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Higher dimensions

  1. Jan 31, 2005 #1
    I wanted to bring this forward from another thread because I think we may be getting at an important point regarding dimensionality. The last post in the thread is quoted below:

    Consider the three spatial basis lines, x, y, z. If there is velocity in x, then there is Lorenz contraction in x. The y and z basis lines are unaffected by velocity in x.

    Real objects are always extended in all three spatial basis lines, as well as in time. For this reason every part of any real object will experience contraction due to velocity. Contraction in x will be distributed uniformly throughout the object. The sphere of the object (obtained by rotating an object through every possible rotation at every possible point in the object) will be flattened in x.

    Since real objects are always extended in x, y, z, and t, we may propose the generalization that real objects are always extended in every available basis line. If there are extra dimensions which cannot be expressed using combinations of x,y,z, and t, then every real object will have some extension in the basis line of that dimension.

    Now when we speak of dimensions as labled [tex]{x_1, x_2, x_3...x_n}[/tex] we may presume that x,y, and z are included in the innumeration, so that perhaps x is the same as [tex]x_1[/tex], y is the same as [tex]x_2[/tex] and so on. Since all the extra dimensions in string theory are said to be spatial, I am presumeing that t is a special case and is not included in the set as an [tex]x_n[/tex].

    Then since there is no distinction made as to which subscript is to represent which basis line (x could as well be [tex]x_2[/tex] and y could be [tex]x_1[/tex] we must assume that all members of the set of [tex]x_n[/tex] are equivalent, so that each [tex]x_n[/tex] is a basis line. Then it follows that if velocity is in the direction of any x, there is contraction in that x, but none of the other members of the set is contracted.

    This is to say that, if the above reasoning holds, every subscripted x is perpendicular to every other subscripted x. That is, they form an orthagonal set. We cannot easily imagine an orthagonal set of spatial basis lines with more than three members, but it seems apparent that this is what we mean when we speak of higher dimensionalities.

    However if we relax the rule that orthagonal axis must be at 90 degrees to each other, we can imagine more than three axis in ordinary space. We can build a representative structure containing more than three axis, with the rule intact that each axis represents an independent dimension, which does not experience contraction when motion occurs along one of the other basis lines.

    In fact, we can represent higher dimensionalities on a two dimensional surface by drawing a circle and dividing the circumference into 2n points, where n is the number of dimensions required. For a three dimensional space, this results in the familiar figure called a hexagram, in which the cord of the endpoints is equal to the radius of the circle.

    In this scheme, motion along one axis will result in contraction along that axis, but no contraction along any other axis. If motion could be carried to the extreme, the axis would contract to a point, and the circle would divided into a figure shaped like the infinity sign. If we were carrying out this operation on a model in three spatial dimensions, the result would be a torus.

    This action would be equivalent to adding a hole to a topological figure, or to punching a hole through the spacetime fabric. Of course we are forbidden in practical terms from doing so with any real object.

    There is however another way to analyse the geometry of higher dimensions. I would like to go on from here to explore the possiblitiy that time is not a special case but is to be included in the [tex]x_n[/tex] set, which makes rational sense, since if there is motion along one of the basis lines, time must be extended along that basis. In fact, time must be extended along any basis line which can accomodate the sense of motion. This is in concordance with the principle of spacetime equivalence due to Einstein and Minkowski. Then we must consider each axis in x as containing within it an axis in t.

    I hope to go on with this idea in another post. Any comments are welcome, especially those directed to challenging this line of reason.


    Richard T. Harbaugh,
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  3. Jan 31, 2005 #2


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    Good idea. I have skipped the copy of my last post to address your concerns freshly.

    Only if the velocity is ONLY along x. The real thing that determines is the velocity vector. If that mixes x, y,and z then there will be proportionate contractions in x, y, and z. This is where the full Lorentz matrices come in.


    This is correct. If the compacted diensions are real, then we extend into them and move through them.

    You are absolutely right about x, y, and z. The t coordinate is represented as [tex]x_0[/tex]

    Right on. Well expressed, too.

