The infinite and the infinitesimal

[Tanelorn]

TL;DR Summary

Infinite space and infinitesimal space are we any closer to explaining these?

Over the years the following has continued to be my biggest question in Cosmology.

In the past couple of years I wondered if we have got any closer to understanding whether our space is infinite or infinitesimal? (By infinitesimal I mean that there is no lower limit to the minimum separation of two points).

We are told that the universe is homogeneous and heterogeneous, and so the conclusion I draw is that the Universe has no boundary and is therefore infinite. However, can a truly infinite universe even be a possibility? Could a big bang 13.8B years ago, which of course happened everywhere, have possibly resulted in an infinite Universe?

Some of the things which I have considered is that space (or time) is not a ponderable thing and so has no objective existence of its own. Therefore there is no problem with space being infinite or infinitesimal, because it has no objective existence of its own, it is nothingness. However, this goes against everything we are used to, because everything we can think of is inside something else.

Regarding infinitesimal distances could we say that the world is quantized at the Planck scale? Or is that simply a high energy measurement problem and there is no lower limit for scale in reality?

At the Observable Universe scale do we have a "get out" to the problem of explaining an infinite Universe by simply saying that beyond the Observable Universe, anything that might be there no longer has any cause or effect on us here, and is therefore no different than if it did not exist at all? However, instead, this would in effect mean that there is an infinite number of casually separated, finite, Observable Universes.

 

[PeroK]

As you may know there is currently no working theory of quantum gravity and no working theory of a discrete spacetime. The cosmological models we have are based on a spacetime continuum.

Summary:: Infinite space and infinitesimal space are we any closer to explaining these?

Regarding infinitesimal distances could we say that the world is quantized at the Planck scale? Or is that simply a high energy measurement problem and there is no lower limit for scale in reality?

There is nothing special about the Planck scale as a lower limit of spacetime distances.

Summary:: Infinite space and infinitesimal space are we any closer to explaining these?

At the Observable Universe scale do we have a "get out" to the problem of explaining an infinite Universe by simply saying that beyond the Observable Universe, anything that might be there no longer has any cause or effect on us here, and so is therefore no different than if it did not exist at all? In effect an infinite number of casually separated finite Observable Universes.

The universe may be infinite or simply very large. The observable universe is continuously extending as time passes, so there isn't anything special about the universe we can currently observe, as of the end of the year 2020.

Whether we will ever be able to confirm that the universe is infinite is a moot point. It's logical that we may one day discover it is finite, but how we would ever be totally convinced it is infinite is a different question.

I don't really see there is any fundamental problem with the current state of affairs: we know the universe is very large, we have no evidence that it is finite and we have valid cosmological models that work if the universe is indeed infinite. I don't see the problem, other than a philosophical one.

 

[kimbyd]

I'm not sure whether either question is knowable. I'll divide the them out:

1) For whether the universe is infinite, the question there is whether the universe is topologically closed. This is a different question than whether it's closed in a curvature sense. Topologically closed means that the universe wraps back on itself at large distances.

But we know that our universe can't do this for at least many times the diameter of the observable universe. And the fact that we can't see very far beyond that makes the chance of observing closed topology pretty much impossible. Certainly if the topology is open, it can never be confirmed observationally.

Our only hope of answering this question, I'm pretty sure, is nailing down the physics that kicked off our observable universe. If that event can only produce a topologically closed universe, then we may have an answer. But we may never be able to answer that question either.

2) For whether or not there is a minimum length, that may also be undetectable. Certainly attempts so far to find measurable effects of such a thing have completely failed. And the energy required to probe distances close enough to the Planck length are effectively impossible to produce, and probably always will be.

Maybe if we're able to determine the correct theory of quantum gravity we'll have our answer. But again, that may be unknowable as well.

So basically it all boils down to the simple fact that we just don't know how much we'll be able to learn in the future.

For my part, I suspect we won't get these answers in our lifetimes. Which is frustrating.

 

[Tanelorn]

Thanks PeroK and kimbyd for your considered replies.

Firstly, thanks for clarifying that space itself is not quantized at the Planck scales, a misconception I have held for decades. Also I thought that the Observable Universe is getting smaller over time?

