25 August, 2007

A Short Introduction to Brane Theory

[EDIT: This entry was never finished, but I'm publishing as is anyway. I apologize in advance for poor formatting and conceptual editing. At some point, I may revisit the topic.]

In this short introduction to Brane Theory, I will be looking at what the many theories deaing with branes that theoretical physicists are currently working on is all about, so that when I speak in future entries about branes, you won't be completely lost. Space as we see it is three-dimensional, meaning that there are three fundamental 'directions', or in a technical sense, that three is the minimum number of co-ordinates required to specify a point in space. Okay, so you realize that space as we see it is 3 dimensional, and time is an additional, yet different dimension, in that it flows in only one direction, and that the universe seems to have a distinct preference for one direction over another. 6:15 PM Einstein showed that space and time are interlinked to such a degree that they complement one another perfectly in just such a way as to make the speed of light constant; if time is dilated, space is lengthed by just enough to keep the speed of light (c) constant. So instead of speaking in terms of 3 dimensions, it is clear that time is extremely similar to the rest except in the respects pointed out above. 6:17 PM Gravity, according to gen rel, is a feature of spacetime geometry itself. It is not so much a force in the classical sense, nor even in the quantum mechanical sense, as it has to do with the geometry of spacetime rather than the propulsion of intermediary particles. 6:18 PM So far, I think you're already very conversant in what I'm saying. But the next step is a recent development, and I'm not sure if you've heard of it. Have you heard of brane theory? Dorek: yes it is a theory of the origin of the universe, as far as i know 6:19 PM i don't know much beyond that me: Yes. It involves extra-dimensionality that isn't kaluza-klein. I'll explain, but stop me if I start on something you already know. 6:20 PM Typically, whenever extra dimensions are thought about, it follows the kaluza-klein theory: that is, that extra dimensions are so small that we can't ordinarily perceive them. It is like a garden hose: from a great distance, it appears one-dimensional, though it's surface actually has two (and volume has three). 6:21 PM A Kaluza-Klein dimension is a fourth spatial dimension that is curved in upon itself (such as a garden hose's surface curves in on itself in one of its two dimensions) but is so small that we do not actually see it. 6:22 PM It is just a theory of course, but if kk dimensions exist, they would have to be on the order of less than hundredth of a millimeter, since extra dimensions would have noticable effects with gravity, and we have verfied general relativity up to a hundredth of a millimeter. Dorek: hold on it's dinner now 6:23 PM but i am interested in hearing what you have to say i will be back shortly me: Okay. Take yout ime. I'll be here awhile. -------------------------------------------------------------------------------- 8 minutes 6:31 PM Dorek: ok me: Back already? That was quick. Dorek: i'm back 6:32 PM me: Must have been a short dinner. Dorek: it lasted from 5:23 to 5:32 6:33 PM me: Where was I? Oh, Kaluza-Klein theory.... Did I give a sufficient idea of what that means? 6:34 PM Dorek: yes me: Keep in mind, of course, that talking about extra-dimensionality is a step away from what we know to a very high degree of likliness is true, and goes into theorizing that won't be experimentally verifiable until the large hadron collider opens in CERN, switzerland, in late 2008. 6:35 PM Then again, superstring theory is completely unverifiable until we get particle accelerators that are a trillion times more powerful than we currently have, and yet superstring theorists get NOVA PBS specials and all kinds of good press. But I digress. 6:36 PM Dorek: those damned liberals and their public tv 6:37 PM me: As recently as the mid-nineties, a new theory of extra-dimensionality came into (second-place) prominence (always after superstrings, of course): brane theory. 6:38 PM The idea is that there are extra spatial dimensions, just like KK theory. But in this case, there are objects within the larger spacetime continuum that have a very peculiar property: 6:39 PM They restrict the movement of certain particles upon the 'surface' of the brane. To explain, let me mention a historical note. 6:40 PM The reason kk theory developed (1920 or so, I think) is because gen rel seemed t work really well in five dimensions. But it is obvious to everypne that there is no fifth dimension: if there were, we could see it, since, presumably, photons would go in that direction, and bounce back tio hit our eyes. 6:41 PM So Kaluza proposed (and Klein later mathematically showed) that the fifth dimensioncould be really small, and we wouldn't be able to detect its existence. Of course, kk theory in terms of gen rel made predictions that turned out to be false, so we know of it now as a failed theory. But this is beside the point. 6:42 PM If you have a dimension big enough, we should be able to see it. So if you propose a new dimension, there has to be a way to keep it invisible to us. Brane theory does this. 6:43 PM In a larger spacetime continuum, there is (presumably) a brane that happens to have three spatial dimensions. On its 'surface' (I use the word in a technical sense), certain particles are trapped. 