    Right again! Have you considered teaching this stuff?

    An example of doing this is the "tesseract" models which represent a (euclidean) four-cube. The four edges meeting at a corner are represented with three of them at right angles to each other and the fourth heading off at an acute angle. It's important to note that although such representations are useful for visualizing some relations they inevitable distort other relations. Generally speaking it's topological features that survive and metric ones that are obliterated.

    Yes I suppose you could do this. I suspect if you work around with it you will find it hard to use it for details. See my comment above.

    Well following up your statement that the contraction is uniform across the body (I think we are both assuming a sphere) it would become a flat disk in 3-space, and a "flat hyperball" in n-space, i.e. it would be a full ball in the (n-1)-space orthogonal to the direction of the relative velocity. I think this is a case where your reductive dimensional model leads you astray.

    This is based on your torus, which is not what happens. See above.

    Close but not quite. The accurate statement is that the direction of a real relative velocity is "timelike". This means the velocity vector is not orthogonal to the time dimension but has a component along it, in addition to its spatial component. So this vector has to point in the cone between the set of v=c vectors and the time axis. The fact that every physical velocity has a non-zero time component is the reason that time and space get mixed up in the Lorentz transformations. You are accustomed to see length contraction and time dilation as separate formulas, but they are each special cases of the true formula. Suppose the velocity vector (relative to an observer) has extension t along the [tex]x_0[/tex]-axis and x along the [tex]x_1[/tex]-axis. Then the dimensions experienced by the observer (they are denoted by t' and x') are given by

    [tex] x' = \gamma x - \gamma\beta ct[/tex]
    [tex] t' = -\gamma\beta cx + \gamma t[/tex]

    where [tex]\beta = \frac{v}{c} and \gamma = \frac {1}{\sqrt{1-\beta^2}}[/tex]
    Last edited: Jan 31, 2005
  4. Jan 31, 2005 #3
    I am expecting company today and tomorrow (they will have to take universal precautions to prevent transfer of the virus which is visiting my house also) so will have small time for now to consider this post. Your comments are very welcome also, and much less discomforting than the hundred day cough, which is going around here.

    I am in fact driven to teach this stuff, which is why most of my close friends groan whenever the topic comes up. They have all patiently suffered being driven into glazed states of semi-coma by my extemporaneous lectures. However in the past two years I have had more success, and even on occassion been specifically invited over to talk about physics! The magnitude of this victory can only be revealed when you know that most of my friends, honored and beloved as they are, are not physicers, nor even scientists, but farmers, laborers, loggers, and small freeholders. A rather anarchous group, we sometimes refer to ourselves as technicolor "amish," because of a preference for communal self reliance and animal husbandry, not in any religious context.

    I have chores to do, but will look forward eagerly to my next opportunity to work with the formulae and corrections to my text which you have given. Thanks! (And yes, I was imagining velocity only in x, a condition which may turn out to be an ideal oversimplification. If all motion is curved, then there is no way to move in only one direction!

    Be well,

    Richard T. Harbaugh
  5. Jan 31, 2005 #4


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    Yes I know the eye-glazed look. My daughter got it yesterday when I described our exchanges here. We can love and admire people who sadly don't share our interests.
  6. Feb 3, 2005 #5

    The sky has cleared after many days of thick cloud, and the stars are very bright. Pegasus is on the low horizon, and Cassiopea not much higher. I have a couple of things on my mind tonight.

    First, there is prescience. Cassiopea is a great summation sign in the heavens, and Pegasus, the great square, is a Greek delta, the sign of difference, change. Between them, like a full colon, the eyes of Draco stare:

    Code (Text):

           *                                *  
                            *          *
    *             *                        *
                            *          *
            *                                 *
    Andromeda, our neighbor galaxy, sits at A.

    The delta, or great square, is the sign of difference, and might be thought of in mathematical symbols as we use them today either as the minus sign, for subtraction, or as the cross multiplication sign, X. The Summa sign of Cassiopea means addition, and Draco, between them, is the indicator for a ratio. What is the ratio of the difference to the sum? And is that not a strange and beautiful symmetry? One wonders what might be found inthe region of B.

