infinite, like a plane, and one finite, like a torus, which is also flat.
Even if I accept that a cylinder is "flat", a torus is not. No way.

-- Jeff, in Minneapolis

Amber Robot
2009-May-02, 07:17 PM
Even if I accept that a cylinder is "flat", a torus is not. No way.

-- Jeff, in Minneapolis
I agree. I always thought "flat" meant that you couldn't cut the piece of paper to make the shape. You can roll a flat sheet of paper into a cylinder, no problem, but there's no way you're making a torus out of it without some kind of cutting or stretching.

But, cosmological topology is not my forte...

01101001
2009-May-02, 07:19 PM
Even if I accept that a cylinder is "flat", a torus is not. No way.

Wikipedia: Torus :: Flat torus (http://en.wikipedia.org/wiki/Torus#The_flat_torus)

The flat torus is a specific embedding of the familiar 2-torus into Euclidean 4-space or higher dimensions. Its surface has zero Gaussian curvature everywhere. Its surface is "flat" in the same sense that the surface of a cylinder is "flat".

Jeff Root
2009-May-02, 08:06 PM
I don't think there's much chance that this is right, but... This so-called
"flat torus" could be curved in one spatial direction and also curved at
90 degrees to that in the time direction to get something that might
look like a cylinder if you ignore the time dimension, but "looks" like a
torus in spacetime. Something sorta like that?

It still sounds like a ridiculous geometry to describe the Universe.
Flat I can go with. Ellipsoidal I can go with. Toroidal, even in four
dimensions, sounds ridiculous.

-- Jeff, in Minneapolis

DrRocket
2009-May-02, 08:36 PM
Even if I accept that a cylinder is "flat", a torus is not. No way.

-- Jeff, in Minneapolis

Sorry, but yes way. It depends on how the torus is constructed, but it is quite possible to construct a flat torus. The easiest is via quotient spaces, or "sewing" as follows, in two steps for some clarity: Identify the edges of the following rectangle by following the edge arrow of opposite sides

|---------->------------>--------------|
| |
| |
|---------->------------->-------------|

If you imagine a grid inside that rectangle then the grid is not distorted by the identifications, and what you get is a flat cylinder (by rolling up the tube to make the identifications or sewing of the two long sides. Angles and lengths transfer to the plane conformally just by rolling the cylinder, as is done with some printing presses. That is what makes it flat.

Now we do a similar thing to get a flat torus, only with a couple of additional identifications.

|---------->------------>--------------|
| |
V V
| |
|---------->------------->-------------|

With these identifications. sewing the long sides together and then the short sides together you get a flat torus.

DrRocket
2009-May-02, 08:46 PM
Richard, DrRocket,

You guys may be smart, knowledgeable, and maybe even right, but I
don't appreciate "answers" that cannot be understood by anyone besides
you two and Ken G. I doubt that the original poster can get anything
useful from your verbiage except keywords to search on. I can't.

-- Jeff, in Minneapolis

Try "Googling" "differential geometry". There is no way to discuss this without either being incomprehensibly vague, using the correct mathematical language, or writing a book on differential geometry and manifolds. Only the second alternative will work in the format that we have.

This stuff is not very amenable to simplifications without losing all of the content.

MessengerM104
2009-May-02, 08:58 PM
Even if I accept that a cylinder is "flat", a torus is not. No way.

J.R. from Minneapolis,
you should think (4 1-ce) of a torus
not as of a doughnut (its surface)
but as of your (presumably) flat video screen:
a spaceship from a primitive vido game
would usally re-enter from the opposite side of the screen
if you had steered it over the (NOT-) edge

you will easily convince yourself
that your space-ship behaves in a way
just as if confined to the surface of a doughnut-torus
(the topologies of that screen universum and a torus *proper* are same)
((and so they topologically are(!) the same))

to (modern!) math (and utterly opposed to physics ;) )
ALL you can define in a logically consistent way
is considered as *existing*

the video-screen micro-verse, that we are speaking of,
consistently describes/defines a *space*
that behaves to all intends and purposes like a torus
and hence can be rightfully considered to be one

now look what a disk-shape object will do
if pushed around on the screen:
(using the obvious *metric*)
((the one inherited from the physical monitor))
the circumference/diameter - quotient will keep exactly pi
and that's what defines a Riemann-ian space of flat curvature!

