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Coordinatefree relativity 
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#55
Sep1211, 12:29 PM

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#56
Sep1311, 10:42 AM

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To say "all measurements are invariant under arbitrary coordinate transforms" is misleading because it does not specify whether you mean "measurements in general" or "any given measurement." It strongly suggests you mean "measurements in general" which would make the statement false. I only acknowledged the statement's truth based on one possible interpretation. 


#57
Sep1311, 10:47 AM

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Also, the unit length cannot be described at a single point. In a unit displacement vector, there is a continuous mapping of some onedimensional space from 0 to 1. 


#58
Sep1311, 12:07 PM

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All geometric quantities are invariant under coordinate transformations. In Euclidean space, geometric quantities include angles and distances. An angle is always measured between two lines at the point they intersect. A distance is always measured between two points along the line that connects them. In Minkowski space, geometric quantities include angles, distances, and relative velocities. Relative velocity is really just the "angle" between two worldlines. I've used the term "relative velocity", but you should note that ALL geometric quantities are already "relative". An angle is always an angle between two lines. One cannot say "The angle of line AB is 30 degrees", that makes no sense. Likewise, a distance is always a distance between two points. 


#59
Sep1311, 03:45 PM

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In any case, as an observer performing an experiment, you may not have the option of measuring the system in a comoving frame. For instance, if you wish to measure the temperature of air passing by at 100,000 miles per hour, you can't simply place your thermometer in and hope to get the result. In all likelihood, your thermometer will disintegrate. If you want to measure the shape of a body passing through our solar system at 90% of the speed of light, you don't have the option to run and catch up and place the protractor on the surface. It's not a matter of naivete. It's a matter of what is convenient and possible. I just measured my own computer screen was 13 and 1/8 inches across. You don't know whether I measured from the left to the right, or whether I was using a yardstick or a ruler. But you do know something about how a length is measured, and you know that I must have placed an object near the screen, most likely that has a zeropoint on it. And I placed that zeropoint somewhere in order to measure the screen. However, even if I used an unnumbered ruler, I still had to count from one end to another of the screen. I used one end or the other as the origin. Or maybe I played a trick on you and counted both directions from the center then added. But that just makes the origin at the center. or hey, maybe I played a really crazy trick on you, counting off random little 1/16 inch segments until they were all marked. So now I've converted this vector quantity into a scalar quantity. Have I now succeeded in describing a distance without having an origin? I don't think so. Because a distance is not made up of discontinuous chunks of ruler. It's made of consecutive chunks of ruler and the continuous space inbetween the atoms. Either way, a unit length, or a unit displacement vector requires space in one dimension to define. 


#60
Sep1311, 06:06 PM

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#61
Sep1311, 09:32 PM

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I am not hiding anything. You are the one who claimed that the mere fact that a measurement was performed explicitly determines an origin. I thus provided you the information that you claimed was required. If you wish to revise your claim, then I will be glad to provide as much detail as you claim is required. Note, however, that there is more to a coordinate system than just an origin, so this is a much weaker claim than the claim that any measurement defines a unique coordinate system. However, since even this very weak claim is false I think it is instructive to pursue it. 


#62
Sep1411, 08:15 AM

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This is conceptually similar to Lorentz's aether. You assert that the coordinate system exists and is necessary even though it has no effect on any physical experiment and any choice is consistent with experimental results. 


#63
Sep1511, 06:52 AM

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I say you have hidden information from me, but you say if I revise my claim, then you will provide detail to me? If my claim is incorrect, why don't you provide detail to me now, and show me that the claim is wrong? But I don't really care about the hidden information, as long as a couple of assumptions hold. (1) the space is not appreciably warped by gravitation where you're taking this measurement, and (2) the origin is stationary with respect to the thing you're measuring However, if you are measuring distance between objects and time between events in Minkowski spacetime, distances DO depend on the origin, because the origin has an intrinsic velocity. 


#64
Sep1511, 11:56 AM

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However, I can describe in detail the measurement and then you can feel free to tell me what information in addition to the mere fact of the measurement is necessary to specify the origin. A and B are two marks on a piece of paper lying on my desk. The marks are stationary wrt the paper but not located at any particularly special location or orientation wrt the paper, and the paper is resting on the top of the desk, but not particularly located in any special position or orientation wrt the desk. The desk is stationary wrt the house, etc. The acceleration due to gravity in my house can be taken to be approximately uniform at 9.8 m/sē. The measuring device is an unmarked standard rod of 8.5" length composed of a piece of standard "letter paper" constructed according to the usual specifications for letter paper. I carefully placed the two appropriate corners of the rod on the marks and noted that the length matched. Thus, the distance from A to B was measured to be 8.5". The rod was not moving wrt A or B during the measurement. 2, there is no origin so since it doesn't exist it is not stationary nor is it moving wrt the thing being measured 


#65
Sep1711, 09:33 AM

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But how would you modify this process if you needed to measure lengths of things that were not exactly 8.5" long? But if you get into general relativity; for instance, the Schwarzschild metric, even your choice of origin will affect measurement of distance, time, and spacetime intervals. 


#66
Sep1711, 02:55 PM

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