    But enough dreaming.

    I want to think about the light cone for a moment. What is the light cone, physically?

    Well, we consider an event, let us say it is a flash of light in an otherwise empty space. It has an origin (my friend Guadalupe has thought a lot about origins, and I am wondering what he would think about this? Is he awake tonight?) at a place. This means that we could think of the first instant of the flash of light as a very small region of space which contains the light, and which is contained itself in a very large region in which the light is not present.

    As time passes, the light expands. It starts out in a very small region of the available space, and ends up filling the available space. In between the beginning and the end, it smoothly increases in size. We may wish to examine this process piecewise along a line in time. In order to do so, we lay out a line of time as if it were a line in space, and then consider the expansion in the other two available spacelike dimensions.

    Two spacelike dimensions makes a surface, and since we have already said that the space is otherwise empty, we may assume that the surface is Euclidian, a flat plane, like an unbounded piece of paper. Well we don't need to consider infinity just yet, so it is ok to think of it as a piece of paper with bounds, that is edges, if we so wish. Let us just say that the surface is large enough to contain the flash of light from its beginning to its ending. And, as my friend Guadelupe would tell you, everything which has a beginning must also end.

    Since we have laid out a line of time along one of the available spacelike dimensions, and we are considering the expansion in the other two available dimensions, the expansion at any instant looks like a circle on the piece of paper. Now we need to stack up really a lot of pieces of paper, in order, to see the cone. Let the origin of the light be a very small circle, and the next page a very sligntly larger circle, and the next page a slightly larger circle, and so on, and after a few billion billion pages, we see the cone developing.

    There are several conditions we must fullfull to see the cone develop. There is the substitution of a time line for a line in space. There is the division of the time line into small equal increments, the thickness of the pages. There is the origin, which must be held in the same place from page to page. And we must remember that the page, flat as it is, is only part of the space in which the flash occurs, since we have deleted the third dimension in space in order to see the development of the cone in time. We see on each page a two dimensional circle, but we must remember that really the circle represents a three dimensional sphere, just that we have flattened it to see it stack up as a cone over time.

    Ok, I believe these are the necessary and sufficient conditions for a light cone. Now let us discuss the origin for a moment, the first page.

    We have assumed that there is a flash of light in an otherwise empty space. The flash of light, however, also must have some conditions, if it is to represent conditions as we actually find them in our space and time. Where did the light come from? How intense is the light? How long does the light flash last?

    The light had to come from somewhere. Maybe it is comeing out of a little hole in a heated box. But then, it would not give us a very pretty light cone, since it would be going in one direction in space but not in all the other directions. The substance of the box would prevent the light from traveling in the direction opposite to the hole. Unless, of course, we add some more special conditions. We might lay out our time line starting at the hole in the box and traveling outward in time in the same direction that the light travels, so we preserve our light cone by choosing a special direction.

    But these conditions are somewhat unsatisfactory. There is that clunky box, and why is it hot anyway, and how do we know in advance which way to stack the papers up to catch the cone as it is emitted? Before the flash, we can't know where the hole is, and after the flash, if it is a short flash, it is gone and we can never catch up to it, since it is light and nothing travels as fast as light. We have assumed some problems with our conditions of origin, if we assume a black box with a little hole, and we see right away that the presense of the box violates our stipulation that the flash occurs in otherwise empty space.

    Maybe the flash is a virtual photon escaping from near the surface of a black hole, while its partner gets swallowed up. This is called Hawking radiation, for anyone reading this who wants to know more about virtual photons near black holes. But then we have more problems. Clearly a black hole is not otherwise empty space. In fact, we might even suggest that a black hole is just a blace box with a hole in it, turned inside out. Well, that is not very satisfactory either then.

