Author: A A Andrews (---.access.as9105.com)
Date: 04-21-05 13:11
Dear Mr Hallum,
I hope the following is of some assistance to you in understanding the problem of simultaneity (or lack of it) in Relativity.
Let us consider the question of the Einstein Express and the two Observers - one on the platform and the other on the train.
To begin with I must state that the following analysis is based upon:
1 - The Speed of Light in free space has the same value in all inertial systems.
2 - Time dilates for an object in motion.
3 - Lengths contract for an object in motion along the line of the direction of that motion.
These three items have been verified time and time again by experiment.
4 - Relativity is what is says it is - relative. This means that in Relative World, if you are standing on a platform, it is the train that moves but if you are sitting in the train, it is the platform that moves.
The next may seem to be a bit long-winded and pedantic but it is essential that all the conditions and circumstances of the Thought-Experiment are defined at the beginning - to bring in another item or aspect halfway through would make a nonsense of any conclusions, so, for this Thought Experiment we have:
a - An Observer (John) standing at the absolute centre of a platform. He knows he is at the centre because he has measured the length of the platform. For the purpose of numerical calculations, let the length be 2 lightseconds. Thus, John is 1 lightsecond from either end.
b - At each end of the platform stands a light which will emit a very short pulse of light along the platform when its' switch is made. This pulse is sufficiently short to 'freeze' any motion, regardless of the magnitude of the motion.
c - When the Driver of the train is exactly level with the far end of the platform, the switch for that light will be made and similarly, at the other end, when the Guard is level with the end of the platform there, that switch will be made.
d - The switches and lights are totally independent of each other and can only be operated by the ends of the train being exactly level with the ends of the platform.
e - John knows that his fiancee Jill will be aboard the train and he knows that she will have measured the length of the train and will be sitting at the exact centre.
f - Finally, John and Jill both have identical cameras which automatically take a photo every time it is struck by a pulse of light from either or both platforms lights, whether those pulses be direct from the lamps or reflected from John or Jill. These cameras also have built-in clocks that imprint the time of the exposure on to the photo.
Start of the Analysis.
John's point of view.
The Einstein Express comes through the station and there is an instant at which John and Jill are exactly opposite each other. At that instant, she reaches out and sets the timer on John's camera to zero and he reaches in and does likewise to her camera. Thus there is no doubt whatsoever regarding the relative positions of the two AT THAT INSTANT and nor is there any doubt that the clocks in their cameras where absolutely synchronised to ZERO.
Now, at the instant John and Jill were opposite each other, the Driver and the Guard were level with their respective ends of the platform and so the switches were made and pulses of light were emitted.Since John was 1 lightsecond distant from each light, he would not be struck by them until 1 second later and the photo from his camera would show him being caught by both lights and a time of 1 second recorded. But of course, during the time taken for the light to reach John, Jill would have moved further up along the platform - to nearer the Driver's light which would illuminate her as it passed her by on the way to John's camera. A little after that, light from the Guards end of the platform would catch Jill up, illuminate her and there would be some light reflected back into John's camera, again taking some time to travel back from where it struck Jill.
So, to summarize, John would have one photo showing himself illuminated by both lights and that photo would also show Jill lit by the Driver's light at a time of 1 secondand one photo, showing Jill lit by the Guard's light at some different and later time.
Now, and I think this aspect is the aspect that causes more confusion than anything else, one must remember that moving objects contract in the direction of motion. John measured the platform as being 2 lightseconds long and, as the train fitted it exactly, he measured that at 2 lightseconds long also but it was moving therefore that was its' contracted length so when we consider things from Jill's point of view, we must remember that she considers herself to be stationary and so she measured the train as being MORE than 2 lightseconds long and is therefore MORE than 1 lightsecond from either the Driver or the Guard. When she measures the platform, because it is , as far as she is concerned, moving, she will measure it at LESS than 2 lightseconds. This means that for her, the Driver will be at his end of the platform and set of the light BEFORE John is opposite her and they synchronize clocks and, of course, John will be well past her before the Guard is level with his end of the platform and sets off his light.
So the first photo from Jill's camera will show her being struck by light from the Driver's end and then, a little later, the second photo will show John being caught by both lamps and herself by light from the Guard's lamp.
The final aspect to consider is that of the recorded times. At first sight they appear to be in conflict because neither John nor Jill can agree as to exactly when events happened. This is simply due to time dilation due to velocity.
If we take the relative velocity to be 0.4c, then, wrt John, he will say he was struck by both lights 1 second after clock synchronization whereas Jill will say it was 1.091 seconds. To make them agree, all one has to do is to multiply or divide by 'gamma' i.e sq. rt (1-k*k) where k = v/c.
To summarize, no problem exists. John sees the lights simutaneously,he is stationary with respect to them, Jill does not as they are moving with respect to her. She sees first the Driver's light and then the Guard's. Both parties are equally right. The difference in their views is due to there being relative velocity between them and the differences can be reconciled by simple algebra and determining the Lorentz Invariant Interval.
This has been a loong-winded explanation of events that are very easy to show on a Space-Time diagram (do not bother with those silly little Puffing Billy drawings that decorate articles purporting to show that Einstein was wrong - just how big-headed can you get?) Were I able so to do I would send you one but my e-mail skills are not that good.
|
|
