- #1
ScottTracy
- 6
- 0
Hi all,
I'm sure that things similar to this have been covered before. Also, I'm sure that a Minkowski diagram would probably clear things up no end but, anyway, please bear with me.
Let's assume that we have our familiar railway carriage packed with observers who are satisfied that they have synchronised their watches and that they are all separated by the same distance as measured by a rigid metal rod. The origin of their co-ordinate system is the centre of the train which can be called O'. Can we say that the ends of the train are at x' = -10 and x = 10? Finally, we place two opposing flashlights at O' in line with the x' axis.
On the platform we have a load of observers with watches and they spread out from the origin O making up the platform frame of reference.
O' is approaching O at some velocity v which has only an x component.
The usual thought experiment is that the flashlights are fired as O' passes O. Let's pretend that this is achieved by some electrical contact on the track that closes a circuit or something. The observers on the train get together and conclude that the flashes reached the ends of the train at the same time. The observers on the platform get together and conclude that the flash of light going in the -x direction reached the back of the train before the forward (+ve x) traveling light pulse reached the front.
This shows nicely that two events (the light pulses reaching the ends of the train) may be simultaneous when viewed from one reference frame but not from another.
Now I think it can be shown that there is a way that both observers could agree that two pulses of light reach the two ends of the carriage at the same time. If we work backwards from the assumption, then I think that the platform observers would have to see the forward-facing flashlight fire before the backwards facing one. This would compensate for the fact that the front of the carriage is moving away from the wavefront, as the platform observers see things.
In practice, maybe this could be achieved by having two separate contacts on the train and two on the track, (i.e. one each on the train and the track for each flashlight). If we placed the contact for the forward flashlight more to the left (-ve x direction) on the track than the other, then this would trigger the two lights at separate times as seen from the platform.
However, the train observers see the light reach the ends of the carriage at the same time and they can easily confirm that the flashlights are in the centre of the train. Given that they measure the speed of light to be constant, wouldn't the train passenger at O' (in the centre of the carriage) confirm to the others that the flashlights fired at the same time ("simultaneously")?
So far, I hope that the example is, if a little strained, still acceptable. The disagreement about the simultaneity of the forward flash and the backward flash are as predicted by the relativity of simultaneity.
What I'm wondering is: If the train passenger at O' sees the flashes simultaneously then they must see themselves passing the two contacts on the track simultaneously (for that is how the lights are triggered). These contacts have been separated by the platform observers by some distance Δx in order to trigger the lights at separate times. Presumably with (Δt = v * Δx). How is it that the observers see Δx' to be zero?
My intuition (I know, a dangerous thing) tells me that Lorentz-Fitzgerald length contraction is probably not the compensating factor. Instead, I have probably just made a logical error.
Anyway, if all of the above is correct, could someone demonstrate the calculation whereby the track contact separation distance Δx is calculated as a function of velocity v so that all observers agree that the light pulses reached the ends of the train simultaneously?
Thanks in advance for the patient explanation and the pretty diagrams
I'm sure that things similar to this have been covered before. Also, I'm sure that a Minkowski diagram would probably clear things up no end but, anyway, please bear with me.
Let's assume that we have our familiar railway carriage packed with observers who are satisfied that they have synchronised their watches and that they are all separated by the same distance as measured by a rigid metal rod. The origin of their co-ordinate system is the centre of the train which can be called O'. Can we say that the ends of the train are at x' = -10 and x = 10? Finally, we place two opposing flashlights at O' in line with the x' axis.
On the platform we have a load of observers with watches and they spread out from the origin O making up the platform frame of reference.
O' is approaching O at some velocity v which has only an x component.
The usual thought experiment is that the flashlights are fired as O' passes O. Let's pretend that this is achieved by some electrical contact on the track that closes a circuit or something. The observers on the train get together and conclude that the flashes reached the ends of the train at the same time. The observers on the platform get together and conclude that the flash of light going in the -x direction reached the back of the train before the forward (+ve x) traveling light pulse reached the front.
This shows nicely that two events (the light pulses reaching the ends of the train) may be simultaneous when viewed from one reference frame but not from another.
Now I think it can be shown that there is a way that both observers could agree that two pulses of light reach the two ends of the carriage at the same time. If we work backwards from the assumption, then I think that the platform observers would have to see the forward-facing flashlight fire before the backwards facing one. This would compensate for the fact that the front of the carriage is moving away from the wavefront, as the platform observers see things.
In practice, maybe this could be achieved by having two separate contacts on the train and two on the track, (i.e. one each on the train and the track for each flashlight). If we placed the contact for the forward flashlight more to the left (-ve x direction) on the track than the other, then this would trigger the two lights at separate times as seen from the platform.
However, the train observers see the light reach the ends of the carriage at the same time and they can easily confirm that the flashlights are in the centre of the train. Given that they measure the speed of light to be constant, wouldn't the train passenger at O' (in the centre of the carriage) confirm to the others that the flashlights fired at the same time ("simultaneously")?
So far, I hope that the example is, if a little strained, still acceptable. The disagreement about the simultaneity of the forward flash and the backward flash are as predicted by the relativity of simultaneity.
What I'm wondering is: If the train passenger at O' sees the flashes simultaneously then they must see themselves passing the two contacts on the track simultaneously (for that is how the lights are triggered). These contacts have been separated by the platform observers by some distance Δx in order to trigger the lights at separate times. Presumably with (Δt = v * Δx). How is it that the observers see Δx' to be zero?
My intuition (I know, a dangerous thing) tells me that Lorentz-Fitzgerald length contraction is probably not the compensating factor. Instead, I have probably just made a logical error.
Anyway, if all of the above is correct, could someone demonstrate the calculation whereby the track contact separation distance Δx is calculated as a function of velocity v so that all observers agree that the light pulses reached the ends of the train simultaneously?
Thanks in advance for the patient explanation and the pretty diagrams
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