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Dear Ken & Alison, Glad you're getting back to me for feedback. This is definitely a good way to check we understand each other. :-) >Date: Sat, 12 Jan 2002 14:28:52 EST>Subject: Re: favoured topology of Universe candidates>To: boud w astro.uni.torun.pl>Dear Boud,>>Many thanks for your prompt reply. We have read through your e-mail and BASI>review, and would like to check that we have understood correctly (not being>experts in this field).>>We understand that you are researching possible shapes of the universe which>are finite, unbounded, and multiply connected (such as a hypersphere,>3-torus, 3-Klein Bottle or other 4D polyhedra). This would imply that we areOK, two comments. (a) science Correct, except that I'm also researching possible *infinite*, unbounded, and multiply connected models. A finite model would be more aesthetically pleasing, but it is best to do observational work with as few prior hypotheses as possible. The Universe is how ever it is, not how I would like it to be. (b) pedagogical, terminology The "shapes" such as the hypersphere, the 3-torus, the 3-Klein Bottle are *not* "4D polyhedra". Technically, they are called "3-manifolds", but since that is rather scary to non-specialists (including many astronomers!), a more friendly term would be "3-spaces" or "3-dimensional spaces" - which can be thought of with the *help* of a 4th dimension - or which can be thought of as 3D polyhedra where certain faces are identified in some way. It is *very* important to explain that when a 4th dimension is used to help think about a 3-space, this is purely a psychological tool, a mental crutch for human beings having difficulties thinking in a multiply connected and/or curved 3-space, simply because we are biased from our childhood experiences and school education to think in terms of Euclidean, simply connected 3-space. It might sound like gestalt psychotherapy ;-) to say this, but as far as a mathematician or physicist is concerned, a 3-dimensional space exists in and of itself, it has no need for the existence of extra dimensions to be itself, even if an extra dimension or too helps it (or beings living inside it!) to understand what it is. In "brane theory", it is true that 4th & 5th dimensions are thought of as true physical dimensions, but the field of observational studies of the topology of the Universe is separate from brane theory. But then again, the 4-dimensional or 5-dimensional models could again be thought of in spaces with an extra "psychological" dimension of no physical meaning, i.e. 5D or 6D spaces, again with the extra dimension just as a thinking aid. So I think the issue of a dimension as a thinking aid with no physical meaning cannot be avoided. I would recommend that in your article you explain something of this issue - the fact that as far as most observational cosmologists interested in either curvature and/or topology are concerned, a 4th dimension is a useful psychological tool for thinking, but is used without any physical meaning. If we go one dimension down, to make things easier, I can refer to Figure 1 of my BASI article http://uk.arXiv.org/abs/astro-ph/0010185 . Here I discuss a 2-space, the "2-torus". There are three different ways (i)-(iii) of thinking about the same space. It's a good exercise to try switching between the three ways and checking that you can understand them as the same thing. In (i) (lower figure), I use ordinary 3-space to help the reader imagine what the 2-space is. In (ii) and (iii), the reader does not need to think at all in 3-space. 2-space is enough, even though it may seem a little weird. Going back up a dimension, it is possible to imagine various 3-spaces, either like (i) within a 4-space or like (ii) and (iii) just within ordinary 3-space, but a little weird. >seeing repeated copies of a small universe (rather than a single very large>or infinite one). Your research centres around studying the sky to identify>repeated patterns of objects (celestial bodies and/or temperature signatures>in the microwave sky) to infer a size and shape for this "tiled" universe.Correct. For more clarity, you might want to put "...studying the sky in three dimensions to identify..." BTW, when you say "tiled" Universe, you're referring to mode (iii) of thinking. >Presumably, in order to decide in which directions, and at which distances,>to look for these repetitions, you must be working from a hypothesis about>the possible shape. For example, according to John Gribbin's New Scientist>article from that date, in 1997 you seemed to be looking for quasar patterns>to support a "twisted torus" or "3-Klein bottle" hypothesis.Nope. I'm a skeptical observer who works the other way around! There is no serious theory for the extension of general relativity which would give a theory of global geometry, and even if there was, the best bet would still be to make observations with as few preconceived ideas as possible. The standard Big Bang model, by the way, is a preconceived idea which I and my colleagues *do* assume, since it (and the theory of general relativity) is (are) extremely well established observationally. Rather than working from a specific hypothesis about the shape, my general approach is on finding the best ways of using existing (or near future) observational data catalogues. It happens that during some of these projects, serendipitous candidate topologies have turned up. I think all of (1), (2) and (3) that I gave you in the previous message are best described as serendipitous, though (2) comes from a (slightly) more systematic approach than (1) and (3). The candidate from my John Gribbin article was found using a very systematic approach, but at the moment it is just sitting in my "list-of-things-to-do". Although it was found systematically, I would probably class its "subjective probability" as 5%. But my subjective probabilities are... subjective. The only serious way to check candidates is by further observational work. BTW, the implicit 3-Klein-bottle-like model was what the observations gave, not what I was looking for. I would have preferred to find a 3-torus than a 3-Klein-bottle-like model! >Your e-mail suggests that your current research focuses on the use of galaxy>clusters and microwave patterns to identify a "model class: 2-torus". Are weHmmm. I don't know if the following is too subtle for a general audience. See how you like it. I would prefer to say: ---- My current research focuses on the use of galaxy clusters and microwave patterns in which serendipitously discovered 2-torus models offer good prospects of observational tests. Later on, if more complicated models are offered as candidates, the experience in testing 2-torus models will be important for testing the more complicated models. After all, if it is not possible to observationally test a 2-torus model, which is relatively simple, how could we possibly hope to test a more complicated model? Candidate (3) is even less ambitious: it involves just one generator, or path between two images of (hypothetically) the same object. ---- [plus a modification of the term "2-torus" as per the comment below] >right in assuming that this refers to an actual universe in the shape of a>(non-twisted?) 