Boundary for an Infinite "Open" Space

[Feeble Wonk]

Please help! I read a statement by Lee Smolin (Time Reborn) that an "open" infinite universe necessarily has a "boundary", through which information would be received, which he used as an argument that cosmological models should prefer a "closed" universe approach. In fairness, he said that this was "counterintuitive", but... Huh? Why a boundary in an "open" infinite spatial universe? I obviously don't understand this concept correctly.

 

[martinbn]

Do you have the exact text? It sounds like he might be talking about boundary conditions.

 

[Chronos]

I suspect he is referring to the maximum distance light could have traveled since the big bang.

 

[Feeble Wonk]

Here's Smolin's quote...
"If the universe is not spatially closed, then it must be infinite in spatial extent. This means, counterintuitively, that there is a boundary to space. The boundary is infinitely far away, but nonetheless, it's a boundary which information could pass through. Consequently, a universe that is spatially infinite cannot be considered a self contained system. It must be considered a part of a larger system that includes whatever information is coming in from the boundary. If the boundary were a finite distance away, you could imagine that there is still more space outside it. The information about the boundary would be explicable in terms of what is coming in from the world beyond the boundary. But the boundary at infinity does not allow us to imagine a world beyond. We are simply required to specify information about what is coming in and going out there. But the choice is entirely arbitrary. There can be no further explanation for the information coming into the universe from the infinite boundary. A choice must be made, and the choice is arbitrary. Hence, we have to concede that nothing can be explained in any model of a universe that has an infinite boundary. The principal of explanatory closure is violated, and with it, the principal of sufficient reason."
The book up until this point has been very interesting, and largely understandable for the most part. But... I don't understand this passage at all. Can anyone enlighten me please.

 

[OCR]

Hence, we have to concede that nothing can be explained in any model of a universe that has an infinite boundary. The principal of explanatory closure is violated, and with it, the principal principle of sufficient reason ...

I don't understand this passage at all.

Neither do I...

Anyway, the book... http://en.wikipedia.org/wiki/Time_Reborn

And a related paper... Temporal Naturalism
http://arxiv.org/abs/1310.8539

 

[Chalnoth]

I'm not entirely sure I understand what he's saying, but if he's saying what I think he's saying is that the behavior of an infinite, open universe is dependent upon the boundary conditions of that universe, even though the boundary is at infinity. If there was no physical boundary, you would expect that any way of writing down the boundary conditions would lead to the same result, but, if I'm understanding what he's saying correctly, this isn't the case. He claims that the idea of a boundary at infinity having an impact on the physics of the system is nonsensical, so this description cannot be correct.

I'm only approximately 60% sure I've caught the gist of this, though...

 

[Chronos]

The statement is illogical. What is the meaning of a 'boundary at infinity'? The phrase '... a universe that is spatially infinite ... must be considered a part of a larger system that includes whatever information is coming in from the boundary.' is similarly insensible. If you tried to express this mathematically it would look something like $$1 + \infty \not\subset \infty$$

 

[ChrisVer]

My question is : the boundary condition of what? of the space? but the space in this case is infinite, so it's unbounded.

 

[Chalnoth]

The statement is illogical. What is the meaning of a 'boundary at infinity'? The phrase '... a universe that is spatially infinite ... must be considered a part of a larger system that includes whatever information is coming in from the boundary.' is similarly insensible. If you tried to express this mathematically it would look something like $$1 + \infty \not\subset \infty$$

You can certainly write down boundary conditions at infinity. His point (if I'm understanding it) is that it doesn't make sense that these boundary conditions should have any impact on the behavior of the system.