(if on the other side you take the (Riemann-)metric
as imprinted upon a doughnut-surface
from the surrounding matrix-space
you will find this particular model-torus curved...)
((so eat it up))
(((and 4-get it)))

publius
2009-May-02, 09:53 PM
Those lines along the left side are supposed go complete the side of the rectangle on the right, but I don't seem to be able to make the picture work when I post it, although it looks OK in the composition and editing window. Sorry, but try to draw the picture yourself with a pencil and paper.

That's the autoformatting -- it removes spaces and other stuff, I think. Trying using CODE tags around it, like so

|---------------------|
| |

But even there, variable width fonts on screen can screw you up

-Richard

speedfreek
2009-May-02, 10:15 PM
There is a diagram showing what you are describing, halfway down the page in the link below.

http://www.geom.uiuc.edu/video/sos/materials/overview/

And from 01101001's last link comes a relevant point:

In 3 dimensions you can bend a flat sheet of paper into a cylinder without stretching the paper, but you cannot then bend this cylinder into a torus without stretching the paper. In 4 dimensions you can (mathematically).

Amber Robot
2009-May-03, 01:18 AM
I didn't think about it in four dimensions. Sigh...

DrRocket
2009-May-03, 02:21 AM
I didn't think about it in four dimensions. Sigh...

In 4 dimensions you just take the cartesion product of two 2-dimensional tori.

You can also do that in other dimensions. The usual torus is just the product of two circles. That not only gets you the torus it gets you the abelian topological group structure as well. The circle, identified with the unit circle in the complex plain under multiplication, is a group.

Jeff Root
2009-May-03, 11:28 AM
Even if I accept that a cylinder is "flat", a torus is not. No way.
Sorry, but yes way. It depends on how the torus is constructed,
but it is quite possible to construct a flat torus.
If I correctly interpret the Wikipedia article Zero One linked and
speedfreek quoted, it can't be done in the ordinary world, only in
an imagined world of more than three dimensions.

Rolling a sheet into a cylinder leaves the grid undistorted, but rolling
the cylinder into a torus stretches the grid. Straight lines are not
parallel anymore.

I don't know how they remain parallel in a four-dimensional world, but
I'm pretty sure they can't remain parallel in a three-dimensional world.

I'll ask a couple of questions that may or may not be relevant.

A sheet can be rolled into a cylinder. There are two ways to roll the
cylinder into a torus. One is by bending it so that the centerline of
the cylinder becomes a circle. The other is by bending the ends of
the cylinder in, as if you were going to form a sphere, but instead of
closing the hole at each end of the cylinder to form poles, you bend
the ends more sharply so that they go inside the cylinder, until the
opposite ends meet. In this case, the centerline of the cylinder
becomes the central axis of the torus.

I think that in three dimensions, these two ways of forming a torus
are topologically identical. I haven't tried to analyze them to see if
they are geometrically identical. My questions are: Does it make a
difference in four dimensions? Is it relevant to the geometry of the
Universe?

-- Jeff, in Minneapolis

DrRocket
2009-May-03, 01:53 PM
I think that in three dimensions, these two ways of forming a torus
are topologically identical. I haven't tried to analyze them to see if
they are geometrically identical. My questions are: Does it make a
difference in four dimensions? Is it relevant to the geometry of the
Universe?

-- Jeff, in Minneapolis

All tori of a given dimension are topologically identical (homeomorphic). It is not a question of "how they are formed" but only what they are. A donut and a coffee cup are the same thing topologically, but not geometrically. Topology doesn't care about curvature, and is basically only concerned with "gobal" characteristics of the manifold. Geometry is intimately involved with curvature and local characteristics, which can have global implications.