6:44 PM Thus, if I shine a flashlight, its photons can never leave the brane. This applies to all particles that physics has discovered: leptons, bosons, and gauge force particles. 6:45 PM At first, it seems to be just a weird way of describing reality, since we could be on a 3-brane and not even know it--it seems, at first, to be identical to living just in 3-space. 6:46 PM But it has a distinct difference: gravity is determined by the shape of space, not be intermediary particles. So gravity in a 3-brane in X-space is fundamentally different from just a 3-space. 6:47 PM In fact, if you determine the properties of gravity sufficiently, that will be the only way we now know of directly understanding the shape of spacetime beyone our brane. In this way, extra-dimensions can be as large as you want, and we still would not have found them already, since almost all of our methods of detection would use particles thatare trapped on the brane. 6:48 PM So far, I'm just repeating what brane theorists have been working on for the last ten years. They were a distinct minority until 2001 in the physics community because the superstring theorists were dominating the field. Dorek : hmm so it is less of an attempt at explaining other possible dimensions rather than an explanation for why we can't see more than 4 6:49 PM besides of course that there might be no more me: Well, it's a model for a universe that could exist, and that has ramifications that, if it were to exist, would be verifiable by experimentalists. 6:50 PM There are many such models that have been accepted and later rejected in physics. Dorek : oh i see me : One thing that goes against simple 3-space with no extra dimensions is that the standard model of physics has 30-something constants that we can't explain nor otherwise account for. 6:51 PM Many of these constants could be explained, if there were extra dimensions to account for the math. 6:52 PM The standard model is accepted as gospel because it works tirelessly: it predicts outcomes to an unprecedented degree of accuracy. But the standard model is just a chart of numbers: there's no explanation, for example, of why the numbers are as they are. 6:53 PM Using extra-dimensionality, theorists are trying to explain these numbers in terms of far fewer constants: string theory, for example, has only one constant, from which every other constant is derivable. 6:54 PM That's what made string theory so beautiful and enticing to many physicists, and it also explains why although year after year more and more evidence accumulates against stringtheory, the physicists still hang on to it, almost as a crutch. 6:55 PM Of course, in 2001, it turned out that string theory is mathematically equivalent to a specific kind of brane theory, and so since then everybody's been working on branes. But string theory limits itself to a very peculiar sort of brane theory, which I am not conversant enough in to go into detail about. 6:56 PM Anyway, I've tried to give a short, somewhat generalized description of brane theory as it stands today. 6:57 PM Obviously, there's much I didn't go into, such as the realization that with brane theory, the dramatic difference in power between the fundamental forces can be explained, and other ideas. But this is enough to start you onto where I had a thought, and explain what I've been working on lately. 6:58 PM So starting here, I'm going to be talking not about what others have done, but what I'm considering. As such, if you have questions, or if you notice inconsistencies, please let me know. 6:59 PM Symmetry has long been an important figure in physics. Symmetry across time, space, rotation, etc. are all fundamental to understanding what we already know. 7:00 PM So much so that when a new type of symmetry was discovered possible, many physicists actuively believe that it is in fact reality. 7:01 PM Supersymmetry was discovered as a possible concept in the eighties, and its predictions cannot be verified as true or false until the LHC comes online in 2008. Yet many, many physicists believe in it actively already. What I thought about was a new kind of symmetry. A dimensional symmetry. 7:02 PM We live in what appears to be 3-space, with three spatial dimensions. 7:03 PM If I wanted to be pedantic and describe something quite meaningless, I could instead say we live in 6 dimensions, but that there are three pairs of symmetric directions. Instead of left-right being a dimension in 3-space, I could instead say that there is left as a dimension in itself, and right as a dimension in itself: but that they are symmetric. 7:04 PM Specifically, when one increases, the other decreases, and by the same amount. Effectively, it is exactly the same as 3-space, but you get the idea, right? Dorek : ok 7:05 PM me: I'm sure many had this thought before, but didn't go anywhere with it, because it doesn't do anything different. But now we have brane theory. So a new thought arises. When you look at each of these 'half-dimensions', you notice certain peculiarites. First, they are direction-specific: in each case, they are always increasing. The slope is always positive when graphed versus time. 7:06 PM This is reminiscent of time as a dimension: it is also uni-directional. 