I share your frustration that we may never know the structure of the Universe at the largest scales and whether it is finite or infinite, and whether there is something beyond which resulted in the big bang. For me these questions are at the heart of Cosmology, with much of the rest at smaller scales instead being more related to Astrophysics. I just discovered this timeline of recent theories:

https://en.wikipedia.org/wiki/Timeline_of_cosmological_theories
https://en.wikipedia.org/wiki/Timel...usters_of_galaxies,_and_large-scale_structure
https://en.wikipedia.org/wiki/Physical_cosmologyOver the years Cosmologists have proposed an increasing number of interesting solutions, but frustratingly it appears that none can be proved, at least yet. However, there certainly is a lot of impressive work being done to try to solve this:
Shape of the universe - Wikipedia
https://en.wikipedia.org/wiki/Brane_cosmology
https://en.wikipedia.org/wiki/Loop_quantum_cosmology
https://en.wikipedia.org/wiki/Cyclic_model
https://en.wikipedia.org/wiki/Conformal_cyclic_cosmology
https://en.wikipedia.org/wiki/String_theory

The idea of Casually Disconnected Observable Universes (a form of multiverse), provided me with some relief at least from this frustration. I think Guth said that the minimum diameter of the whole universe is 10^32 times larger than our Observable Universe. However, at that distance and relative velocity, it no longer matters to us here whether they share the same physics as us here, or whether they even exist at all. Therefore perhaps we can than say that our Universe or Observable Universe is finite because what is beyond is completely casually disconnected from us.

My gut tells me that due to the fact that the Universe is temporally finite with an age of 13.8B years that there is something beyond, a cause resulting in an effect, but that is all I have.

 

[Klystron]

...
Some of the things which I have considered is that space (or time) is not a ponderable thing and so has no objective existence of its own. Therefore there is no problem with space being infinite or infinitesimal, because it has no objective existence of its own, it is nothingness. However, this goes against everything we are used to, because everything we can think of is inside something else.
...{bolding added}

While not intended as a cosmological model but as an aid to understanding, consider different topological objects such as a Klein bottle derived from connecting Möbius strips or bands. From the article:

Whereas a Möbius strip is a surface with boundary, a Klein bottle has no boundary. For comparison, a sphere is an orientable surface with no boundary.

I found that learning the mathematics (and constructing physical models) of these objects also
helped me understand space-time geometry and related theories. Pardon if the wikipedia entries become too technical for a B-level thread, but the general explanations can be useful.
Examples:

Klein bottle Mobius strip

 

[PeterDonis]

I thought that the Observable Universe is getting smaller over time?

No, it isn't. Why would you think it is?

 

[MikeeMiracle]

He is probably referring to the number of objects we can see due to universal expansion pushing them outside of our observable universe.

 

[PeroK]

He is probably referring to the number of objects we can see due to universal expansion pushing them outside of our observable universe.

That, I believe, may be an inflationary misconception, to which the obvious cure is:

https://www.physicsforums.com/insights/inflationary-misconceptions-basics-cosmological-horizons/

 

[Tanelorn]

Yes, that was what I was talking about, I did indeed think that objects are receding from us and that the most distant objects will eventually no longer be visible. I believe I have even read articles which suggest that ultimately only our local group of galaxies will be visible. Has this now been debunked? If so what is the current prediction for the diameter of the observable universe over the coming billions of years?
Observable universe - Wikipedia

On the subject of Cosmology theories, is Brane cosmology well liked? I like how it contains the three large dimensions of our Universe. Does Brane cosmology offer insights into what is happening inside black holes (singularities)? Back in the day Black holes were supposed to be gateways to other dimensions in the popular press.
Brane cosmology - Wikipedia

 

[PeterDonis]

I did indeed think that objects are receding from us and that the most distant objects will eventually no longer be visible. I believe I have even read articles which suggest that ultimately only our local group of galaxies will be visible. Has this now been debunked?

No, but some things should be clarified:

(1) The size of the observable universe, in terms of distance, is increasing. That is simply because, the older the universe gets, the more time light has had to travel to us, so the greater the distance light that reaches us can cover. Note that this would be true even if the universe were not expanding at all.

(2) Most of the objects that are now in our observable universe, in the sense that we can see light from them, will not always remain in our observable universe, in the sense that there will be some point on their worldlines after which light emitted by those objects will never reach us--so, from our perspective, we will only observe the history of those objects up to a certain point, and not after that. This only happens in a universe in which the expansion is accelerating, as it is in ours.

This will happen for any objects that aren't part of the same gravitationally bound system as we are. I don't think that is just our local group of galaxies; our local group is part of a larger galaxy cluster that, AFAIK, is one gravitationally bound system.

 

[Tanelorn]

Thanks Peter, yes I was thinking a few hours after my original post that I should have clarified that I meant the distance to the objects and not the size of space itself.