Manifolds need not be considered as embedded in anything. The most useful way to think about manifolds is as intrinsic manifolds, that is manifolds that are considered as free-standing mathematical objects without regard to any embedding. That is the way that space-time is viewed.

There are embedding theorems, you can look them up on Google if you like, that address the question of embedding of manifolds in higher dimensional spaces. In the case of topological embeddings the relevant theorem is the Whitney embedding theorem. In the case of geometric embeddings (isometries) of Riemannian manifolds the relevant theorem is the Nash embedding theorem. There is a theorem similar to the Nash embedding theorem that applies to Lorentzian manifolds, and other manifolds with metrics that are not positive-definite, but I don't know of an on-line reference (there is a monograph by Greene that contains such theorems).

These issues are not germane to the geometry of the universe, because the geometry is independent of any embedding. The ability to study the geometry of a manifold without reference to any embedding is the beauty of Riemann's theory. Embeddings are irrelevant to either the geometry or the topology of the manifold.

The point of the constructions that were mentioned earlier is not that one must realize the torus or the cylinder in that manner, but that you can, and that those constructions show a way, not necessarily the only way, to realize the relevant topologies in a flat geometry.

astromark
2009-May-03, 07:23 PM
You do not make this easy... or as has been said. Understanding of this subject is way off the scale. I do not understand your answers. , and I do not think any one can. Its as much about perception as understanding. None of it matters. Its the biggest wast of energy I can imagine. If you can not explain or answer this question so as I can understand you what is the point.? and yes I am so arrogant as to state my understanding and concept of reality is as good as it can be. The scale of this question appears to be beyond me. I can see in my three dimensional world the universe stretching away from me to infinity.. Having excepted that finite but unbound describes this as well as it should. Obviously my understanding of this curved flat is suspect. Help ?

agingjb
2009-May-03, 07:51 PM
I wonder if the apparent flatness of the observable universe means that the global topology is flat or that the universe is very big indeed so that its curvature is hard to measure in our restricted view. My garden is "flat" but it's part of the surface of a sphere about 6 orders of magnitude larger.

I also wonder if toroidal flatness would imply that some directions are, in some sense, special.

Jeff Root
2009-May-03, 08:47 PM
I think that in three dimensions, these two ways of forming a torus
are topologically identical. I haven't tried to analyze them to see if
they are geometrically identical. My questions are: Does it make a
difference in four dimensions? Is it relevant to the geometry of the
Universe?
It is not a question of "how they are formed" but only what they are.
I asked based on the notion that spacetime consists of three
dimensions of space and one dimension of time. If a "flat torus"
can only be "flat" when it is formed in four or more dimensions, then
that suggests its curvature in different dimensions might have very
different effects. If it is curved only in spatial dimensions, then it
can't be flat, so it must be curved in the time dimension in place of
one of the spatial dimensions, leaving that spatial dimension flat.
Since the two rollings of a sheet to form a torus (done sequentially)
are different from each other (the second causing deformation), it
seemed plausible that the two curvings might be fundamentally
different as well, which they would be if one is a curving of space
in two dimensions and the other is a "curving" of time.

In that case, the two different ways of forming the torus might
correspond to two different combinations of warped dimensions,
one being "flat" and the other not, depending on which dimensions
get "rolled".

A donut and a coffee cup are the same thing topologically, but not
geometrically. Topology doesn't care about curvature, and is basically
only concerned with "gobal" characteristics of the manifold. Geometry
is intimately involved with curvature and local characteristics, which
can have global implications.
I mentioned topology in addition to geometry because Joseph Silk
used both terms in his ESA interview.
ESA: Interview with Professor Joseph Silk (http://www.esa.int/esaSC/SEMR53T1VED_index_0_iv.html)
If you could explain the meaning of both terms as Silk applied them
to the Universe as a whole, without depending on terms that are more
arcane than they are (such as "manifold"), that could really help.

I would like to know why Silk used used each term rather than the
other in each instance. I know that topology and geometry are
entirely different things, yet it isn't clear from his useage that they
aren't interchangeable, at least in this context.

