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Coordinatefree relativity 
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#19
Sep611, 04:09 PM

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#20
Sep711, 06:38 AM

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Can you show me your proof? (or tell me where I can see a similar proof.) 


#21
Sep711, 06:45 AM

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#22
Sep711, 06:46 AM

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#23
Sep711, 06:52 AM

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Also in practice, in order to do any real GR calculation, you have to vastly reduce the number of coordinates so people use symmetry arguments to reduce the complexity of the problem before actually trying to solve it. One thing about coordinate free arguments is that imposing a coordinate system in curved space is not easy, so you have to start by figuring what coordinates you can and cannot use, and what the properties of those coordinates are. 


#24
Sep711, 07:18 AM

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Rendering the image on a computer screen does require a coordinate system, but drawing a picture on paper does not invoke a coordinate system. You can use whatever coordinate system you like to describe the image on the paper. Using your eyes to view the world does give you an origin, but it does not require you to use a spherical or Cartesian coordinate system to describe things in the world, (though those are the most convenient.) It would still be correct to say there is no way you can picture anything, or describe shape, location, or size, without an origin (your eyes), and without picturing it at a certain distance away from you. It's just that you aren't constrained to use any particular form of coordinates to do so. 


#25
Sep711, 07:54 AM

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#26
Sep711, 09:55 PM

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Also imagine a sponge. If you try to imagine putting a coordinate system on a surface of a sponge in which you have to take into account every little nook and cranny, you will go insane, and you will find that you *can't* create a mapping from the surface of a sponge to a global coordinate system. Now imagine, a thousand black holes.
What you can do is to magnify the sponge and create a local "flat" coordinate system for a tiny piece of the sponge, and the tape a thousand of those coordinate systems together to describe the sponge. The description of the sponge then becomes a description of how you "tape" the local systems together. Once you do that, you realize that you don't need the local coordinates at all. All you need is a description of how to "tape" different pieces of the sponge together. One other way of thinking about how coordinatefree geometry works and why it is useful is to think about LEGO's. I want to tell you how to build a gas station out of LEGO's, and I *could* give you a bunch of X,Y,Z coordinates for each part of the gas station, but you'll go insane. Or I can tell you to connect piece one to piece two. Connect that piece to another piece etc. etc. By specifying how two pieces connect, you can build complex structures that don't involve coordinates. In GR, the "LEGO's" are things called tetrads. 


#27
Sep711, 09:58 PM

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One problem that people have with GR is that people are trying to fit it into their intuition of how threespace works. For example, the questions of "where did the big bang happen" or "what does curved space curve into" are questions because people are trying to picture thing using the rules that work for things they see everyday. The important thing is that those rules don't apply. Space doesn't need to curve into anything. The big bang was everywhere and nowhere. Mathematically, you can describe spaces that aren't obviously connected to your daily visualization, and learning how to visualize those things is why math is hard. One of the hard part of doing the math is to "let go" of your intuition and learn a new set of rules for how space works. 


#28
Sep711, 10:11 PM

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Besides which, angles are observer dependent anyway, so you can't actually unambiguously state what any angle is without deciding on the velocity of the reference frame you're using. 


#29
Sep711, 10:19 PM

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Maybe if you went with a Klein Bottle or something like that, I would agree with you; I might go insane with that one. But not with a sponge. 


#30
Sep811, 06:57 AM

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By observing a figure with my eyes, what coordinate system have I defined? Where is the origin, is it in my center of mass, or is it in my right eye, or my left eye, or somewhere between them, does it matter if I am right or left eye dominant? Am I implying spherical, or cylindrical, or Cartesian, or some other arbitrary coordinate system? Which way are the axes oriented? Is the coordinate system righthanded or lefthanded? Does my handedness make a difference? Is it orthonormal? None of this is specified, therefore you have not invoked a coordinate system. A coordinate system is a 1to1 differentiable mapping from points in the manifold to points in R(n). Looking doesn't uniquely define such a mapping. 


#31
Sep811, 07:29 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. I'm still curious about your argument here: 


#32
Sep811, 10:43 AM

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I was also reminded of this video, at about 2:10, people drawing straight upanddown lines on a piece of paper passing by;
http://www.youtube.com/watch?v=igOWR_BXJU 


#33
Sep811, 01:07 PM

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#34
Sep811, 01:41 PM

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As Dale is hinting, a reference frame and a coordinate system are not the same thing.
A coordinate system is a map from an open subset of manifold into R^n. A reference frame is a collection of n linearlyindependent vectors at a single point. A local inertial frame is the GR analogue of an orthonormal frame: it is a collection of n mutually orthonormal vectors at a single point. 


#35
Sep811, 11:00 PM

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Well, regardless of the definitions of manifolds, reference frames, and coordinate system, let me just reiterate my point:
There is no way you can picture anything, or describe shape, location, or size, without an origin (your eyes), and without picturing it at a certain distance away from you. 


#36
Sep911, 01:52 AM

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