OK, please let me begin by declaring that, YES, I have read around this topic, but I am still stumped. It continually plagues me. I even had a friend blurt out of the blue the other day, "What's with the universe? Does it just keep on going forever? Or does it stop? If so, what's beyond it?" I could sense he was troubled by the whole cosmic concept (as we ALL should be!!)... and so am I. All I have read is convoluted ambiguous stuff about definitions of "universe" and "nothingness" etc = word games, no less. I read a recent thread here about "metrics"... which still doesn't do it for me. I'm not interested in an artificial concept, I'm only interested in the Reality of the whole situation. Is the universe finite? (= implies a "Beyond") Is the universe infinite? (= seems nonsensical, and counter-intuitive to "Big Bang" theory) PLEASE help! This is a very important & fundamental thing, for ALL of us (even if we aren't aware of it). Thanks! :yuck:
Both your parenthetical additions involve a mistake. Finite does NOT imply a beyond. A ring, for example, is a finite one-D universe with no beyond. a sphere surface is a finite 2-D universe with no beyond. Imagine that there was no space besides that spherical 2D surface. a traveler in that surface would never meet a boundary----no wall would mark the edge of his world the analogous 3D thing is called "ess-three" and written S^{3} it is finite, and has no boundary. if the universe space is finite, it is possible that it is a boundaryless space like S^{3} It could also be infinite and why do you say that would be counterintuitive? Bigbang theory does not say that the universe began as a point. Popular books that say that are misleading the public. Bigbang theory actually does not specify whether the initial singularity was finite or infinite. The space-finite versus space-infinite issue is undecided among cosmologists at present and there is some controversy about this heating up. Many cosmologists prefer to assume space-infinite because it simplifies the math or for various other reasons. Indeed to make it even simpler to calculate, they assume it is spatially FLAT as well as infinite---just your vanilla flat 3D graphpaper space! Others say, since we don't KNOW for sure that it is infinite or flat, you shouldn't assume it because it introduces errors in the analysis of data. They say you should analyze data in a way that leaves open both possibilities. Two guys in particular have been arguing forcefully for this viewpoint. Edward Wright and Bruce Bassett. they each have written papers recently about it. According to Ned Wright, the best-fit finite-space picture would have space be S^{3} with a radius of curvature about 130 billion light years.Then the bigbang singularity could be described in a finite localized region. Space is very very big now (but still not infinite) and it could have been a smallish compact space when it started expanding. the other version, with infinite-space, would assume that the initial bigbang singularity was already also spatially infinite---extending straight out to infinity in all directions. Ned Wright and Bruce Bassett do not favor one over the other. They don't say that the infinite space picture is WRONG. They just warn that you can misinterpret observational data and get trapped in circular reasoning if you ASSUME the infinite case when we dont actually know which is right. So far, allowing for observational uncertainty, the data is consistent with either case.
Thank you for the detailed response! I think my problem boils down to a basic inability to visualise/conceptualise, say, a 1D "ring" without the 2D paper it is drawn on. Likewise, a 2D spherical "surface" without any 3D "inside" or "outside". But let me play some more, maybe I'll ask more qu's down the track. Thanks again!
