The Hawking 4D closed manifold

[Schlofster]

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:

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?

 

[Schlofster]

I have been reading your sticky post about the balloon analogy.
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.

 

[Chalnoth]

I have been reading your sticky post about the balloon analogy.
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.

Yeah, I'd have to say both of your points are entirely valid. We can try, of course, but really it's not reasonable to expect that we will actually succeed.

As for how Stephen Hawking's no boundary conjecture can be squared with the current expansion, well, I think his description works as well as any here. One can think of the Earth as having a single point that is furthest North, but there's nothing strange that goes on there: it's just a feature of what we label "north". Similarly, there may be a point that is furthest back in time, but, if Hawking's conjecture is correct, then such a point would also be nothing strange: just a feature of how we perceive the flow of time.

 

[Chronos]

Your questions are inconsistent with your sources, Schlofster. The balloon analogy is exactly that - an analogy. Every scientist who has used that analogy has warned it is not mathamatically rigorous, merely a convenient way of characterizing the point in layman terms.

 

[Schlofster]

Your questions are inconsistent with your sources, Schlofster. The balloon analogy is exactly that - an analogy. Every scientist who has used that analogy has warned it is not mathamatically rigorous, merely a convenient way of characterizing the point in layman terms.

Yes, that is why I said this:

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.

My point is that it is not a very good analogy (even if it is the best that we have).

 

[Schlofster]

After some reading, I think that I may have a handle on it 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 Earth's 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 a 2 or 3 dimensional (excluding time) surface, but a 3 dimensional (excluding time) space.
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?

 

[Peter Watkins]

One little puff of wind sufficient to lift our ant, and he would immediately think, "Bless my soul, there exists another dimension!" This realisation would render his two dimensional world useless, both as a reality and a concept.

 

[Chalnoth]

How does this relate to our universe?
Our universe is not a 2 or 3 dimensional (excluding time) surface, but a 3 dimensional (excluding time) space.

Yes, but as far as the mathematics are concerned, this is mostly an irrelevant distinction. There are some significant differences, but not in terms of understanding the expansion.

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.

Um, only if the universe has a topology which wraps back on itself. And if the topology is pretty complex, it may take more than a few trips around the universe before getting back to your starting point (if you travel straight the whole way). It is possible that the universe does not wrap back on itself. We can't currently say a whole lot about the topology of the universe when our vision is limited to only a small patch of the whole.

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.

Correct.

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?

Yes, it's accurate.

 

[Schlofster]

One little puff of wind sufficient to lift our ant, and he would immediately think, "Bless my soul, there exists another dimension!" This realisation would render his two dimensional world useless, both as a reality and a concept.

Well Peter, if you are referring to the ant directly, I would point out that this is merely an analogy as I have been repeatedly reminded on this forum.
If you are referring to us by implication, then I would say:
Although what you say is correct, it is merely speculative conjecture.
Until there is empirical evidence supporting your puff of wind, we should approach the problem with skepticism about other dimensions while not completely disregarding the possibility of their existence.

 

[Schlofster]

It is possible that the universe does not wrap back on itself.

Would this not imply that the universe has an edge or a boundary?

 

[Chronos]

How would you detect this 'edge'?

 

[Chalnoth]

Would this not imply that the universe has an edge or a boundary?

Not necessarily. It may be infinite. Or it may taper off asymptotically to infinity.

 

[Schlofster]

Not necessarily. It may be infinite. Or it may taper off asymptotically to infinity.

Yes, but in that case, Hawking's 4D closed manifold would be wrong.

 

[Schlofster]

How would you detect this 'edge'?

Well, I suspect that you couldn't, but then it could be argued that one would have just as much difficulty detecting the actual topology of the entire universe especially if it is infinite or asymptotic.