Consider this thought experiment:

Point a super telescope in one direction of space and observe some galaxy sufficiently far away so that we are close to the time of the big bang. Let us call this galaxy A.

Now, point this telescope in the opposite direction and make another observation of some galaxy B.

Now, both of these are so far away and also far back in time that they are near the early universe which was supposedly very small and so therefore A and B while we measure the distance to each of them to be around 13-14 billion lightyears away so that we measure the distance between them to be around 27 billion lightyears apart they were supposedly both existing at a time when the universe was much smaller and consequently the distance between them must also be much smaller. Doesn't this indicate that the universe is not euclidian? I mean if space were essencially euclidian and not curved their distance would literally have to be around 27 billion lightyears apart at a time when the universe wasn't even 27 billion lightyears big?

So how is one to understand statements such as "the curvature of space is near 0 and therefore implies an almost euclidian space"?

Can someone please enlighten me?

salte

Locally, space can be considered Euclidean. This is used to solve many problems.

On the large scale, your guess is as good as mine as to the shape or curvature.

If the angles of a triangle add up to 180 degrees, the triangle must be situated in flat space. If the angles add up to more or less than 180, the space is curved one way or the other. These clever astronomers measured the angular size of the temperature variations in the CMB, and after some steps of logic, essentially determined the "triangle angles" added up to 180! At least from the time of decoupling, all the space in the universe must therefore be flat!

Here's a pdf of the more technical stuff....

A Flat Universe from High-Resolution Maps of the Cosmic Microwave Background Radiation -- P. de Bernardis, et al.

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Everyone is entitled to his own opinion, but not his own facts.

What Cougar is saying is that the original poster, salte, is correct in
saying that the visible Universe is apparently Euclidean overall.

salte,

For realism, rather than galaxies 13 or 14 billion light-years away, we can talk
about light from the hot plasma that filled the Universe 13.7 billion years ago,
which we see now as the cosmic microwave background radiation. It was
emitted from a sphere surrounding our current position a few million light-years
in diameter. (A more accurate figure was posted here just recently but I don't
recall what that figure was.) Because of the expansion of the Universe, in
which the matter that emitted the light and the future position of Earth were
moving apart at high speed, the light travelled a distance of 13.7 billion
light-years to reach us.

If that is hard to visualize, the standard analogy is to imagine a balloon with
two points marked on it. As the balloon is inflated, an ant starts walking from
one point to the other. At first the progress the ant makes toward the second
dot is less than the increase caused by the expansion. But eventually the ant
covers enough distance that the progress it makes is greater than the
expansion, and it starts getting closer to the second dot.

-- Jeff, in Minneapolis

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"I find astronomy very interesting, but I wouldn't if I thought we
were just going to sit here and look." -- "Van Rijn"

"The other planets? Well, they just happen to be there, but the
point of rockets is to explore them!" -- Kai Yeves

The CMBR was roughly 40 million light years away when the light was emitted, and is roughly 46 billion light years away today.

__________________
If something is in me which can be called religious then it is the unbounded admiration for the structure of the world so far as our science can reveal it... of the manifestations of the profoundest reason and the most radiant beauty, which are only accessible to our reason in their most elementary forms...
Albert Einstein

Was the universe Euclidean when it was highly compressed in the moments after the Big Bang?

Not necesarily - it may well have been curved. One of the 'problems' inflation was proposed to solve was flatness.

For our current observable universe to appear flat would have required very fine tuning of the initial conditions. Inflation allowed for the curvature of the earliest universe to be more natural (i.e. not fine tuned) then inflation 'swelled' whatever curvature the universe might have had to seem Euclidean.

For example if you were to stand on a bowling ball, it's curvature would seem very apparent, but not nearly so apparent if you inflate the bowling ball to the size of earth. The tiny embryonic universe swelled from something billions of times smaller than a proton to 600 billion miles, in a thousandth of a second - that's a bloody big bowling ball!