    Maybe the flash is a proton decay in a tank of heavy water deep under the Tower Sudan mine. That is not really empty space, but it is very regular space, which is maybe not much different from space if it were empty. Anyway the water is transparent to the light so as far as the light is concerned it is as if it were empty. Then we surround the tank with photon detectors and catch the light as it tries to escape the empty space....well. So much for the flat stack of papers. The detectors have to be curved around the tank, and once the photon goes into a detector, it is ended. We can't really know what happened to the photon between its beginning and its ending. In fact, since it is a single photon, and has a direction, it doesn't make a nice circle on the paper at all, only a little point.

    It seems we are having problems finding conditions that realistically approach those we imagined for our light cone. Well nevermind. It is still a good idea, and there are many good ideas we can imagine that we can't exactly reproduce in the laboratory. That is why we have thought experiments, and it seems this discussion is turning into one of those.

    Well ok. Flat sheet of paper, first one in the stack, the origin. We won't wonder how we knew which way to lay the paper, and we will say that the flash at the origin is of some intensity, so it is composed of many photons going off in all directions at once. Next page, small circle. Next page, slightly larger circle. Next page and so on. Jump to the end of the chapter. Bigger circle. End of next chapter, even bigger circle. Actually, in order to see the light cone forming, we have to make our pages pretty thick. Well, thick in time, not necessarily in space! We can use really thin paper, but we have to remember that light travels fast, so if we want to see the cone in a small enough space to appreciate that it is a cone, we have to imagine that we are taking the circles and sort of spreading them out along the time line, that is, making thick pages. Even though the pages have to be thick in space, so that we can see the cone, they have to represent very thin slices of time, so that we can catch the expansion before it gets away from us.

    And it is good that we make the pages thin, in space, because they have to be very wide pages, again because light moves so fast. We need really a lot of pages, and they have to be really wide pages, and so they had better be really thin or we will quickly run out of resources!

    Lets skip over all the in-between pages and turn directly to the end of the book. What will we find? Does the circle ever go off the edge of the paper? Does it fade away, losing intensity until we cannot see it any more? Do the pages have to get thicker or thinner to accomodate the scale of time and space as it approaches infinity? Ooops. There's infinity again. I didn't really want to talk about that. But it just keeps on showing up. Nevermind.

    Well, we had really a lot of photons to start out with. We had to start with really a lot of photons because we saw that a single photon travels in a single direction and so does not make a nice circle for our light cone. Very well. But we are not going to assume an infinite number (ooops again) of photons! That would be impossible. The universe, our universe anyway, can only contain some limited number of photons, surely?

    So as we get to the end of the book, what happens to the circle? Does it eventually divide up into individual points, each point an individual photon? And then if we keep on going, say we decide to follow one arc of the circle because it gets too big to draw the whole thing on a single piece of paper. And then we eventually repeat that process until we are following only one photon, and it is a single line. What happened to the light cone? It seems this brings us right back to the single photon in otherwise empty space. And we have already seen that a single photon in otherwise empty space cannot create a light cone.

    Well, I am not trying to say that the idea of a light cone is not an otherwise interesting and useful object. It is nice to be able to talk about angles and velocities and wonder about what lies outside the cone or before the cone or after the cone. Maybe we can just set our pages to a thickness so that the speed of light is represented by fortyfive degrees of angle, and that is pretty cool. Then what lies outside of fortyfive degrees? Why, nothing. After all, nothing can travel faster than light, can it? All right, nevermind tachyons, nevermind ignorons, nevermind Unruh radiation, and for gosh sake, lets not talk about positrons and if they might be traveling backwards in time. All of that is clearly not part of the discussion. Is it?

    But, anyway, back at the circle "A" ranch, a bunch of us cowpokes were wondering, well, now, those circles, don't they have to be somewhere? So doesn't there have to be something outside of them? I mean, even if it is at an angle that would have to be greater than the speed of light. I mean, just because Andromeda is some exponent lightyears away, so the light we see is coming from some exponent number of years ago, do we really have to say that something as big as a galaxy that was there so many years ago isn't really there in any meaningful sense, "now", whatever "now" means. I mean it doesn't seem likely that Andromeda has just gone away, leaving her skirts of light trailing behind her? She has to still be there, somehow, somewhere, doesn't she?

