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Coordinatefree relativity 
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#37
Sep911, 06:19 AM

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#38
Sep911, 08:38 AM

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Ben says "A coordinate system is a map from an open subset of manifold into R^n."
What are examples of open subsets of manifolds? Are the chalkboard or the computer monitor not fair examples of subsets of manifolds? What are examples of R^n? If you describe the tension in each contractible muscle in your arm, is that not an example of R^n? If you describe the direction of an image in front of your face, wouldn't the natural inclination be to describe this either in terms of left, right, up and down, forward and backward; either in a rectangular or spherical coordinate system? Is there any way to describe distance without invoking some kind of numerical measure. (I can acknowledge that a dog may or may not invoke numbers in estimating distances, but if not, he also cannot communicate to other dogs where something is. On the other hand, bees are known to communicate quantitatively about distant locations.) Let me try some questions and see if you can answer them without invoking any kind of coordinate system.
What is it exactly about this statement 


#39
Sep911, 08:43 AM

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Upon mentioning bees in my last post, it occurred to me that another way of describing things is with landmark based geometry. Instead of stating a distance, you just say from this landmark go to that landmark.
You need not mention direction, distance, shape or size. 


#40
Sep911, 11:19 AM

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JDoolin Coordinates are almost like a religion to you. The point is not "Is it possible to use Coordinates?" but "Is it helpful to use Coordinates?". If we do decide to use Coordinates we must decide which ones to use and how many. A reasonable answer is none to both. Even when possible, using coordinates is not always worth the trouble they cause. Often even a pro coordinate zealot will be say things like
"Imagine that we have some coordinates, but we do not know anything about them, but they are there really they are, they are really messed up, but that is okay, we love them anyway, they are really complicated, we do not know how to get any numbers, if we had numbers it would not help because there would be so many and there would be so much error and the calculations would be so impossible, that is okay though we are not going to use them anyway, also these coordinates require that we embed the object we are working with into a much more complicated object which might be impossible..." Why would we want to introduce coordinates if (in a particular situation) we are not going to use them and they are not helpful? At best we have wasted time, and introduced needless complications. 


#41
Sep911, 11:59 AM

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Maybe something that would help understanding is to emphasize the point that, in general, if you are given two different sets of coordinates of some objects (which may or may not be the same object), it is extremely difficult to tell if the sets of coordinates are describing the same object. So, we describe objects in a coordinatefree fashion so that we never have to worry about that problem.
Using a programming analogy, this is (I believe) an equivalent problem to the halting problem, since the halting problem is equivalent to the word problem for groups, and group presentations are 'coordinates for groups'. For examples of how mathematicians think about things that have no concept of size, distance, etc. consider topological spaces. None of these notions exist until you define a metric. 


#42
Sep911, 01:09 PM

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JDoolin, the concepts of "distance" and "angle" do not require the use of coordinate systems at all.
Think back to the classical Greek geometry you did in highschool. Suppose I have a triangle ABC composed of three lines AB, AC, and BC. There is an unambiguous notion of the angle A that exists independently of any coordinate system. There is an unambiguous notion of the distance AB that exists independently of any coordinate system. In fact, there is no need to use coordinate systems at all; relying on a simple set of axioms, one can derive all geometrical facts using only the pure geometrical concepts of distances and angles. It was Descartes who invented (or reinvented) the notion of "analytic geometry": that is, marking the points of the triangle ABC by some coordinate system and then using the coordinate system to derive facts about the triangle ABC. This method makes some geometrical proofs more straightforward, but it is certainly not necessary to use a coordinate system, and in many cases it adds unneeded complexity. For example, try to show that the socalled "conic sections" are actually sections of a cone. Using coordinate systems, this is an algebraic nightmare. Using pure geometry, there is an elegant trick. 


#43
Sep911, 04:18 PM

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2) ~19" from the monitor to the tip of my nose 3) The center of the monitor is straight ahead and at a ~100º angle from vertical 4) I move them all over 5) ~15" away Note that none of the above required the specification of a coordinate system. 


#44
Sep911, 05:17 PM

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But once you have decided that you are using ANGLES and DISTANCES to describe the location of landmarks and features, you have implicitly defined a coordinate system. I don't know what you mean by "pure" geometric structures. But I can say that you need to look a little deeper for the "impurities" and ambiguities that really do exist in these lofty concepts. 


#45
Sep911, 05:38 PM

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Geometry studies what happens once you define a notion of "distance" and "angle". 