I'm glad it was helpful. As far as being able or not able to imagine a finite volume boundaryless space which is NOT embedded in an abstract space of higher dimension...I have some advice. My advice is relax and not to worry. Personally I think it is perfectly fine to think of a 2D sphere embedded in an abstract surrounding space. but if you live in one along with other 2D creatures you have to remember not to BUG the 2D astronomers by asking about the abstract centerpoint and talking as if it has some location or even some physical existence. If we lived in a 2D expanding balloon then the CENTER of the expansion would not be a point in the balloon. It would not be a real physical location for us, it would only exist in some higher (3) dimensional space that we can't navigate or triangulate or peer at with a telescope. So it is better not to talk about it. ======================= So by analogy imagine we live in an ESS-THREE (symbol S^{3}) which is the 3D analog of the balloon. Quite possibly we do. the cosmologist Ned Wright posted a paper earlier this year where he gives a best-fit which allows one to estimate the radius of average curvature of the ess-three as 130 billion lightyears. BUT YOU CAN'T POINT AT THE abstract CENTER, BECAUSE ALL THE DIRECTIONS YOU CAN POINT ARE IN OUR 3D SPACE. there could be this abstract higher dimension space that our ess-three is engulfed in and it might have a center but that is all non-science fantasy because you can't do an experiment about it. you can't point at it. it has no practical existence. So you have to remember not to bug astronomers and such by asking about it. suppose they come next year or in 10 years and they way "well! we have measured more accurately and now we are pretty sure the universe is finite and has a certain intrinsic curvature which you can measure by summing the angles inside big triangles and our best guess is that the universe is spatially an ess-three with a radius of curvature of 130 billion LY." Then what you must absolutely not do, this is an absolute no-no, is ask they "Where is the center?" the center would only be an abstract idea, you could not point at it because all directions are in your world, there would be no scientific evidence that it exists. Some people cannot imagine a 3-sphere with no inside and no outside and some people CAN imagine a 3-sphere with no inside or outside (by imagining the experience of inhabitants who go exploring and get back to the same place and stuff like that). whether one can or cannot imagine does not cut the ice----what counts is whether it can be defined mathematically, and it can. So the advice is to imagine it anyway that works for you but don't ask the astronomers where the center is
hmm I must admit I havn't followed your reasoning fully here Marcus, but I think there is a confusion of ideas. The question of whether the Universe is flat or curved and the question of whether it is finite or infinite are unrelated. You seem to be implying that a positively curved ('closed') Universe is necessarily finite which it not true. The 'ess-three' analogy you make with the surface of a balloon and 2D observers can trick you into thinking that the finite volume that someone crawling on the balloons surface can visit implies that the total volume of the Universe is also finite. This is not true, all it tells you is something about a given observers future light cone, i.e. that it converges. There can be infinitely more observers whose allowed region of travel does not overlap yours. All FRW solutions are valid only for infinite universes, including flat, closed and open ones. We don't really know how to write the GR solutions for finite Universes, which in itself is no reason not to consider the possibility, but it is important to realize that 'finite universe' effects that may at some point be observable are unrelated to issues of spatial curvature.
Hi Wallace, are you talking about infinite in a 4D sense? I was only talking about spatial finiteness. I think that is what the questioner was asking about.
I'm not quite sure what 'infinite in a 4D sense' means? In a positively curved FRW universe there is an infinite amount of space, however there is only a finite amount of space that a given observer can visit, due to the convergence of future light cones. In most (see here for some pedantic details) flat universes, given enough time, an observer can visit an infinite amount of space, by which I mean that they can reach any arbitrary point in space a given distance from their starting point (the pedantic issues arise because 'distance' is tricky as we know!). Now in our curved space, even given infinite time there is a finite distance from the origin that an observer can go to before, despite their best efforts, they find themselves coming closer to the origin once more (by the balloon analogy, they are on the opposite of the balloon to the origin). However, what you must not do is assume therefore that this maximal distance the observer can travel to defines the limit of the spatial volume of the Universe. It only defines the finite volume that is accessible to the given observer. There is nothing in this that prevents an infinite amount of space to exist, just that an observer can only get to a finite amount of space. So just because there is a maximal distance that you can go from the origin does not mean that there is no space at greater than that distance. As I say, all FRW solutions describe universes with an infinite amount of space, this includes curved ones.
I can see all that Wallace, as soon as you allow time into the picture it changes things. I was talking about a single spatial slice. I think that was what the questioner was wondering about
I think it is simply untrue that you can't have FRW solutions for a universe that is topologically equivalent to R x S^{3} that is a 3-sphere cross a time axis. there are plenty of metrics that are solutions and live on that. So we should try not to say anything that appears to contradict that. I was talking very informally about how one might imagine the experience of living in a SMALL ess-three from the inside. Then one can explore all around, perhaps in a few minutes or hours or days. I want to make the point that one can imagine these spaces from the inside without picturing a surrounding higherdimensional space. I was not talking about exploring the fullsize universe there one is limited by the speed one can travel. it isn't practical ================= maybe we can figure out some way to operationally check if our universe is spatially finite by having helpers in a chain of galaxies that go all the way around the equator. never happen but as a Gedanken experiment.
If you don't want to talk about time then you should also avoid talking about motion (in this case the wandering on the Balloon). I don't see that the OP asked anything about constant time slices, I'm pretty sure you introduced that idea ;) The question was pretty simple and fair. Is the universe finite or infinite. It is an interesting question but one that is not related in anyway to spatial curvature.
