Is the universe bounded or unbounded?

[otseng]

In the Light, stars, and creationism thread, I proposed a theory to reconcile a young earth with being able to see stars that are billions of light years away. The theory assumes that the Big Bang is true, however, it also assumes that the universe is bounded. In typical cosmology, it is assumed that the universe is unbounded.

Bounded means that the universe has a boundary to it. There exists an "edge" to the universe in which beyond this boundary, our universe does not exist.

In an unbounded universe, there is no "edge". The universe "wraps" around itself. So, if you are to go in any direction in a straight line, you will eventually come back to the starting point.

This is hard to conceptualize, but can be explained like a surface of a sphere. On the surface of a sphere, if you start at any point and then go in a straight line, you will eventually come back to the starting point. Now, instead a 2-D surface on a sphere, the universe is a 3-D topology that curves in on itself.

The ramifications of either of these two assumptions make for drastically different cosmological conclusions.

So, the questions are:
1. Is the universe bounded or unbounded? Why?
2. What are the ramifications of whether it is bounded or unbounded?

[Alien]

I've posted this quote elsewhere, but I wanted to post this here also since it's apropos to this thread.

Big Bang Theory:

Astrophysicist George F. R. Ellis explains: "People need to be aware that there is a range of models that could explain the observations….For instance, I can construct you a spherically symmetrical universe with Earth at its center, and you cannot disprove it based on observations….You can only exclude it on philosophical grounds. In my view there is absolutely nothing wrong in that. What I want to bring into the open is the fact that we are using philosophical criteria in choosing our models. A lot of cosmology tries to hide that."

Otseng, I am coming to two conclusions about Mr. Ellis: in one he is right and in the other one he is wrong:

1) his spherical model with a geometrical centre of the expansion in 3D does fulfill Hubble's Law. This is, I confess, unexpected.

2) on the other side, this geometrical centre in such a model is a nonsense because ANY point in such a model can be considered as the centre of the expansion.


Please follow me (if you like):

Let's assume the expansion of the universe is centered in the Earth. Earth is at the centre of a expanding universe, in which every point P goes apart from Earth.

Earth is therefore the origin of a 3D environment that can be modelled with three Cartesian axes x, y, z.

A generic point P1 has a position vector with coordinates P1 = (x1, y1, z1). Earth has coordinates (0, 0, 0).

In this model, the velocity of any point is then proportional to its position (distance from the centre). Let this linear coefficient be a scalar number "a".

This means that the point P1 has a velocity V1 = a * P1

Therefore the velocity, expressed as vector, has components

V1 = (a x1, a y1, a z1)

The modulus of the velocity is

V1 = a * SQR( x1^2 + y1^2 + z^2 )

Hubble's Law states that the recession velocity of a galaxy is proportional to the distance.
Let the distance from Earth (origin of x, y, z) be "d". This distance is therefore computed as d = distance of P1 from origin zero.

d = SQR( x1^2 + y1^2 + z^2 )

Here we can see that V1 is proportional to d (by the factor "a"), ie the recession velocity is proportional to the distance d from Earth: Hubble's Law is fulfilled with a mathematical model of a 3D spherical expansion centered on the Earth.

But, what about for example the situation on Andromeda? If we consider what people in P1 see with reference to another point P2, what would they observe in this model?

Centering in P1, the distance of P2 from P1 is a vector with components

d = P2-P1 = SQR[ (x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2 ]

The velocity is also the difference V2 - V1, ie a vector DV with components

DV = V2-V1 = a * SQR[ (x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2 ]

Now, surprise! DV is again proportional to d. Hubble's law is fulfilled also for Aliens living in P1 (Andromeda?). These people is geometrically and cinematically fully entitled to say that THEY are the centre of the universe and not poor Earthlings.

All this is to say that, even in a model with a centre for the Big Bang, it is a nonsense to talk about the centre itself. This notion of centre would be totally relative and referred to any point in that 3D model.

