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No definite viewpoint for the accelerating traveler? 
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#19
Feb2113, 08:26 PM

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#20
Feb2113, 09:03 PM

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PF Gold
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Tell me, why would I want to use such a noncoordinate system reflecting a nonobservables in a way the does lead to nonsense that has no observable or logical basis? Instead, I can use any valid coordinate system to compute any observable, and conceptually model reality in a consistent way. [edit: Why absurd? I know that NYC blows up once; every possible observation I can make indicates it blows up once; I have a plethora of valid coordinates consistent with SR that model it blowing up once. Why should I choose a method that constructs mathematically invalid coordinates and models it as blowing up at two different times of my history? Really??] 


#21
Feb2213, 08:33 AM

P: 999

That said, I can certainly understand how you would want those properties to hold for useful coordinate systems. And I can understand that you might want to adopt terminology requiring this to be the case. Topology is not my strong suit, but what I think you are saying is that you want a "mathematically valid" coordinate system to be one that embodies a homeomorphism between a manifold and cartesian nspace. Any coordinate system which assigns multiple coordinates to the same point cannot (of course) be a homeomorphism because it fails to be a bijection. Suppose that I am driving east when the NYC blows up. Let's say that it blows up at 2:30 pm EST. I glance at my clock and see that it reads 1:30 pm CST. But I am not paying careful attention and don't know whether I've crossed the time zone line yet. I can label the NYC blow up at both 1:30 or 2:30 using "my personal time zone" coordinates. This does not entail that NYC blew up twice. 


#22
Feb2213, 08:42 AM

Mentor
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#23
Feb2213, 08:49 AM

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#24
Feb2213, 09:00 AM

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As for definitions, the following is one common definition of coordinates: "In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element on a manifold such as Euclidean space" To a mathematician, the pole in polar coordinates is not covered by the coordinate system. In fact, this feature, in the case of a sphere, is the quintessential example used to show that there are manifolds such that no single coordinate system (patch) can cover the whole object. 


#25
Feb2213, 09:10 AM

Mentor
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Q: If we call the tail a leg, how many legs does a horse have? A: Four. Calling a tail a leg doesn't make it a leg. 


#26
Feb2213, 09:12 AM

C. Spirit
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#27
Feb2213, 09:31 AM

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See ch 2. (especially around p. 3437): http://arxiv.org/abs/grqc/9712019 


#28
Feb2213, 09:36 AM

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On the same page that you reference, one sees the following: "Schemes for locating points in a given space by means of numerical quantities specified with respect to some frame of reference. These quantities are the coordinates of a point. To each set of coordinates there corresponds just one point in any coordinate system, but there are useful coordinate systems in which to a given point there may correspond more than one set of coordinates" Emphasis mine. 


#29
Feb2213, 09:37 AM

C. Spirit
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#30
Feb2213, 09:41 AM

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#31
Feb2213, 09:41 AM

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#32
Feb2213, 10:05 AM

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#33
Feb2213, 10:10 AM

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A function need not be injective mate but it cannot map a value in the domain to two values in the range which is what you are doing by assigning two different events to the same point on the manifold for a single observer.



#34
Mar313, 04:01 PM

P: 24

I've been looking at a lot of the posts that Mentz recommended to me in another thread:
http://www.physicsforums.com/showpos...&postcount=287 That post showed what someone who is going around and around in a circle, at constant speed, would say is the current age of some inertial person who is located some distance away from the circle. That post was interesting to me, because it is very similar to an example that I saw in a NOVA program called "the fabric of the cosmos". There, Brian Greene gave an example of someone on a planet in an extremely distant galaxy, who is riding a bicycle around and around in a small circle, and who says that for each of his loops, the time here on earth is swinging back and forth over centuries! Brian Greene got that result by using the spatial threedimensional "simultaneous time slices" of the sequence of inertial frames that are momentarily comoving with the bicycle rider. Brian also has essentially the same example in his book that has the same title as the NOVA show. As far as I can recall, in both the TV show and in his book, Brian didn't seem to be presenting his method as "just one of several different possible answers" to the question of "What is the current age of the inertial person, according to the accelerating person?". My impression was that he seemed to present it as THE answer. On my CADO forum search, I also found this link, https://sites.google.com/site/cadoeq...eferenceframe that seems to give a pretty good summary of all the CADO stuff. The CADO equation explained in there gives the same result as Brian Greene's "momentarily comoving inertial frame" method, but it's quicker and easier. 


#35
Mar313, 05:30 PM

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#36
Mar313, 06:35 PM

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If the question is "how can you do physics in a noninertial frame" then it is not even an answer let alone THE answer. If the question is "what is the naive noninertial simultaneity convention which most novices tend to adopt and which is useless for any actual physics" then I would agree that it is probably THE answer. 


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