I think we can put the question clearly something like "does the universe have a compact spatial slice?" maybe you or one of the others would like to help rephrase it. here is a link to something I posted earlier about Bruce Bassett's paper https://www.physicsforums.com/showthread.php?p=1326040#post1326040 ============== my guess is that we get a lot of questioners here who have the same question in mind if you could take an instantaneous snapshot of space-----as Wallace said, a "constant time slice"----or as I said a "spatial slice"----- and that is how many people think of space----then would that be finite or infinite? It could be that mattex, who started this thread, had this kind of question in mind. It is sometimes hard to tell until you talk what the question is. ============== People also wonder about whether, if you could take an instantaneous snapshot of space would it have a BOUNDARY, a kind of wall, or skin? A lot of people's questions are fundamentally TOPOLOGICAL in nature. they ask how can space be finite (i.e. topologically compact) and not have a boundary? So we need to be prepared to answer. How do you suggest to someone to imagine a compact boundaryless space? (at that point the finite speed of light does not enter, it can be something you explore in a few days or years, like the surface of the earth which is finite yet boundaryless)
But that's not the question that was asked, and it is not what lay people generally mean when they ask 'is the Universe finite or infinite'. We can change the question to make it easier to answer, but then we are answering a different question!
How would you paraphrase mattex' question? I am interested. And what do you think lay people generally mean when they ask "is the Universe finite or infinite?"
I would paraphrase it like this, is there an infinite number of stars/galaxies/atoms etc in the Universe? The answer is the same regardless of the overall geometry of the Universe.
Great! I might not have thought of that. I think to make that question meaningful you may need to say something like is there a finite number NOW AT THIS MOMENT if particles can change from one kind to another, and time goes on forever, then the history of the universe might have an infinite number of different particles even though there were allowed to be only, say, TEN particles at any one time.
here is the original question that we are paraphrasing I would say that this refers to an instantaneous snapshot of space. Does it keep going on forever or does it stop (i.e. have a boundary, and something beyond)at the present moment? If you include time in the discussion then it quickly boils down to asking about does time go on forever or will everything end say in 50 billion years from now? That is an an interesting question too, but I don't think he was asking about a possible end of time. I think the question was analogous to someone who asks about the surface of the earth. "Does the surface of the earth go on forever? Or does it stop, at some kind of boundary, and have something beyond it?" People used to ask this question. they imagined the surface of the earth ending at a cliff. with some danger of falling off. I think we are hearing a similar sort of curiosity concerning the extent of space. Maybe we should start a thread and ask people!
Right, so as you pointed out well in your reply there are so fundamental misconceptions about what either option (finite or infinite) would imply. Having sorted them out I don't see how the remaining question pertains to spatial curvature? Whether or not the universe 'goes for ever' is not a question who's answer depends on the overall geometry. Whether we could travel in one direction forever and not end up where we started is such a question, but it is a different one. Edit: Okay so since posting this the previous post from Marcus has grown! I can't see how the issue of time comes into this at all? Regardless of whether the universe is curved (positively or negatively) or is flat the question of spatial finiteness has the same answer whether or not you 'consider time'. That answer is independent of the geometry! In a constant time slice of a curved FRW universe there is an infinite amount of space. When you throw time into the mix you find that there is a finite volume of space that can be visited by an observer. If you want to talk about truly spatially finite universes you need to come up with a different solution than FRW, one that does not obey the cosmological principle.
This is a terrific discussion! You people have certainly forced me to re-evaluate my fundamental assumptions. Yes, I suppose I was talking about a "spatial slice" of the universe - not taking into account time, or light-cones, or speed-of-light limitations, etc. I suppose what I'm really after is a "God's eye view" of the universe, HERE and NOW - which, in hindsight, is simply wishful thinking. I sincerely hope this discussion can continue!
Wow! we may have to leave it at that, at least for the time being. by positive curved I understand Omega = 1.01, for instance, say. by "amount of space" I understand you to mean volume. the slice I am thinking of is a spatial slice. I translate finite to mean compact, so for instance the whole spacetime could be topologically R x S^{3} What I gather is that you (whom I regard as expert) consider that impossible.
I, too, am somewhat confused here. If constant instants of cosmological time are used to foliate a Freidmann-Robertson-Walker spacetime into spacelike sections, then (as marcus has posted) [itex]\mathbb{R} \times S^3[/itex] results when the spacelike sections have constant positive spatial curvature with respect to the spatial metric induced on the the spacelike sections by the spacetime metric. In this case, each spatial section is compact and has finite volume [itex]2\pi^2a^3.[/itex] See Box 27.2 of Misner, Thorne, and Wheeler.