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Author Topic: Is the shape of the Universe the same thing as its geometry?  (Read 3849 times)

Offline chris

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I've been sent a book to review; it's rather good and addresses the question of dark matter and the search to identify its real nature. Later I'll post a review and share my thoughts on it with you all.

However, first, it has provoked a question.

There's a chunk of the book dealing with the geometry of the Universe; three models are proposed - flat, hyperbolic (saddle shaped) and spherical.

What follows is not clear and the deductions not well explained, leaving me a bit confused.

So is "geometry" maths-speak for "shape"? And does "flat" quite literally mean flat, like a 2D-sheet? And if that's true, how can the rule of homogeneity apply?

Thanks

Chris


 

Offline JohnDuffield

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This relates to the "shape" of the universe on Wiki which gives this picture:


Image courtesy of NASA, see Wikipedia

Yes it's confusing all right, because at least two out of three above were always going to be wrong, and because we aren't dealing with the shape of the universe at all. And we aren't dealing with its geometry. We aren't even dealing with the geometry of spacetime. It's more like "the geometry of spacetime at this moment". That's why the Wikipedia article says "Cosmologists normally work with a given space-like slice of spacetime". But then there's a problem in that people think this is the geometry of space, when it isn't. If spacetime is curved light will curve*, but that doesn't mean space is curved. See Baez and note this:

"Similarly, in general relativity gravity is not really a 'force', but just a manifestation of the curvature of spacetime. Note: not the curvature of space, but of spacetime. The distinction is crucial."

Instead it means space is inhomogeneous. See Einstein's Leyden Address for that:

"This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that "empty space" in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials gmn), has, I think, finally disposed of the view that space is physically empty."

Also see http://iopscience.iop.org/0256-307X/25/5/014 which says inhomogeneous space is essentially what curved spacetime is. Now look a bit further down the Wikipedia article. See where it refers to the FLRW model? Follow the link to the FLRW article and what you can read is this: "The FLRW metric starts with the assumption of homogeneity and isotropy of space". It assumes space is homogeneous, so it assumes spacetime is not curved. So when WMAP found that the universe was "flat", it shouldn't have been a surprise. If you have two parallel light beams, they don't curve round in some great circle, and they don't diverge or converge. They just go straight. As for why Einstein though the universe was closed like the sphere above, I don't know. It's as if his confidence and intuition failed him when it came to cosmology. If he hadn't modelled the universe using dust, but instead used space alone, I think he wouldn't have made his "greatest blunder".

As for the shape of the universe, the actual shape of it, well that's a whole different kettle of fish. I think it's spherical myself, with an outer edge, but you won't find many people saying that. Instead there's a tendency to say it's infinite, and airbrush away any issues about how it can get to be infinite in a finite time, or whether it was infinite at the time of the big bang, or how an infinite universe can expand. I expect this infinite universe assertion won't persist, because it's what Phil Plait would call "bad cosmology" that IMHO leaves the door open to unscientific multiverse hypotheses. See George Ellis on that. To answer you particular questions:

Quote from: Chris
So is "geometry" maths-speak for "shape"?
I'd say it's maths-speak for the disposition of spacetime, but it isn't much to do with the shape of the universe. 

Quote from: Chris
And does "flat" quite literally mean flat, like a 2D-sheet?
No. It means light goes straight. 

Quote from: Chris
And if that's true, how can the rule of homogeneity apply?
I'm not clear what you mean, but light goes straight when space is homogeneous.


* actually, light curves when spacetime is "tilted", but it's got to be curved to acquire a tilt. The plot of gravitational potential can't get off the flat and level if there's no curvature.
« Last Edit: 26/09/2014 08:25:50 by JohnDuffield »
 

Offline yor_on

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Offline JohnDuffield

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I read it, yor_on. I'm afraid to say it confuses curved spacetime and curved space. See this:

"Whenever you can find three points in space, and join them with laser beams, and find that the triangle doesn’t have the expected 180 degrees, that means that space is curved."

It isn't true. You could do this sort of thing in a gravitational field, where your light beams are curved. But they're curved because space is inhomogeneous, not because space is curved. Again see the Baez article:

"Similarly, in general relativity gravity is not really a 'force', but just a manifestation of the curvature of spacetime. Note: not the curvature of space, but of spacetime. The distinction is crucial."
 
 

Offline PmbPhy

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Quote from: chris
So is "geometry" maths-speak for "shape"?
The term geometry is defined as study of figures in a space of a given number of dimensions and of a given type. Myself and others use it when referring to shape, so yes in that sense and context.

Quote from: chris
And does "flat" quite literally mean flat, like a 2D-sheet?
The term flat refers to a physical space in which Euclid's parallel postulate holds true. For example; if two geodesics (i.e. a curved of extremal length) start out parallel then they will remain parallel, regardless of where they start and he direction the proceed in.

Quote from: chris
And if that's true, how can the rule of homogeneity apply?
I wish I could tell you but I have no idea what the rule of homogeneity is or what it means.
 

Offline evan_au

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A bit of history:
Traditional geometry, as we were taught at school was based on the work of Euclid, who based it all on 5 Postulates. His fifth Postulate says:
Quote from: Euclid
Within a two-dimensional plane, for any given line ℓ and a point A, which is not on ℓ, there is exactly one line through A that does not intersect ℓ.

