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Author Topic: Hawking's 4D Closed Manifold  (Read 4096 times)

Offline schlofster

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Hawking's 4D Closed Manifold
« on: 15/05/2009 22:26:38 »
Hi,
I am struggling to understand Stephen Hawking's view of the universe as a 4D closed manifold.
In a recent interview, I believe he had this to say:

Quote
SO IT WOULDN'T BE POSSIBLE TO SAY THAT REALLY THAT THE UNIVERSE HAS A BEGINNING OR END, OR WHAT WOULD BE POSSIBLE TO SAY ABOUT BEGINNING AND CAUSATION?

    The universe... the universe would have a beginning and an end in the same sense that degrees of latitude have a beginning and an end at the north and south poles respectively. There isn't any point with a latitude 91 degrees north. And similarly, there isn't any point in the universe which is before the big bang. And the, but the north pole is a perfectly regular point of the earth's surface, it's not a singular point. And similarly, I believe that the big bang was a perfectly regular point of space time. And all the laws of physics would hold at the big bang. And if that is the case, we can completely predict the state of the universe from the laws of physics.

    SO WHAT CAN YOU SAY ABOUT, ...ESSENTIALLY MEANS THAT THE UNIVERSE, YOU CANNOT SAY THAT THE UNIVERSE IS CAUSED, RIGHT? IS THAT CORRECT?

    That's right. The universe does not have any beginning or end. It does not have any cause or consequence. It simply is.

What I don't understand, is how this theory is compatible with the scientific observation that the universe is expanding?
I have 2 questions:
1) what is it expanding into? - the word expanding contains an implicit reference to some reference point external to the thing that is expanding?
2) the word 'expanding' contains an implicit reference to time, how does this fit into his theory?

I have also been reading about the balloon analogy (stickers stuck to a balloon as it is inflated)
for explaining parts of the "Lambda-cold-dark-matter" mathematical model.

I am starting to think that non physicists like me find this so difficult for two reasons:
1) it seems like natural language is not rich or precise enough to fully describe the mathematical model
2) there doesn't seem to be a geometric interpretation of the mathematics that would allow me to create a mental model of what the universe actually looks like.
Even imagining warped spacetime involves merely a projection onto a 3D surface.

It seems as if our evolved brains are just not capable (or maybe just mine) of visualising it because we have never encountered anything like it until now, and have thus not needed to be able to create an accurate conceptual model to allow us to survive in our environment.

Is it possible for an average human mind (like mine) to understand it without going into the math?


 

Offline Soul Surfer

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Hawking's 4D Closed Manifold
« Reply #1 on: 16/05/2009 10:38:40 »
We are very familiar with our three dimensional closed approximately spherical surface on the earth I am assuming that this presents no problem to you.  It is quite possible to travel for ever and you may eventually return to the same spot but could go round the earths surface many times always crossing new territory without ever coming back to your starting point.  Now all this travel takes time so imagine the earth is like a balloon continually being blown up.  The 4D space time manifold is a bit like this. It is a structure that is changing with time but it is always a closed structure and you cannot escape from it at any time or in any position.  This is exactly what the inside of a black hole is like!
 

Offline schlofster

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Hawking's 4D Closed Manifold
« Reply #2 on: 16/05/2009 21:55:00 »
OK then, so I think that I understand now - please correct me where I am wrong.

If someone asks the question 'what lies beyond the edge of the universe?'
it is kind of like a member of the flat earth society asking 'what lies beyond the edge of the earth?'
The difference being that the earths surface is just that - a surface, and the universe is a space.

I think that it would help non physicists like myself to clearly define which parts of the balloon
analogy is pertinent when trying to describe the mathematical model of the universe and which
parts are not.

I think it works like this:

The Balloon Analogy
-------------------
"All points on the surface of a balloon get further away from each other as the balloon is inflated."

The pertinent parts of this analogy:
- The distance between any pair of points on the surface of the balloon increases with time because
new surface is being 'created' evenly (at the same rate) at all points on the surface between the two points.
- If one compares any two pairs of points on the surface of the balloon with differing distances between each point in the pair
  - If the balloon was not expanding, a journey (at a finite speed) between the pair of points
    that are further apart, would take longer than the journey between the pair of points
    that are closer together.
  - If the balloon is expanding, during the extra time that is taken to journey between the
    pair of points that are further apart, more surface is created than would be
    created during a journey between the closer pair of points.

The parts of this analogy that are not pertinent:
- The fact that a real balloon exists within a larger space, and as it expands it occupies more volume (thus a greater portion of this larger space) is not pertinent.
- The fact that in the case of a real balloon, the new surface is 'created' by thinning out the wall of the balloon and using that material to 'create' new surface is not pertinent.