7:07 PM So I thought: what if these half-dimensions are the true reality, and they're all perfectly simliar and symmetric. Yet time's counterpart happens to not be on this particular brane. 7:08 PM The idea is that we are actually in higher dimensional space, and time's symmetric counterpart is not on this brane. But the implication is strong: _this would explain why time is considered different from lateral dimensions_. 7:09 PM The idea is that it wouldn't matter what dimension you plot against, so long as you only plot against one single 'half-dimension', so a definite direction can be shown. Now that you get the basic idea of my premise, I'll explain the math of it, and show you how far I've gotten with it. 7:10 PM In gen rel, time is treated as a special dimension, and in every equation where other dimensions appear as variables, time always appears alongside c^2 and some other stuff. 7:11 PM Without this, the equations don't work. So I've started to work out combinations of variables such that we can rewrite x^2 + y^2 + z^2 + c^2t^2 as: 7:12 PM f(x1,x2)^2 + (f(y1,y2)^2 + .. etc. 7:13 PM The idea is to come up with a mathematically valid way of rewriting the equations such that each instance of a dimension can be written instead as a function of two dimensions that look similar to how time looks all on its own. 7:14 PM I admit that finding such would not be ironclad proof by any means, but if such a matematical expression exists, then it would at least allow the possibility of my idea to be true. 7:15 PM Once the math is found, then it will be easy to point out that my idea follows the rest of physics exactly in most every respect, while introducing a new symmetry, and it also would have an observable consequence. 7:17 PM If true, and time does have a counterpart, and if that symmetric relationship is identical to the relationship between lateral dimensions, then gravity must 'flow' toward those directions as well. 7:18 PM In other words, if you looked at gravity in a 5-space world with a 4-brane (spacetime), you'd see a peculiar effect of gravity that isn't possible unless we really are a brane in 5-space. 7:19 PM Similarly, if we looked at gravity in a 6-space with a 4-brane, you'd see a similar, but quantitiatively dfifferent peculiar effect of gravity. When the LHC comes online in 2008, we will be able to actually measure this peculiar effect. 7:20 PM Brane theorists today hope that the measurement (if it occurs at all) will correspond to the dozens upon dozens of different dimensionalities that are proposed today. Whichever it agrees with will be what physics has to rely on, because we have to follow what experiment tells us. String theorists are hoping, by the way, for evidence of 10 or 11 dimensions, because one of those is required by string theory. So here's the thing: 7:21 PM I want to work out the math on this, and quick, before the LHC comes online. Afterward will be too late. If it happens to be true (probably not, but whatever), then its strength as a theory depends on whether or not the resultsa are predicted before they are obtained. 7:22 PM I need to work out the quantitative peculiar effects before the LHC experimentally shows us what they are. 7:23 PM Presumably, the quantitative effects for my idea would be in-between the effects seen for predictions of a 5-space or a 6-space, because in my idea, there would actually be 5.5-space or 6.5 space, owing to the extra half dimension of time. But I don't know enough math to determine such effects yet. 7:24 PM Dorek : hey sorry my foot just hit the reset button accidentally me: No prob. 7:25 PM Dorek : the last thing you were saying is that you need to work this out before the experiment comes online me: I need to work out the quantitative peculiar effects before the LHC experimentally shows us what they are. Dorek: yes me: Presumably, the quantitative effects for my idea would be in-between the effects seen for predictions of a 5-space or a 6-space, because in my idea, there would actually be 5.5-space or 6.5 space, owing to the extra half dimension of time. But I don't know enough math to determine such effects yet. 7:26 PM Dorek: ok me: And it's pointless to even try if there is no way to rewrite currently accepted equations as functions of two 'half-dimensions'. Why go through all the work of finding out the quantitative predictions if the idea can't work in the first place? So I need to verify first that it's at least _possible_, in theory. 7:27 PM Only then would I start on working out the quantitative effects, in order to predict possible results that we may get in 2008. 7:28 PM me: The thing is: this idea could have occurred to any number of physicists already, yet they're not saying anything publicly. So either they haven't worked out the idea enough to make a prediction yet, or it's not even possible in theory, so they've discarded it. Or, I guess, as a distant possibility, I may have been the first to think of it again since brane theory developed. 7:29 PM Either way, I need to do a lot of work to get this done, and it's been what's preoccupied my time for the last week or so. Even if it's completely wrong in reality, if it's workable in theory, then it will be enough to actually publish in a journal. 