This has been investigated in various ways:
  • for example, on the surface of the Earth, all lines eventually intersect (ie there are no parallel lines). This was formalised by Riemann.
  • Bolyai & Lobachevsky formalised a geometry where there are many parallel lines; this happens on a horse saddle, for example.
  • These are non-Euclidean 2-Dimensional surfaces, but the same thing can occur in 3D (or higher dimensions).
  • These were developed as mathematical models in the 1800s, and applied to cosmology in the 1900s.
It's easy for us to visualise curved 2-D surfaces, but very hard to visialise in 3 (or more) dimensions. So diagrams in relativity and cosmology often imagine space as being a stretched 2D rubber sheet, onto which is placed a bowling ball (the Sun), while a marble (the Earth) is rolled on a circular or elliptical orbit. The surface of this stretched sheet does not follow Euclid's "flat" geometry.

But models of the solar system (or even an entire galaxy), are mere dimples on the overall shape of the universe.

Cosmologists would like to know if the entire universe as a whole is flat, spherical or hyperbolic, as this determines the ultimate fate of the universe. To simplify things, they assume that the universe is homogenous, ie that the matter is spread out evenly throughout the universe, and ignore the real clumpiness of galaxies and superclusters. This rule of homogeneity is expressed in the Cosmological Principle*.

Cosmologists then make theoretical models of this homogenous universe, and try to compare its predictions with measurements results such as the WMAP space probe. It seems that the universe as a whole is pretty close to Euclid's flat model - but this does not exclude a slight spherical or hyperbolic curvature.

The discovery of Dark Energy has required modifications to some earlier theories; more information on the behaviour of Dark Energy will help "tune" current theories.

*As I understand it, the "rule of homogeneity" is not really a law of physics, but more an approximation that lets us solve the equations of cosmology on today's computers.
« Last Edit: 26/09/2014 11:59:17 by evan_au »
 

Offline yor_on

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Well, I would also say that is based on what scale you measure. Like everything else, as you move further back to look, things tend to 'smooth out', become even. And it also depends on what ones assumptions are, as whether you think of the universe as finite or infinite. At large scales the universe is homogeneous and isotropic, but when you 'magnify' it you might want to define it differently. I assume it to be infinite, meaning that wherever you stand it will look the same as it does from here. That solves that definition nicely I think :)
 

Offline evan_au

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The mathematical study of shapes is called topology.
 
The discussion above has focussed on the curvature - flat, spherical or hyperbolic.

Another aspect of shape is whether it has any "holes". For example, a sphere and a doughnut both exhibit the same type of curvature. However, the path is different if you travel far enough in a single direction. Topology considers shapes with different numbers of holes to be distinct shapes.

At present, experiments are focussed on determining the curvature of our vicinity and the universe; at this time it is not clear how you would detect or count any "holes".
 

Offline yor_on

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Myself I prefer it infinite, no extra 'dimensions' needed, more than those we have found. And a Big Bang doesn't necessarily need to start from 'a singular pinprick' inflating into a 'common seamlessly existing' universe.  http://www.astro.ucla.edu/~wright/infpoint.html . It's enough with each observer finding it so, the same way each observer today, no matter where he is, will have a 'same' equivalent view of the universe, ending at approximately 13.7 billion light years away from each unique observer.
 

Offline yor_on

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Then again, that is a question of degrees of freedom firstly, to me that is. The degrees we have 'inside' is four. Length, width, height and 'time'  as our local arrow. You can see the universe two ways. One is the one assuming it to be a whole 'common universe', then defining one arrow for it (equivalent to a entropy). The other is strictly local, then connecting that local arrow to 'c'. If you do you will need something more defining this 'common wholeness' we agree us to exist in, but assuming there should be a way to define that, we end up with the universe we actually can measure on. The first expression does not.
=

One more subtlety to defining it locally is that constants will hold. It's not a question of 'islands of time and space'. And why they hold is very simple. Because your measurements will tell you so, wherever you go.

and actually, any try for those 'islands of time and space' will meet so many problems as the universe is observer dependent. It's a third way to define a universe, not having to do with the two I mentioned before. Also it will demand a principle for what then makes up our 'common universe, as you no longer can use a arrow for it all, furthermore refuse the local definition instead presuming some universe only able to be defined from its 'outside'. What I call the 'eye of a god'. Very complicated to get to work, as I see it impossible. Defining it that way entropy becomes something 'observer dependent' too, also have to be assumed to be real and 'co-existing' from each observers point of view as we find each observer defining those 'islands of space and time' differently, the local definition elegantly avoid that (in my view:) by using constants.

and it connects to what I described as a way to look at a 'Big Bang', using observers, then, and now.

Such a universe I expect to be defined by 'observers', and we also need to assume that a 'observer' should include anything able to interact, as in communicate, with something else. And that should take care of the discussion if it is consciousness that defines a universe. You need it to be observers interacting, and as far as I know there was no 'consciousness' at the moment of that 'Big Bang'. Unless you get more mystical than I can be naturally :)
« Last Edit: 26/09/2014 22:27:53 by yor_on »
 

Offline chris

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God this is complicated! Glad you lot are here to help me...
 

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