__________________
If something is in me which can be called religious then it is the unbounded admiration for the structure of the world so far as our science can reveal it... of the manifestations of the profoundest reason and the most radiant beauty, which are only accessible to our reason in their most elementary forms...
Albert Einstein

It does seem to me that a change from one manifold to another is unlikely (but who knows?) so that however the universe started out, it would retain the same overall topology.

Thank you for the answers guys.

However, the inflation doesn't really cut it for me. An inflation can make the universe much larger than original size in a short moment but can it really make a curved space flat? Wouldn't that be equivalent of some asymptotic curve actually reach infinity? I mean no matter how large the x is you get larger and larger values but you never actually reach infinity. IOW the space may appear flat and be almost flat because it is so huge but it isn't really flat, it has just a very small curvature that is near 0 but not exactly 0. Isn't that the most reasonable result of an inflation?

It also occurs to me that it is hard to measure something and determine it to be exactly zero, you will always get some margin of error so to conclude then that space is flat appears non-sensical to me. At best we can say that it is so huge that it appears near flat to us. Yet to conclude that it IS flat would be equivalent of saying that an asymptotic curve that approaches infinity as x goes towards infinity has reached infinity for some finite value of x and that just doesn't make sense to me.

Any comments?

salte

I've never seen a reference to the topology changing (but like you say who knows). On another thread we were looking at the search for clues to topology in the CMBR, as would be indicated by matching patterns in the anisotropies. It's looking less likely patterns will be found, ruling out that we live in a small universe - that is a universe where the fundamental domain is smaller than the particle horizon. We might not ever be able to observe the topology of the entire universe. (At least by that method. I haven't heard of other methods, unless gravitational waves reveal something new.)

__________________
If something is in me which can be called religious then it is the unbounded admiration for the structure of the world so far as our science can reveal it... of the manifestations of the profoundest reason and the most radiant beauty, which are only accessible to our reason in their most elementary forms...
Albert Einstein

That sounds about right to me.

Right again. The figure I have seen quoted for the margin of error is 2%. So the universe is observed to be flat within 2% out to the surface of last scattering.

__________________
If something is in me which can be called religious then it is the unbounded admiration for the structure of the world so far as our science can reveal it... of the manifestations of the profoundest reason and the most radiant beauty, which are only accessible to our reason in their most elementary forms...
Albert Einstein

Absolutely. My previous post was much too... absolute in its assertion of flatness. Without checking, I believe the margin of error is fairly significant in the CMB temperature variation power spectrum test.

__________________
Everyone is entitled to his own opinion, but not his own facts.

Large Cat, 2 to 3 percent is quite reasonable. Ever more accurate measurements have only reduced the error bars. The groans you hear in the background are from cosmologists with careers based on exotic shapes. See the April '08 issue of Astronomy magazine, it's the story on the shape of the universe. I was amused, they carry on at great length about speculative shapes, but in the diagrams of flat space, hyperbolic space, and closed space, the flat space caption says that that's the way the best measurements are to within 2%. The other shapes have no supporting observations.

If you want to do serious math about the observed universe, assume it's flat. And has only three space and one time dimensions, while we're at it.

Good question, OP, thanks.

Oh, I thought it was much larger than that, but your cited April 2008 publication is obviously much more current that whatever it is I'm (vaguely) recalling.

__________________
Everyone is entitled to his own opinion, but not his own facts.

But we must make sure we don't confuse the flatness of space with the topology. A 3-manifold like a 3-torus, for example, is finite and flat (it has euclidean geometry and zero curvature) but still "wraps round". This might be one of those shapes you mention that has no supporting observations, but it is as valid as any other in that it is flat.

On the other hand, a Seifert-Weber manifold is dodecahedral and has negative curvature, so this seems less likely a candidate if we think our observable universe is within 2% of being flat. But that 2% margin of error means we can only say with any certainty that if space is curved it has a radius far larger than our observable universe and to all intents and purposes it seems like our observable universe is flat.

Although we might think that space is euclidean, space-time as described by general relativity is non-euclidean and this is where Salte will find the answers he seeks.

Hmm. If it's Euclidean, then it's flat, but then it's either infinite or has boundaries (and anyway it's not locally Euclidean if GR holds, and what was it when it started?).