    Well, you may say, it isn't that Andromeda isn't there any more, it is just that the information we have about Andromeda is a little dated. I mean, we can calculate the speeds of the stars in Andromeda and observe their directions and come to the conclusion that if they were there then, they must be over here a little further now, unless they have been swallowed up by a black hole or gone supernova or something. But there are a lot of stars in Andromeda and it seems unlikely that all of them have gone supernova already (now that would be a flash of light!) or that all of them have been swallowed up by a black hole or something, so Andromeda as a collection of stars must still be there, not?

    So outside the circles of our cone of light there must be something there, and even if we cannot know exactly what it is, we can use our powers of observation and our knowlege of physics and our faith in an isotropic, homeostatic universe, to predict with some certainty, with a nod to Heisenburg, that there is a universe, kind of, outside our light cone. And even behind our light cone (uh oh, now I have gone into deep hot water) there is something. Something was there before the flash, it is just that information (light) from before the flash cannot ever catch up to us as we go on turning the pages. In other words, the past is still there! We just can't see it because information from the past cannot catch up to us.

    But wait a minute, if information from the past cannot catch up to us, what is memory? If information from the past can't catch up to us, how do we know that there is any past there? Light from nothing in an otherwise empty universe? Ho ho ho, of course not. That was just the conditions we started with so we could see our idea of a light cone, remember?

    So, when the learned doctors tell us that nothing exists outside the light cone, much less behind the light cone, we have to remember that they are insane and have stared at the stars too long instead of remembering the importance of cuddling up in nice warm soft blankets here by the campfire. With friends, if possible. Ah friends, the importance of friends.

    Now, what was it about that symmetry that was bothering me so much? Nevermind. Pegusus has set, and Andromeda is low on the horizon, hiding behind her skirts, and the sum of all things is looking up, and the dragon is laid out flat, and Spring is surely coming. I have to decide. Should I plant a garden? Or study the stars? My friends are laughing.

    Be well, yes, and love,

    Last edited by a moderator: Feb 3, 2005
  7. Feb 3, 2005 #6
    I should very much like to plant a garden this spring, and have time to look at the stars. Perhaps I can find a way to do both. My garden friends may be willing to care for the garden if I am away for, say, a month at the height of the growing season. Or maybe my star friends will get a clue and schedule their stargazeing meetings at a season more suitable for gardeners. Well it doesn't really matter. It is unlikely, as A. Zee says, that the fate of democracy hangs in the balance.

    Anyway, last night's post was a little rush of ideas and I got a bit loose with throwing papers around and so on. For example there was the idea about a hole in a black box being kind of the inverse of a black hole...that wasn't very well stated. And as for the learned doctors being insane from staring too long at the stars, well, certainly that thought could be taken as ill-considered. I meant no disrespect. I was just doing a little anti-elitist teasing, a kind of tickle under the blankets, and I hope none of the learned doctors will be offended. As it happens, I have worked with certified insane individuals from time to time, and they can actually be good, stimulating company. I have a sort of fondness for crazy people.

    I recently re-read "The Man Who Loved Only Numbers" by Paul Hoffman about the stellar mathematician, Paul Erdos. Now there was a case in the vertice. It seems he would have starved to death if people weren't there to prepare his meals. Forget about laundry. It seems amazing, reading his story, that he did not fall into the bands of a nice, quiet asylum. It seems we must take him as a kind of savant, because his daily behavior was extreme, and must have at times bordered on the intolerable to his friends. Lucky for him, he had plenty of freinds, and he spread his peregrinations out fairly thinly among them.

    But what I really wanted to talk about was the light cone again, or rather, a sort of inversion of the light cone. Instead of thinking about light radiating out in all directions, we might find it instructive to imagine light falling in from all directions...a condition called Infall. Then we won't have to worry so much about the source of the light...maybe it is big bang thermal energy or something like that. And that annoying question about what happens to the light way out in the outer limits goes away...we know what happens to the light, it falls in until it reaches some central object, and then it stops. And, since we have a central object, it is easy to pin our papers to it and draw our circles. Pretty much any central object will do, we do not have to worry about if it has a hole in it or not.