Now, taking the pertinent lessons from the analogy, and ignoring the parts of the analogy that are not pertinent, think of the following scenario in the trimmed down analogy:
Imagine a very small ant that lives on the surface of the balloon, and imagine the balloon is very large in relation to the ant.
The ant's 'universe' is two dimensional (excluding time) as far as he (or she  ;)) can see, but in his reality, it is actually curved, but the curvature is too gradual to be seen by him.
This two dimensional surface is all that exists in the trimmed down analogy.
We could say the following things:
- If the balloon wasn't expanding, the ant could start walking in any direction, and assuming the he lived long enough, he would end up at his starting point.
- If the balloon is expanding, the ant could only complete one complete one circumnavigation of the balloon, if he was able to cover at least as much surface as is created during the duration of his journey around the whole circumference of the balloon (added to the circumference of the balloon when he started his journey) at his maximum walking speed.
- If anyone asked the ant "what is beyond the edge of this universe?", the ant would have to reply "the universe".
- If anyone asked the ant "where is the edge of the universe" , the any would have to reply "there is no edge, but every point on the surface acts like an edge because new surface is created at every point at every time".
- If anyone asked the ant "is the universe infinite in size?", the ant would have to reply "no, but is has no boundary or edge".

How does this relate to our universe?
Our universe is not two dimensional (excluding time), but 3 dimensional (excluding time).
If our universe wasn't expanding, we could take a rocket journey in any direction, and assuming the we lived long enough, we would end up at our point of departure.
Our universe is expanding though, so whether we could journey 'around' the universe and reach our starting point, is dependent on it's current size (distance that we would have to travel to reach our staring point again), and it's rate of expansion.
As far as I know, for the current values of:
- the size of the universe
- it's rate of expansion
even at the speed of light, we would still not be able to cover enough space to reach our starting point again even if we traveled for ever.
To be more verbose, the rate at which space is currently being created between our departure point at time t0 (start of our 'round trip' journey) and our departure / arrival point at time t1 (end of our 'round trip' journey) is greater than the rate at which one can cover distance at the speed of light.

Is this roughly correct, or is it still way off the mark?
« Last Edit: 16/05/2009 22:44:56 by schlofster »
 

Offline schlofster

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Hawking's 4D Closed Manifold
« Reply #3 on: 16/05/2009 21:56:10 »
p.s. extreme verbosity would not be an baseless criticism  [:I]
 

Offline Soul Surfer

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Hawking's 4D Closed Manifold
« Reply #4 on: 17/05/2009 11:34:19 »
On the earth's surface our world is three dimensional but gravity bends our notion of up and down into a sphere (the stable shape of the earth). As we can only see a part of it the earth appears flat to us.  This is precisely the same with the universe.  The overall gravitational field bends the universe into a vast closed space but we can only see a tiny part of it as we look to the farthest times and distances so it looks perfectly flat.  The cosmic microwave background has some slight hints that space may be bent slightly but as yet measurements are not accurate enough to be certain.  My personal guess is that the universe is more likely to be doughnut (toroidal) shaped rather than spherical but we will have to wait and see on the experts.
« Last Edit: 17/05/2009 11:36:01 by Soul Surfer »
 

Offline schlofster

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Hawking's 4D Closed Manifold
« Reply #5 on: 17/05/2009 19:03:02 »
OK thanks, but how accurate is my understanding of expansion?
Is new space created between two points with the passage of time like in the balloon analogy?
 

Offline LeeE

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Hawking's 4D Closed Manifold
« Reply #6 on: 17/05/2009 19:35:05 »
If the universe is a closed 4D manifold then I think both space and time would need to be finite.  In a closed universe it would be possible to get back to when and where you started by only traveling forward.  The universe seems open and not closed to me.
 

lyner

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Hawking's 4D Closed Manifold
« Reply #7 on: 18/05/2009 13:34:30 »
Soul Surfer
My personal guess is that the universe is more likely to be doughnut (toroidal) shaped rather than spherical
I think you are being over-simplistic about this model. Are we not talking of a 'hypersphere' which involves other dimensions? A toroid would, necessarily be asymmetrical and I don't think that this has been observed.
 

Offline Vern

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Hawking's 4D Closed Manifold
« Reply #8 on: 18/05/2009 13:58:54 »
Quote from: Soul Surfer
This is exactly what the inside of a black hole is like!
Wow ! I wonder if Hawking knows this ;D
 

Offline Soul Surfer

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Hawking's 4D Closed Manifold
« Reply #9 on: 18/05/2009 23:01:04 »
Sophie and Vern the reason is that I reckon that it is most likely that our universe resides on a 3d rotating brane inside a black hole but we can only see a tiny bit of it so it looks flat. OK this could well be wrong but it is the best logical solution to avoid the discontinuity problem associated with the big bang and achieve a fractal structured universe of indefinite extent in space and time.  Black holes are probably a bit like the tardis bigger on the inside than the outside. I am working on this on my website but would like to get someone who really understands relativistic orbits in high gravitational fields interested in the problem. some of the stuff is in the Evolutionary Cosmology topic in the New theories area.
 

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Hawking's 4D Closed Manifold
« Reply #9 on: 18/05/2009 23:01:04 »

 

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