[Curious]

Otseng, I am coming to two conclusions about Mr. Ellis: in one he is right and in the other one he is wrong:

1) his spherical model with a geometrical centre of the expansion in 3D does fulfill Hubble's Law. This is, I confess, unexpected.

2) on the other side, this geometrical centre in such a model is a nonsense because ANY point in such a model can be considered as the centre of the expansion.


Please follow me (if you like):...


...All this is to say that, even in a model with a centre for the Big Bang, it is a nonsense to talk about the centre itself. This notion of centre would be totally relative and referred to any point in that 3D model.

For those members not so mathematically inclined would it be a fair simplification to say that earth would appear to recede from Andromeda at the same rate as Andromeda appears to recede from Earth? In this case it is a fair assumption that you make as we observe from a single data point.. If on the other hand we were able to correlate the data obtained from numerous different points throughout the universe, the data might be used to build a model that is consistent with all observations.
At the moment it would be just as plausible that the data might be interpreted to indicate that we are all on the "edge" of the universe and are looking outwards in all directions towards the centre (although this itself seems illogical).

[otseng]

In my first posting I quoted the Cosmological Principle which said that no matter how far we travelled the Universe would always look the same.
This means that the Universe is unbounded, this is not quite the same thing as infinite as the Universe could be a hypersphere of finite dimension.

I reread your post about the Cosmological Principle and the universe being unbounded. However, I'm still not persuaded that it is unbounded. Could you also address the other arguments that I made in this thread that the universe appears to be bounded?

2) on the other side, this geometrical centre in such a model is a nonsense because ANY point in such a model can be considered as the centre of the expansion.

I'm not sure you can go from:
V1 = a * P1
to:
V1 = (a x1, a y1, a z1)
It seems like it should be something more like:
V1 = (a' x1, a'' y1, a''' z1)

But, since we have mathematicians on this forum, I'll let them verify this.

Also, if you are to use equations for an unbounded universe, Euclidian math will not apply. Since I have never studied non-Euclidean geometry, I'll defer this also to the mathematicians.

All this is to say that, even in a model with a centre for the Big Bang, it is a nonsense to talk about the centre itself. This notion of centre would be totally relative and referred to any point in that 3D model.

I do not believe so. Suppose there is a center (universe is bounded) and its position is (0,0,0). We have star A at (10,0,0) and star B at (-10,0,0) and star C at (30,0,0). Each star is going away from the center. Stars B and C are the same distance from star A. But, the redshift would be different for B and C on star A. The only point where the redshift would be proportional to distance for all stars is if the observer is at point (0,0,0).

[Bro Dave]

Actually, the universe in both defined and infinite. The answer to this seeming paradox, is that while the universe is still expanding and adding to it outer edge, at any moment there IS and outer edge, but only instantaneously so. I hope this explaination from the Urantia Book helps clarify a bit.

The Grand Universe is the present organized and inhabited creation. It consists of the seven superuniverses, with an aggregate evolutionary potential of around seven trillion inhabited planets, not to mention the eternal spheres of the central creation. But this tentative estimate takes no account of architectural administrative spheres, neither does it include the outlying groups of unorganized universes. The present ragged edge of the grand universe, its uneven and unfinished periphery, together with the tremendously unsettled condition of the whole astronomical plot, suggests to our star students that even the seven superuniverses are, as yet, uncompleted. As we move from within, from the divine center outward in any one direction, we do, eventually, come to the outer limits of the organized and inhabited creation; we come to the outer limits of the grand universe. And it is near this outer border, in a far-off corner of such a magnificent creation, that your local universe has its eventful existence.

UB paper 12 section 1 paragraph 7

Bro Dave

[Alien]

For those members not so mathematically inclined would it be a fair simplification to say that earth would appear to recede from Andromeda at the same rate as Andromeda appears to recede from Earth? In this case it is a fair assumption that you make as we observe from a single data point.