4D 3-torus, which is modelled as a 3D 2-torus?OK, I used an abbreviation without explaining it, sorry! Firstly, as I explained above, the 3-torus is really 3D, and the 2-torus is really 2D, even if an extra dimension is used as a psychological crutch. The model I've called "2-torus" for my observational work on the real Universe (as opposed to pedagogical explanations with one dimension subtracted) is really the 3-torus, but one side length is considered bigger than the horizon diameter, so big that observations made that far away would require the light to have been emitted before the Universe was born. If you called it a "3-torus with one very long side" or a "3-torus with one presumably infinite side length" that would be correct, would probably help avoid confusing readers, and would avoid experts in the field getting confused about what I'm doing. >We would also be interested in your views on other possible tessellated>polyhedra models. In particular, we are interested in any topologies whichWell, again, I would prefer to say "other possible 3-spaces" or "other possible, multiply connected 3-spaces, which can be thought of as tesselated polyhedra models". >may be consistent with the recently emerging field of braneworlds. For>example, Michael R. Feltz's website - http://www.cyburban.com/~mrf/ ->identifies brane theory with the Riemannian hypersphere topology.Well, I quite like the idea of non-experts writing web pages (if I hadn't been job-hunting for the past few years I probably would have made more effort to interact with non-specialists, and maybe sometime in the future I will be able to...), but I'm afraid there is some confusion in Michael Feltz's page: :The discussion here attempts to answer this question as it relates to:the often hypothesized, but little explored, "finite but unbounded":universe. The formal name for this model in topology and cosmology is:a "closed cosmic hypersphere".There are many finite but unbounded models other than the hypersphere, e.g. the 3-torus. :Otherwise the days of an expanding universe are numbered because this:high density model will expand only to certain maximum size and then:contract into a "big crunch" ("big bang" in reverse) at some future:date at least tens of billions of years from today.Correct. And the geometry of *this* model is a 3-sphere, also known as a hypersphere. :The alternate model explored in this series of essays is the long:suspected and often hypothesized "closed cosmic hypersphere" which:incorporates a fourth spatial dimension.The hypersphere provides one of the three possible curvatures, plus the assumption of trivial topology. Calling it the "closed cosmic hypersphere" is OK. But it does *not* incorporate a fourth spatial dimension. A fourth, psychological, dimension is just one possible way to think about it. And it is not an "alternate" model. It is the model which (for reasonable values of the local cosmological parameters, which show that the Universe is "approximately" flat, like any continent on the Earth is "approximately" flat) will expand to a maximum size and then contract into a "big crunch". Back to your question! My views on other possible "tesselated polyhedra" models (and now you know they're physically only 3D, not 4D), are that I'm totally open to any of these multiply connected models, whether for a flat Universe (where the angles of a triangle add up to 180°, and this includes the "3-torus with a very long side" models - this is a flat model!), for a spherical Universe ("hypersphere", where the angles of a triangle add up to more than 180°) or for a "saddle-shape" Universe ("hyperbolic", where than angles of a triangle add up to *less* than 180°). Of course, in either the spherical or the saddle-shape cases, the observable part of the Universe would be approximately flat, again like any continent on the Earth is approximately flat. >Given that the readers of Astronomy are mainly non-scientists, we would be>very grateful if you could provide as simple an explanation as possible.I know very little about brane-theory, but I think it is still much too theoretical to have any serious links with observational work. I think the best link you could make between observational cosmic topology work and brane theory could be something like the following: ---- Scientists trying to measure the 3-dimensional shape of space (with a type of non-expanding map of the Universe called "comoving" - although in reality the Universe is expanding, in this special map, the Universe can be thought of as static) sometimes use a psychological 4th dimension to explain or think about different possibilities for the shape of space, e.g. a 3-space which seems to be tesselated by many copies of the Universe. Other, more theoretical scientists, working on "brane theory", think that 4th and 5th dimensions might have real physical meaning (of course, there's also time, making a 6-dimensional world). However, the scientists trying to measure the 3-dimensional shape of space (also called the topology and curvature of space) prefer a more conservative, purely empirical approach and would like to measure the shape of 3-space without any preconceived notions, apart from the standard Big Bang model. They hope to detect any of many possible shapes of 3-space, whether it's the 3-torus with one very long side or any other. ---- For more on brane theory, you might want to ask Nathalie.Deruelle w obspm.fr, who gave a very nice explanation during a workshop in Paris last year, but unfortunately I was too tired and stressed to really listen properly :-(, though she did seem to give a good explanation. She's definitely an expert in brane theory, and is very supportive of work in cosmic topology. And she's probably one of the best people around who can make an intelligent comment on whether there's any link between the two. >Many thanks once again for your help,My pleasure :-) - feel free to keep passing draft texts to me for comment and/or other questions. Boud boud w astro.uni.torun.pl Torun Centre for Astrophysics, University of Nicolas Copernicus, Torun http://www.astro.uni.torun.pl/ (affiliation by the time the article is published!)

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