#46
Sep911, 06:40 PM

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#47
Sep1011, 06:29 AM

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Your first answer maps an open subset of the space in your room to R^1. Your second answer maps an open subset of the space in your room to R^1. The third answer is a projection of a vertical plane in your room onto [0,360). Your fourth answer only succeeds in avoiding a coordinate system by failing to be specific. Your fifth answer invokes three dimensions, since you are probably describing the average distance between the surface of the chalkboard and the surface of your body. 


#48
Sep1011, 06:57 AM

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If you are using Euclidean plane geometry (i.e. nothing is moving; nothing has any relative velocity), then I would have to agree that angle and distance have unambiguous meanings. However, I think it is also interesting (if we are talking about relativity) to consider objects that are moving. 


#49
Sep1011, 07:19 AM

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However, for the sake of argument, even considering a 1D embedded manifold in my room (so that we can map open subsets of the manifold to open subsets of R^1), a measure of the distance between two points does not establish a coordinate system. First, the measure of distance is invariant under shifts of the origin. Is the origin on me or is it on the monitor or on some other point elsewhere? Second, the measure of distance is invariant under reversals of the basis vector. Do coordinates increase from me to the monitor or from the monitor to me? Third, the measure of distance in inches does not preclude the use of a coordinate system using other units. Do the coordinates change at a rate of one coordinate per inch or one coordinate per meter? Fourth, the measure of distance does not indicate if the coordinate system is uniform. Do the coordinates change at a linearly decreasing rate as a function of distance? A measure of distance simply does not establish a mapping to R^n. There are many unspecified details. By telling you the distance from me to the monitor is 19" you cannot tell me unambiguously what is the coordinate position for me nor what is the coordinate for the monitor nor what are the coordinates for each point between us. Nothing less constitutes a coordinate system. 


#50
Sep1111, 02:16 PM

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I believe some use the word "clopen" to describe such subsets as a line or a plane through space. Even if you use an unnumbered ruler to measure the distance, you still must determine the "from" point, and by doing so, you have defined an origin. Yes, you can arbitrarily designate your origin, but that does not mean you can make the measurement without choosing an origin at all. A coordinate system need only have dimension high enough to measure whatever quantities you are interested in. If I ask you for the diagonal length of your computer screen, we only need a one dimensional coordinate system. If I ask you the dimensions of the computer screen, we need a twodimensional coordinate system. If I ask you where your left eye is in relation to the computer screen, we need a threedimensional coordinate system. If I allow for the fact that your eyes may be moving at a relative velocity wih the computer screen, and I ask the same question, we need a fourdimensional coordinate system. I think you could define a coordinate system without determining any distances, (so I see there is a difference), but I don't think you can go the other way, and determine a distance without defining, at least, a onedimensional coordinate system. 


#51
Sep1111, 02:58 PM

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You seem to be under the misapprehension that a coordinate system is the same as making a measurement that results in a number. That is incorrect, all measurements are invariant under arbitrary coordinate transforms. So the existence of a measurement does not require nor inform you as to any coordinate system. Nature does not come equipped with a required set of coordinates, regardless of what measurements you may make. You should know the definition of a coordinate system by now. Can you unambiguously define a unique coordinate system from the fact that the distance between A and B is 6"? *The almost is that in order to measure the distance between A and B you do not need to identify one as "from" and the other as "to". The measurement of distance is invariant under that choice. 


#52
Sep1211, 07:04 AM

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You can describe the distance between two points without defining a coordinate system.
But in order to measure the distance between two points, you must define an origin and unit length, which is the same as defining a coordinate system. 


#53
Sep1211, 07:38 AM

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The angle between the two marked paths is 90 degrees, but from the perspective of the dot in the middle, the angle between the particle paths is 180 degrees. To claim that "all measurements are invariant under arbitrary coordinate transforms" is one of those true, but misleading statements. The measurement of the angle depends on the velocity of the protractor that measures the angle. Sure, no matter what reference frame you're in that protractor will measure the same angle, but some observers will note that the protractor is distorted, and the measurement is actually incorrect. Just because the measurement is invariant, but the actual observation is very different. And I want to reiterate what I said in my previous post: While it is possible to describe things without defining an origin, it is impossible to measure things without explicitly defining an origin and unit length, and it is impossible to visualize anything without implicitly defining an origin. Even my ability to describe things without defining a coordinate system: ( "My monitor is 20 bloots across" ) actually conveys no useful information, until I define what a bloot is. 


#54
Sep1211, 08:31 AM

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