Yes, but I am saying a little more than that. The “Ellis-Otseng” mathematical model does imply the existence of a geometrical centre, origin of the expansion, but this geometrical centre can be put in any place in the universe, just because we can put in any point the “zero” of our coordinate system, and the model remains consistent.

Therefore, the Earth is not the only possible geometrical centre: there are infinite points such this in space.

If on the other hand we were able to correlate the data obtained from numerous different points throughout the universe, the data might be used to build a model that is consistent with all observations.

Yes.
These models are already consistent with observations because observations must be the same in each point. Even the Ellis-Otseng model says that.

At the moment it would be just as plausible that the data might be interpreted to indicate that we are all on the "edge" of the universe and are looking outwards in all directions towards the centre (although this itself seems illogical).

Sorry, what you say here is not clear to me.

This geometrical model discussed by myself is my understanding of the model from Mr. Ellis, somehow supported by Otseng. This model does not say anything about an edge of the universe. Mathematically, and geometrically, it works in the same way with an infinite universe or with a finite universe with an edge.
The difference with the classical Big Bang model is that the Big Bang model refers to a finite universe with no edge, whilst, as said, the Ellis-Otseng model may refer to an infinite universe or to a finite universe with edge.

If in the Ellis-Otseng model we consider a point close to the edge (if the edge exists), then the observations in the direction towards the edge would be undefined. What I have said about the “nonsense” of talking about one single centre is valid until you stay at a certain distance from the edge, or, if there is no edge and the universe is infinite. The edge introduces a discontinuity in the geometry of the model.

But, in both options, the Ellis-Otseng model is making an act of faith because a) it fails in describing an edge b) it assumes an infinite universe (and the concept of infinite is not very scientific because it implies an infinite quantity of matter/energy in the universe). Please note that the classical Big Bang model avoids both these problems (no edge, and no infinite universe).

I'm not sure you can go from:
V1 = a * P1
to:
V1 = (a x1, a y1, a z1)
It seems like it should be something more like:
V1 = (a' x1, a'' y1, a''' z1)

It should be a legitimate operation, because "a" is not a vector, but a scalar, ie a pure number, a pure proportional factor.

Also, if you are to use equations for an unbounded universe, Euclidian math will not apply.

The model I am talking about is described by a 3D Cartesian reference. Of course, this is a simplification and it does not take into account any space curvature. It refers only to a "flat" universe. But, even in its simplification, shows that a geometrical centre does not exist in 3D. Even if Hubble's Law is fulfilled. I think it is also independent on an existing/non-existing edge. The existence or not of an edge in 3D is only linked to the amount of matter (finite quantity = edge; infinite quantity = no edge).

The edge discontinuity disappears when you put the centre in a fourth dimension, like the classical Big Bang model 3D/4D.

In addition, I think that if you increase the complexity of the Ellis model as I discussed it, for example by considering space curvature, then the concept of geometrical centre will disappear further and the model will develop towards the classical Big Bang one.

[Curious]

At the moment it would be just as plausible that the data might be interpreted to indicate that we are all on the "edge" of the universe and are looking outwards in all directions towards the centre (although this itself seems illogical).

Sorry, what you say here is not clear to me.

Ok, perhaps I was a little unclear. What I mean is that (from the point of view of either Earth or Andromeda) either point, when taken alone, could be considered to be central. If we take into consideration the data from the other origin we see that either origin could be thought of as central. If we consider the position of Andromeda from the point of view of Earth we might interpret that Andromeda is less central (or nearer the "edge") and therefore is looking towards the universal centre when observing the Earth. Likewise, the converse would appear a valid assumption when considering the position of Earth from the perspective of Earth observing Andromeda from the perspective of scientists on Andromeda. So observers from position A would assume that all positions that observe A look towards the centre while from B,C,D etc. these B,C,D scientists would assume that A is observing them while looking towards the centre.
I hope this makes it a little clearer.