-- Jeff, in Minneapolis

robross
2009-May-03, 10:45 PM
"Flat" just means parallel lines stay parallel, and perpendicular lines stay perpendicular, no matter how long you extend the lines, even if they come back to their original starting place.

The "surface" of a torus is just one way of curving a 2-dimensional plane into three dimensions such that you can have parallel lines, unlike what happens on the surface of say, a sphere, where lines do not remain parallel.

Our 3-dimensional space is curved into an (at least) 4-dimensional space such that if you were to shoot out two parallel laser beams, they would remain parallel no matter how far they traveled across the universe. That's all that is meant by "flat." There are many ways of curving a 3-d space into higher dimensions that keep parallel lines parallel, but I'm not sure of any definitive way to "prove" what that higher-order shape might be.

DrRocket
2009-May-03, 11:29 PM
I asked based on the notion that spacetime consists of three
dimensions of space and one dimension of time. ...
I mentioned topology in addition to geometry because Joseph Silk
used both terms in his ESA interview.

If you could explain the meaning of both terms as Silk applied them
to the Universe as a whole, without depending on terms that are more
arcane than they are (such as "manifold"), that could really help.

I would like to know why Silk used used each term rather than the
other in each instance. I know that topology and geometry are
entirely different things, yet it isn't clear from his useage that they
aren't interchangeable, at least in this context.

-- Jeff, in Minneapolis

There is no really easy and elementary answer.

Space-time in general relativity does not have 3 dimensions of space and 1 dimension of time, except locally. That is important. There is no way to explain this without using the notion of a manifold. There is no way to understand general relativity without the notion of a manifold either.

Topology and geometry are not the same thing and the terms are absolutely not interchangable. Two manifolds with the same geometry will have the same topology, but the converse may not be true. That was the coffee cup and donut example.

Try looking at some of these sites to get a feel for differential geometry. It is not the easiest subject, but perhaps you can get a flavor without trying to master the details.

http://en.wikipedia.org/wiki/Manifold

http://en.wikipedia.org/wiki/Differential_geometry

http://en.wikipedia.org/wiki/Basic_introduction_to_the_mathematics_of_curved_sp acetime

http://en.wikipedia.org/wiki/Basic_introduction_to_the_mathematics_of_curved_sp acetime

Cougar
2009-May-03, 11:42 PM
I wonder if the apparent flatness of the observable universe means that the global topology is flat or that the universe is very big indeed so that its curvature is hard to measure in our restricted view....

Measurements from our restricted view have increased considerably in sophistication, but it still appears there's next to no deviation from flatness (counting 70% dark energy). Just to clarify, flatness = overall, gravitationally neutral. (?)

astromark
2009-May-04, 06:36 AM
I have been told that there must be as many as seven dimensions and, there could be as many as eleven... Reluctantly I will concede to the greater minds at work in this field. I am not happy about it. Having listened to the pod cast of the proph...
I can not see it. I like most of the people around me, perceive the world we live in as a three dimensional spacial fact. Time being the other part of this equation. Flat or curved ? I am not coming to grips with. Yes two lasers aligned in parallel will not spread or merge or other wise bend back to eventually to go back whence it came... If the signal strength has the required energy then on to infinity they go.
In this questions and answers part of this forum many interesting questions have gripped my attention. Only this subject leaves me wanting to know why I can not understand any of this other dimensional understanding. I have concluded that none of it is real and, is in fact just an effort by those mentally agile to belittle my humble three dimensional mind. or not...:)

MessengerM104
2009-May-04, 07:15 PM
Yes two laser(beam)s aligned in parallel will not spread
yes, they will (don't forget your Hubble)

I have concluded that none of it is real and, is in fact just an effort by those mentally agile to belittle my humble three dimensional mind.

the be-littli-ng game is played anywhere
(and everywhere)
((science-d:cool:m included))
(((...there are :)bviously beTTer games to play...)))