7:31 PM At this stage, I'm just looking for ways to translate all the accepted equations involving dimensions into equations involving half-dimensions, such that at least one of the halves is similar to how time is treated in the equation. The problem is I can't use c, since it's in terms of space per time. So I have to create an analog of space per space, which sounds weird at first, but is perfectly doable. 7:32 PM It's just that no one's ever done it before. At least not that I've researched. So... Any comments? Dorek: hmm 7:33 PM i understand what you are seeing 7:34 PM but unfortunately my background in physics and mathematics are not good enough to grasp the idea of putting the concept into mathematical formulas 7:35 PM i guess i would first need to know the current accepted formulas 7:36 PM me : Not counting the math, then, do you feel like the idea is conceptually sound? No glaring difficulties that I haven't noticed? 7:38 PM If I were to send you a list of accepted formulas involving dimensions, would you be interested in looking at them thoroughly enough to where you could help with my project? I haven't talked to anyone else about this yet... Dorek: no, in fact when you first expressed time as being a simple half-dimension that is located outside the dimensions that are located fully within our space it seemed pretty ingenious 7:39 PM if i understood what you were saying correctly sure i'd help you me: That sounds great. I was hoping that the first time I told someone about this, I'd get a positive response. 7:40 PM I was afraid to go to an actual physicist first, because I hadn't done any of the math yet, and they could easily pre-empt my idea. Dorek: if you send me the formulas, i'll study them me: But if you're willing to slog through the math, I'd really like to coauthor a paper with you. Dorek: also, if you could give me any material that would catch me up to you, in terms of physics and possibly math i'd like to look at that as well me: Sure... Dorek: books, or anything 7:41 PM me: For a historical treatment of brane thoery from one of the physicists that worked on during the surperstring years, written in a good style, I might recommend... hold on, I forgot the name of it. one sec. 7:45 PM Warped Passages by Lisa randall, published 2005. 7:47 PM For a broader view, so you can get to know physics from a philosophy of science point of view, I recommend: The trouble with Physics, by Lee Smolin, 2006. 7:48 PM These are not textbooks, so you should have little difficulty getting through them. 7:49 PM As for explaining the math, what I'll do is send you a collection of accepted equations that involve dimensions, and for each one, I'll tell you where it's from, and what it means. 7:50 PM All that has to be done with them is figure out a way to rewrite them so that every dimension term is separated as a function of two half-dimension terms that are inversely proportional to one another. 7:51 PM AND at least one of these half-dimensions must look similar to how the time dimension is treated in that same equation, so we can make the argument that we could restructure the whole equation in terms of any half-dimension we so chose, as long as it's complementary dimension is not in this brane. 7:53 PM You know enough math to realize that there are probably an infinite number of ways to restructure an equation in terms of whatever the hell you want, but some restructuring can't be done, like arithmetic with infinity, or division by zero or some other such thing. And since this involves inverse proportionality, I'm mainly worried about avoiding division by zero in the restructuring. 7:55 PM Once that's accomplished, the next part will be the really hard part: we'll have to make quantitative predictions using methods that I don't yet understand that brane theorists are currentlyutilizing to make their predictions. 7:56 PM But once we've shown that it can be done, I think that will be enough motivation to go out there and start learning what's needed to do that next step. And even if it turns out to be nothing more than a neat mathematical trick, it'll still be a fun and interesting way to spend our free time. (Or at least that's how I view it.) 7:57 PM Dorek: it sounds like quite a task 7:58 PM but i've been recently thinking that i've not been utilizing my free time productively enough i'll read up on the material you send, as well as the two books i may need more than that, but i'll start with what you given 7:59 PM me : Excellent. I'll be sending along those equations soon; I'd rather not have to type them in, and I don't have access to a scanner, so I'll do go back to my original sources to find a few webpages that will give them to you. I'll make sure to make it explicit where on the pages they'll be located. 8:01 PM Hopefully, the local library may carry those two books; if not, I'm sure you can find them at B&N, if you care to sit in the cafe while reading, or you might just buy them altogether. Dorek: awesome me: At worst, you could use an intelibrary loan. Dorek: also, if you could cut paste and email me the text of this conversation i lost half me: You can see the text in chats, on the bar on the left side of the screen. Dorek: i was going to go to bn tomorrow afternoon anyways

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