[Curious]

Actually, the universe in both defined and infinite. The answer to this seeming paradox, is that while the universe is still expanding and adding to it outer edge, at any moment there IS and outer edge, but only instantaneously so. I hope this explaination from the Urantia Book helps clarify a bit.

Oh yes, it really clears it up, thanks Dave

[Alien]

Ok, perhaps I was a little unclear. What I mean is that (from the point of view of either Earth or Andromeda) either point, when taken alone, could be considered to be central. If we take into consideration the data from the other origin we see that either origin could be thought of as central. If we consider the position of Andromeda from the point of view of Earth we might interpret that Andromeda is less central (or nearer the "edge") and therefore is looking towards the universal centre when observing the Earth. Likewise, the converse would appear a valid assumption when considering the position of Earth from the perspective of Earth observing Andromeda from the perspective of scientists on Andromeda. So observers from position A would assume that all positions that observe A look towards the centre while from B,C,D etc. these B,C,D scientists would assume that A is observing them while looking towards the centre.
I hope this makes it a little clearer.

Understood and agreed.
I would say a little more: you said in word what I was analysing mathematically, but the interesting part is that this can be applied both to the classical Big Bang model and to my analysis of the Ellis model (geometrically centered).

In other words, I am still curious to see a mathematical model that implies a geometrical centre of the universe in 3Dimensions and that fulfills Hubble's Law.

The question about the edge of the universe comes on top of this, I think.

[Curious]

I am still curious to see a mathematical model that implies a geometrical centre of the universe in 3Dimensions and that fulfills Hubble's Law.
The question about the edge of the universe comes on top of this, I think.

Well, what we "know" about the universe, if we assume a finite mass, we can "conclude" that rather than a fixed geometrical centre of the universe there is , in all probability, an unfixed gravitational centre of the universe. This would be far beyond my ability to calculate though (even had I the lifespan to finish such a vast equation)and would provide a snapshot in respect to that particular configuration of matter, spatial curvature, temporal distortion yadi yadi ya.

[Ian Parker]

I am still curious to see a mathematical model that implies a geometrical centre of the universe in 3Dimensions and that fulfills Hubble's Law.
The question about the edge of the universe comes on top of this, I think.

Well, what we "know" about the universe, if we assume a finite mass, we can "conclude" that rather than a fixed geometrical centre of the universe there is , in all probability, an unfixed gravitational centre of the universe. This would be far beyond my ability to calculate though (even had I the lifespan to finish such a vast equation)and would provide a snapshot in respect to that particular configuration of matter, spatial curvature, temporal distortion yadi yadi ya.

The reference

http://scienceworld.wolfram.com/physics/HubbleLaw.html

gives the Friedman Laimaitre model, this has been rigorously derived from General Relativity and obeys the Cosmological Principle. As I have explained with a Cosmological Principle there can be no edge, or rather no edge which is seen by any observer. We can go an infinite distance since after each journey the Universe looks the same. Viewed from outside the Universe there may be an "edge" travelling at c, but each observers frame of reference shows the same Universe with no edge. This is like Poincare's Universe which meant that no matter where you were the edge was still the same distance away. You could never reach it. A Universe 13.7 billion light years across and expanding at EXACTLY c is an infinite Universe even though it does not seem to be so at first sight. Poincare is certainly infinite.

What is probably of more interest ion "debating Christianity" is whether there are multiple universes and whether the Universe runs in cycles. A big rip has been postulated for 20 billion years time. The big rip in fact looks suspiciously like the big bang. The anthropic principle whereby the constants of physics are set to lead to life is circumvented if we have a hyperuniverse each with slightly different physical constants.

I would like to ask whether or not there is any slid evidence for a hyperuniverse. Pope Benedict has reacted rather cooly to the idea of a hyperuniverse or multiverse.