my humble three dimensional mind

humble is good
plus
your brain is three dimensional (n0t counting time 41ce)
there also should be a (small) fractal dimension to the thing (i do not know of)

your... M I N D ...is too multidimensional to count

(but it could be done...)
((andwill!))

astromark
2009-May-05, 08:46 AM
Messenger M 104; Its kind of you to be thinking of me but, stop.
I just want clarity, ( not the wine ). How do I perceive that which I do not comprehend. What I want is an explanation of other dimensions. How, where, Can it be explained to a three dimensional thinking being that there might be other ways of folding space and time... as at present my three dimensional mind does not except input. I might be of topic with this but I do not understand so conclude it as nonsense. Is the universe flat ? No. Its a huge mass of galactic objects including you and me. I wish to understand the main stream scientific explanation of multi dimensional space time. That is if I do not already know.

01101001
2009-May-05, 01:35 PM
Is the universe flat ? No. Its a huge mass of galactic objects including you and me.

What's your definition of "flat" here? It doesn't sound like the one cosmologists use.

DrRocket
2009-May-05, 02:08 PM
What's your definition of "flat" here? It doesn't sound like the one cosmologists use.

That is an important point. Cosmologists use a rather idealized version of general relativity, one that admits a rough means of separating "space" from "time" in a global way -- a foliation by a one-parameter family of space-like hypersurfaces of constant curvature. In that idealization it is possible for the space-like hypersurfaces to be flat in the sense of differential geometry. It is likewise important to recognize that cosmological "time" is a reflection of the parameter, since general relativity tells us that it is not really possible to separate time from space in a curved geometry.

It is abundantly clear that space-time is not actually flat. If it were there would be no gravity, and certainly no black holes. Galaxies, and other matter make true flatness impossible. So one must keep in mind that the notion of flatness applies to an idealized model that is used to approximate the universe on the largest of scales.

astromark
2009-May-05, 07:14 PM
Yes... 01101001.; I was just making that point obvious. Only in a general galactic frame is the universe actually flat. Yes it is. Down here in the real world it is apparently not flat. Dr Rocket,; has covered it well...
From the school of the obvious,. Saturn's ring system is flat while Saturn is not.
I am well aware of the cosmologist's different use of this term. Flat. I just do not like it. It harps on to the 'dark mater', Dark energy. Black hole... None of those things are what the word used would suggest. For me this is an issue.
So when I say the universe is not flat. It is because the universe is very much, not flat. is it.?

DrRocket
2009-May-05, 08:40 PM
Yes... 01101001.; I was just making that point obvious. Only in a general galactic frame is the universe actually flat. Yes it is. Down here in the real world it is apparently not flat. Dr Rocket,; has covered it well...
From the school of the obvious,. Saturn's ring system is flat while Saturn is not.
I am well aware of the cosmologist's different use of this term. Flat. I just do not like it. It harps on to the 'dark mater', Dark energy. Black hole... None of those things are what the word used would suggest. For me this is an issue.
So when I say the universe is not flat. It is because the universe is very much, not flat. is it.?

Unfortunately, the word "flat" can be easily misconstrued. It has a very precise meaning in the context of differential geometry, as do all words when used in conjunction with rigorous mathematics. That is the meaning when used in the context of general relativity. But flat in the terminology of differential geometry is not the "flat" of everyday conversation, as is evidenced by the flat cylinder and flat torus.

There are other terms which may be confusing when not understood in terms of their narrow meaning in mathematics. Compact is one. A proper definition requires a knowledge of topology, but for most purposes in physics "closed and bounded" gets across the meaning. But unfortunately closed in the context of manifolds means something else -- a closed manifold is a compact manifold without boundary. The reason that this is important is because what is often referred to as a "finite" universe is really a universe that as a manifold is closed.

I hope this helped a bit. But while your complaint has merit, it is also the case that the words used in mathematics and physics often have precise meanings somewhat at odds with that of everyday useage. There is really not a good alternative other than to learn the specialized language of the subject. Worse, I don't know a good way to do that without a considerable amount of study in the subject.