Quote:
Originally Posted by modest
My confusion is right now all in my head and I'd have to work at putting it down in a post... if you think this is the wrong thread for discussing that metric then let me know.
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No, it is a fine place to discuss your difficulties. Please go and read the opening post to this thread. You will find almost exactly your diagram in that opening post (with some subtle but very important differences).
Quote:
Originally Posted by modest
But, in another thread you're saying that the moving train frame can decide to use the fixed frame of the platform in defining simultaneity. By your own objection, this would require the person in the train to know they are moving and thus not be a valid approach.
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Then perhaps we are confusing the meaning of the term “know” here. English is not really a very exact means of expressing ideas. If you substitute the word “presume” for the word “know” perhaps the confusion would kind of dissipate a bit. What we are really talking about here is the ability to perform a valid change in our frame of reference.
It is Einstein's contention that there exists no preferred frame of reference (insofar as the laws of physics are concerned). That is an assumption and cannot be proved (that is why he “postulated” that the speed of light is a constant). It is nice in that we need not worry about how we are moving relative to “a preferred frame of reference” however, we can still always do all our physics in “a preferred frame of reference” if we wish (that is, in fact, the central issue of relativity itself).
Quote:
Originally Posted by modest
Defining tau loosely as what clocks measure and x as what a ruler measures and the metric  where c=1 and y & z are omitted we might plot something like:
S is not moving in x while S' is. The red lines are light emitted from S and detected at S'. Taking things slowly, I'll just ask one question: what is the change in tau between the detection events for S'? Is there enough information (as I've said nothing about time) to answer this question? |
Your diagram is fine except for one very important issue: all three entities being referred to are momentum quantized in the tau direction (even the photon which just happens to have a tau momentum of zero) thus their position in the tau direction has an uncertainty of infinity. It is thus an error to presume that their path (and thus their interactions) can be explicitly represented in your diagram you have displayed. There is nothing wrong with the coordinate system you have laid out; the problem is that you have to very very careful in deducing the positions where interactions can occur.
Time which I define to be an interaction parameter (things can interact if they are in the same place at the same time) is not a measurable thing but rather a hypothesized parameter used to divide the future from the past (something which can not be defined except at the point of interaction). Since both interacting elements positions in the tau direction have an uncertainty of infinity, so does that point of interaction.
The only factor which we have to offset the difficulty of establishing when and where an interaction can occur is the fact of my fundamental equation which is a wave equation with a velocity of v
?: i.e., the distance between interactions is explicitly v
?t where t is the time change since the previous interactions. For convenience, if we ignoring the momentum quantization and examining only the vectors which show the actual direction of the momentum we need to talk in terms of interactions themselves. We can start with the assumption that S and S' interacted at the origin of your coordinate system. This a good reason to show the vectors for the momentum of S and S' as originating there and establishes t=0 for both entities.
Now let us describe the phenomena you are attempting to show in your diagram. The next interaction of significance occurs after a delay of one unit of distance in the coordinate system (v
?t=1): after waiting one unit of time (or one unit of tau as measured on his clock: i.e., he defines time so that c=unity) S emits a photon towards S'. Then, after waiting another unit of time (essentially identical to the first period) S emits a second photon towards S'. The difficult issue is to establish exactly the position in your diagram where those two photons interact with S'. Those interaction points are very definitely not at the end of the red lines you show.
Interaction can only occur when the length of the path of both S' and the photons (plus their delay on S) are exactly the same. Since you have shown S' to be at the angle of 45 degrees, that point is easy to calculate (actually it is not difficult for any angle but this one I can do in my head). The point where the first photon interacts with S' (in the rest coordinate system) will be when x+1 (essentially the age of the photon) is equal to x divided by the cosine of 45 degrees which is
or
.
Which is, of course, the point at which S' is at the point x=2.4155 (remember, both the photon and S' are smeared out in the tau direction). Since S' is moving at an angle of 45 degrees with respect to S and their clocks agreed at the origin the clock on S' will read exactly 2.4155 at the moment of receipt of the photon (assuming S' also defines time such that c=unity).
The second photon will be intercepted at a distance where x+2 is equal to
or
.
Thus an observer on S' will actually see the period of the clock on S (which is being used to time the photons being emitted) as having a period of 2.4155 times the period of his clock. Now all you have to do is figure out the Doppler effect the S' observer attaches to that timing (essentially, the delay he attributes to the fact that S is moving away from him).
You really need to follow the second part of my opening post on this thread. The details of this kind of calculations are worked out in detail.
I hope I have cleared up the significant issues here. Looking forward to hearing from you again.
Have fun -- Dick
