No definite viewpoint for the accelerating traveler?

PAllen

Multiple mapping is prohibited for coordinates by definition. Do you really think it makes sense to say that NYC exploded at 3pm on my watch and also at 4PM on my watch, even though I only see it explode once, and no observation I can make is consistent with it being simultaneous to two points on my world line? Instead, don't you think it is better to say that these completely unobservable lines of simultaneity that Einstein never used have limits on their applicability. They are useful (as are other simultaneity conventions) when they don't lead to absurdities.

(It would be different if there was some observation consistent with multiple simultaneity. But there is none.)

jbriggs444

Yes, I do think it makes sense to say that NYC exploded at two distinct coordinates. And you are incorrect about this giving rise to inconsistencies. It's just a coordinate. It's not an observable.

PAllen

There are accepted definitions of coordinates. They label a point on a manifold (physically, an event) once. Thus, such a system is not a coordinate system.

Tell me, why would I want to use such a non-coordinate system reflecting a non-observables 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??]

jbriggs444

Not all definitions of coordinates require that the things being labelled be manifolds. Not all "coordinate systems" on a manifold need be bijective, topology-preserving or even cartesian.

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.

I can use polar coordinates to refer to the north pole without worrying overmuch about nonsense ensuing. Mind you I agree that using such coordinates to label the north pole is one thing. Using them to model physics at the north pole would be more difficult.

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 n-space.

Any coordinate system which assigns multiple coordinates to the same point cannot (of course) be a homeomorphism because it fails to be a bijection.

The question you posed did not ask whether you should use a coordinate system that happens to have multiple coordinates for a single event. You asked whether I thought that it would make sense. I think that it does make sense. It's not an example that lends itself easily to a coordinate system that labels the same event twice, but one can contrive a labelling that does so.

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.

Nugatory

Do you reset your watch as you're driving? If you do, you're changing from one coordinate system to another. If you don't, there's only one label you can assign to the blowup event, and that's the time on your watch when the blowup happens.

jbriggs444

I have one hour of coordinate values to play with. If I choose to use them to double-label events, that's my business, not yours.

PAllen

Why do you think it makes sense to use a model that has a feature that is counter-factual to all observations, especially what there are a plethora of lmodels consistent with both observations and SR to choose from?

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.

Nugatory

As long as you don't call them "coordinates", sure. You can even call them "coordinates" if you want, but....

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.

WannabeNewton

You are being quite rude with your later posts to PAllen for no reason at all as he is correcting a very absurd statement you are making. A coordinate map is an n - tuple of coordinate functions on an open subset of the manifold representing space - time. You are claiming the coordinate map can take a point in this region and map it to two different coordinates (the events that characterize the space - time aspect of the manifold for a given family of observers). This is not even a problem of physics, you are going against the very definition of a function which is a basic set theoretical object.

DaleSpam

Those used in GR do. If you disagree, please provide a good reference supporting the use of non-bijective coordinates in GR.

See ch 2. (especially around p. 34-37): http://arxiv.org/abs/gr-qc/9712019

jbriggs444

Regarded as a mere labelling of events, it is not counter-factual.

As I read that definition, it requires that for every coordinate there is exactly one point. i.e. that the coordinate mapping be a function from coordinates to points, but not neccessarily either an injection (mapping to any given point by at most one coordinate) or a surjection (mapping to every point by at least one coordinate)

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.

WannabeNewton

Where exactly did you learn about coordinate charts and manifolds? You have all the wrong ideas about topological manifolds.

WannabeNewton

It is utter nonsense.

More nonsense. Let [itex]M[/itex] be a topological manifold. [itex]\forall p\in M[/itex] there exists a coordinate chart [itex](U,\varphi )[/itex] where [itex]U\subseteq M[/itex] is a neighborhood of [itex]p[/itex] and [itex]\varphi :U\rightarrow \mathbb{R}^{n}[/itex] is a homeomorphism.

DaleSpam

These are not valid coordinate charts in GR. Other disciplines may make use of such coordinates, but not GR.

jbriggs444

Rather than continue this distraction, I will bow out, acknkowledging that I am at the very least using standard terminology incorrectly for purposes of this context.

WannabeNewton

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.

Alain2.7183

I've been looking at a lot of the posts that Mentz recommended to me in another thread:

I also did a forum search on "CADO", and found some more recent posts, including this one:

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 three-dimensional "simultaneous time slices" of the sequence of inertial frames that are momentarily co-moving 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...eference-frame

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 co-moving inertial frame" method, but it's quicker and easier.