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Locally Lorentzby JDoolin
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#37
Dec2210, 01:20 PM

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You're making kind of an odd claim, though, regarding this reference frame. You say that the number of wavelengths is an invariant as long as we always use the same line of simultaneity, which is of course true, but it's rather like saying, "the radio will always be at the same volume, so long as we never turn the volume knob." Another issue I see with your "given line of simultaneity" is that it can only apply when you have the emitter and receiver moving on parallel worldlines. If the emitter and receiver were moving toward or away from each other, there is no builtin line of simultaneity which asserts itself as "THE ONE" 


#38
Dec2210, 06:56 PM

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Of course, the "distance" I've just defined is observerdependent; if I go through the same process in the receiver's frame, I will come up with different answers if the receiver is moving relative to the emitter. See next comment. 


#39
Dec2210, 07:10 PM

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Which of course leads to... 


#40
Dec2410, 03:01 PM

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From your description of what you think I mean, I can tell that this ascii picture was woefully insufficient to get my idea across, so I took the time to attempt to remedy the situation with the attached diagram. As you can see, in the diagram on the left, the emitter is chasing the receiver, and viceversa on the right. I'm not saying something about their motion relative to each other but about their motion relative to a given observer. I don't want to go through and respond to your last response pointbypoint, because if you've missed this idea, you won't make sense out of anything else I've said. However, I do want to clarify a couple of ideas. One, the difference between a distance between events, and a distance between objects. If you draw a line of simultaneity in some observer's reference frame, intersecting two object's worldlines, this represents the distance between objects. This is an observer dependent quantity. If those objects happen to have parallel worldlines, this distance is subject to the lorentz contraction. If you have the spacetime coordinates of two events, (regardless of whether their separation is null, timelike, or spacelike) you can calculate the distance between events by using the distance formula, (sqrt(x^2+y^2+z^2). This is also an observer dependent quantity.The distance between two events, is to be distinguished, also, from the spacetime INTERVAL between the two events, which of course is an invariant sqrt(t^2x^2y^2z^2). There is one other aspect of your argument where I think you should take a bit more care, because at one point, I think you have suggested that the "distance between events" is meaningless unless those events have a spacelike separation. Yet, you were able to fairly well describe the distance between timelike separated events from the frame of the car, and from the frame of the ground, and the frame of the snowflake, so I know you cannot really believe these ideas to be meaningless. 


#41
Dec2410, 07:11 PM

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I think you're also somewhat misstating the role of symmetries in a physical situation. For the special case we've been discussing, where the emitter and receiver are at mutual rest, there is a special state of motion in which the problem looks simplest: the state of motion which corresponds to the mutual rest frame of emitter and receiver. This is what I meant by saying that this particular problem (*not* the general problem, which allows emitter and receiver to be moving relative to one another) has a "symmetry"it looks simpler if you pick a particular reference frame in which to describe it. In that frame, all the different kinds of distances we've been discussing are equal: the "distance between observers" is the *same* as the "distance between emission and absorption events". This distance is also the "proper distance" between the emitter and the receiver, and "proper distance" has the same kind of special status in relativity as "proper time" doesit qualifies as an "invariant", because it can be defined in a coordinateindependent way, just as proper time can. (It's true that it can't be the general invariant that appears in the light intensity law I referred to above, because it only works for this special case, as I noted in a previous post. The general invariant, I think, is a generalization of this one, but I haven't completely worked it through yet.) 


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