OK, apparently flat but perhaps not infinite, so what manifold? Hypertorus would be flat, but surely some anisotropic indications? Can a hypersphere be flat? There's that parallelisability, but who knows, and anyway surely more anisotropy.

I really ought not to speculate, but I'd guess hypersphere, but very large so that it is flat to the limits of our possible observations.

Clues to topopolgy (the 'matching circle pairs') only show up in CMBR anisotropies if the fundamental domain is smaller than the surface of last scattering, If I've understood correctly. As far as I can tell the research is close to ruling out a fundamental domain that small. (Of course 'small' is relative!)

If the fundamental domain is larger, then they never overlap within our observable universe, so they won't find patterns in the CMBR where light traverses the 'multiple connections' where fundamental domains tile space. All the subtleties regarding angularity of the anisotropy harmonics and whatever escape me for now, but they also seem to be saying 'flat'.

'Very Large Hypersphere' seems satisfying to me - circles (or at least ellipses) and spheres seem to be favoured by nature. 3-torus also seems a good fit and would allow a smaller (entire) universe that still fits our observations, but who knows, really, so I'm gonna settle for a flat euclidean observable universe for my purposes - I don't know of any features of space or time that I'm currently interested in where that would pose a problem.

__________________
If something is in me which can be called religious then it is the unbounded admiration for the structure of the world so far as our science can reveal it... of the manifestations of the profoundest reason and the most radiant beauty, which are only accessible to our reason in their most elementary forms...
Albert Einstein

Can you explain how this works Speedfreek? Its only when we add in time that our common perception of 3d space becomes non-euclidean, right? And I guess we'd have to say that space exists without time only in our man-made models - one doesn't exist without the other in reality?

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If something is in me which can be called religious then it is the unbounded admiration for the structure of the world so far as our science can reveal it... of the manifestations of the profoundest reason and the most radiant beauty, which are only accessible to our reason in their most elementary forms...
Albert Einstein

Well I can't say I fully understand how it works yet (hence the "shifty" emoticon!), but I can point you to a relevant wiki page on Euclidean Space.

"In modern mathematics, Euclidean spaces form the prototypes for other, more complicated geometric objects. For example, a smooth manifold is a Hausdorff topological space that is locally diffeomorphic to Euclidean space. Diffeomorphism does not respect distance and angle, so these key concepts of Euclidean geometry are lost on a smooth manifold. However, if one additionally prescribes a smoothly varying inner product on the manifold's tangent spaces, then the result is what is called a Riemannian manifold. Put differently, a Riemannian manifold is a space constructed by deforming and patching together Euclidean spaces. Such a space enjoys notions of distance and angle, but they behave in a curved, non-Euclidean manner. The simplest Riemannian manifold, consisting of Rn with a constant inner product, is essentially identical to Euclidean n-space itself.

If one alters a Euclidean space so that its inner product becomes negative in one or more directions, then the result is a pseudo-Euclidean space. Smooth manifolds built from such spaces are called pseudo-Riemannian manifolds. Perhaps their most famous application is the theory of relativity, where empty spacetime with no matter is represented by the flat pseudo-Euclidean space called Minkowski space, spacetimes with matter in them form other pseudo-Riemannian manifolds, and gravity corresponds to the curvature of such a manifold.

Our universe, being subject to relativity, is not Euclidean. This becomes significant in theoretical considerations of astronomy and cosmology, and also in some practical problems such as global positioning and airplane navigation. Nonetheless, a Euclidean model of the universe can still be used to solve many other practical problems with sufficient precision."

It seems to me that the non-euclidean nature of space-time presents itself to us through various effects like the observed time-dilation of GPS satellite clocks and the apparent recession speeds of distant galaxies (which is what the OP was questioning).

Well, no, it could be finite and unbounded, like the surface of a sphere, or any number of other compact topologies.

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Everyone is entitled to his own opinion, but not his own facts.

I don't see, but I shall be told, how a space can be (globally) Euclidean, finite, and unbounded. I would not count a hypertorus as Euclidean (by which I would mean R4).