    The only problem seems to be to decide on a location for the time line and to decide on the thickness of the paper slices. The rest pretty much takes care of itself. So let's lay out our time line in, say, that direction. Then the papers are laid out perpendicular to the time line, or we could say, the time line is normal to the surface of the papers. Every page has a little dot at the center of it and that is our collection point. On the first page, just the dot.

    Then on the next page, a tiny circle. That circle represents all the points on the page which are close enough to the point for light to reach from them to the point in the time represented by the thickness of the paper. You see it is just the opposite of the light cone idea, in that light is now falling in to the center of the circle rather than going out toward the edges of the paper. One condition, and this is a rather stange one and might bear some thinking. That is, the central point.....it isn't really staying in one place any more, is it? That is, it seems to be moving through the book from page to page, present right there at the center on each and every paper. That is, it is moving through time. Well it has to be, doesn't it? It could hardly be said to exist if it did not move through time. It has to have the quality of duration. But you see that this is a change from the light cone, which after all did not have anything at the center of each page, just the origin of the light on the first page, and after that, emptyness, the innermost reach of the expanding circle. Now we have a little dot. Well it isn't much difference really. We still have the circle, and it is still expanding page to page, as we advance through the book, forming a sort of cone. If you cut out each circle on each page and stacked them up, you would see the cone shape very clearly.

    Now we can still adjust the thickness of the paper so that the cone expands outward at fourtyfive degrees if we wish. That is, the fortyfive degree angle still represents the speed of light as it falls inward in our cone. The circle represents the region of everything which can be known by the point in the time since the origin. This is convenient. What is outside the circle? Could be anything. We just don't know.

    We don't have to worry about the probable position of Andromeda as we chop our way through the snake, because Andromeda won't be part of the picture for millions of years anyway, and when we do get to Andromeda, as Andromeda crosses the boundary of our Infall cone, we can just make a note of it, and not be concerned about where and if it was somewhere before it fell into our cone. There is no history of Andromeda, no physics known outside the cone, no need to speculate about black holes swallowing up galaxies and so forth. The only thing that matters, the only thing there is, is the information that exists inside the circle. Outside the circle is completely unknown, just as if we were one of Schroedinger's kittens, waiting to be born. (That's an oblique reference to a book I saw at the bookstore, but I havn't read it yet.)

    The future is where it belongs, in the unknown and unbounded. I imagine Heisenburg would be pleased.

    What is outside the cone? I dunno. We can't know. No one can know. What is behind the cone? Before the cone? Same answer. There be dragons. We can speculate that what is outside the cone must be similar to what is inside the cone, but that is all we can do, and there is the lurking uncertainty. Anything could be out there, and we just havn't encountered it yet. Could be anything. Could be nothing. There is really no reason or grounds to wonder. Something completely against all our known physics could be just beyond the horizon. All the stars could decide to gather in one corner of the sky and spell out "Eat At Betty's Pies." It isn't impossible, even if it is a blatant plug. (Acutally it is the plot gimmick of a story in an old sci-fi magazine from my youth...Analog or Amazing or Strange something or other. I don't remember the author but I hope he or she would be pleased that at least I remembered the gimmick.)

    So now after all this I have a little suggestion, and I hope you may find it amusing, or even interesting. Here it is. The realm of general relitivity is the light cone. The realm of quantum mechanics is the inverse light cone, may I call it the Infall cone. So the relationship between GR and QM should be an inverse one. That's all. I hope it is useful.

    Now there are two ravens in the pine tree outside my wintery window, and the lake and the sky are very very blue, and the shadows on the bending drifts of melting snow are very long, and I get to work tonight dissolving grease with organic solvents and polishing steel with soft rags and aerosols of mineral oils, pays my way, and if the dish doesn't run away with the spoon, I may be back tomorrow with questions about what's a light cone good for anyway?

    Be well,

    Richard T. Harbaugh,
    Last edited by a moderator: Feb 3, 2005
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