Modelling Einstein's Thought Experiments (Trains + Mirrors)

[DunnyBasher]

OK, I have an assessment task to create a 3D model of SOME of Einstein's thought experiments involving trains and mirrors, and present it to the class. Therefore a minimum of 2 thought experiments. The task is unique (ie each student is required to model a different part of the course), and I'm fairly confident with my understanding of SR, so I won't have many problems explaining it.

My current plan is to model the following two thought experiments:

1. Imagine a train consisting of two mirrors with a light pulse reflecting between them. The path of the light viewed by an observer on the train would be vertical. However, if the same clock is viewed by an observer on the platforms the train travels by at half the speed of light then the path of the light pulse would by diagonal according to Pythagoras' rule. The two observers then see the pulse of light traveling different distances in one second.
   
   Fairly easy to model, HOWEVER I need to model it in the best possible method. Two current ideas..:
   • Simple box on wheels, 1 side open, hole in the top with a string attached to a ball. Box is projected, string is released, ball falls in a parabolic curve to the observer out of the box. Then go on to explain that the ball to an observer in the box sees it fall vertically. Then explain that as light isn't affected by gravity it would travel in a diagonal line, therefore traveling a further distance to an observer resulting in different 'clocks'.
   • Volunteer sits on a swivel chair, which is spun and a ball is tossed vertically, which they see fall vertically. Repeated but with the volunteer projected at a constant velocity, to which they would see the ball fall in a parabolic curve. As with the other alternative, explain how light would travel diagonally and what the effect of this is. HOWEVER, this isn't really a practical model, as my entire presentation is limited to 6 mins and for it to be a viable model I would have to use every person (although it is a very small class) as a volunteer.
2. Imagine that you are sitting in a train facing forwards. The train is moving at the speed of light. You hold up a mirror in front of you, at arm’s length. Will you be able to see your reflection in the mirror? Yes, the reflection will be seen because, according to the principle of relativity, it would not be possible for the person in the train to do anything to detect the constant motion with which he or she is travelling. However, a person watching this from the side of the track should see the light from your face traveling at twice its normal speed. But, since the speed of light is constant, distance and time become relative. this means that time passes differently for you on the train and for the person at the side of the track.
   
   This is the harder one to model, but I think it would be best done using two projectiles traveling horizontally, one traveling at 2x the speed of the other, and explain that as the speed of light is constant to all observers, the observer looking in the mirror would see the light travel their relative time to reach the mirror, while the observer on the platform would observe the same speed but over a different time.

If anyone would be able to offer feedback, suggestions, criticism and whatnot, or even another of Einstein's thought experiments involving trains and mirrors which would be easier to model, it would be greatly appreciated.

 

[JesseM]

For #1, doesn't it confuse the issue a little to get parabolic curves (due to gravitational acceleration) involved, when the thought-experiment only involves straight-line paths? I realize the curve is supposed to be an analogy for the path of the light beam in the frame where the apparatus is moving, which looks like the top section of a triangle, but is there any way for you to get the speed up and down to be close to constant? Maybe you could have a blindfolded person on a wheeled chair trying to draw a vertical line up and down on a chalkboard while someone slowly pushed the chair sideways? The ideal thing would be to use some sort of machine that lifts an object up and then moves it back down at constant speed, but I can't think of any easily-accessible machines that do that.

 

[robphy]

On a moving cart, rather than tossing a ball vertically, try rolling a ball horizontally from left-to-right (perpendicular to the direction of motion). You can capture the track of the motion with a camera above the scene.

 

[DunnyBasher]

Maybe you could have a blindfolded person on a wheeled chair trying to draw a vertical line up and down on a chalkboard while someone slowly pushed the chair sideways?

That's a great idea...I'll discuss with my teacher whether just that model would suffice or if I should use the ball dropping in a box as well to model the experiment, not just the principle.

On a moving cart, rather than tossing a ball vertically, try rolling a ball horizontally from left-to-right (perpendicular to the direction of motion). You can capture the track of the motion with a camera above the scene.

I've considered that previously, however the presentation is limited to 6mins and MUST be performed for the class (ie. I can't show them a video I've made previously) so it wouldn't be particularly practical.

Anyone got suggestions for the 2nd thought experiment? Seems to be simple to explain, but more difficult to model...while trying to keep the model as accurate as possible (ie. trying to model an object traveling at the same speed for all observers, but covering a greater distance for an observer.

 

[JesseM]

Imagine that you are sitting in a train facing forwards. The train is moving at the speed of light. You hold up a mirror in front of you, at arm’s length. Will you be able to see your reflection in the mirror? Yes, the reflection will be seen because, according to the principle of relativity, it would not be possible for the person in the train to do anything to detect the constant motion with which he or she is travelling. However, a person watching this from the side of the track should see the light from your face traveling at twice its normal speed. But, since the speed of light is constant, distance and time become relative. this means that time passes differently for you on the train and for the person at the side of the track.

This is actually a badly-stated thought-experiment, since relativity doesn't actually tell you what things would be like from the point of view of someone moving at the speed of light, the only valid inertial frames are ones moving at sublight speeds (and the 'principle of relativity' referenced in the thought-experiment only applies to these inertial reference frames). It would be fine if the train was said to be moving at 0.5c relative to the ground, or 0.8c, or 0.99999c, but doing experiments while moving at exactly the speed of light just isn't possible in SR. Perhaps you could ask your teacher for clarification?

 

[DunnyBasher]

doing experiments while moving at exactly the speed of light just isn't possible in SR. Perhaps you could ask your teacher for clarification?

I realize that the thought experiment is impossible in SR, but it's the thought experiment which Einstein 'performed', so I assume that we are not allowed to alter it...so I think I'll have to model Einstein's interpretation of the thought experiment. I'll try and clarify with my teacher what is expected of that model ASAP.

 

[JesseM]

I realize that the thought experiment is impossible in SR, but it's the thought experiment which Einstein 'performed'

Well, this was a thought-experiment Einstein came up with at age 16, long before he thought up relativity, and it seems more to have been about showing problems with existing ideas about light than coming to any positive conclusions as to the solution. This page has a discussion of what Einstein's thinking may have been about this thought-experiment, suggesting that he may have been using it to reject an "emission theory" of light in which the speed of a light wave depended on the speed of its source (this is different from the 'aether theory' which was popular before relativity, in which light was a vibration in a physical medium called the aether, and light waves would always travel at a constant speed relative to the rest frame of the aether, so that observers moving relative to the aether could see light moving at different speeds, but the speed of a given light wave would not in any way depend on the speed of the source of that wave).

 

[DunnyBasher]

OK, I'll remember that for my presentation..

Back to the point, would anyone be able to improve the 2nd model? Seems to be lacking significantly...

 

[JesseM]

Did you ask your teacher about it? It's hard to come up with a good model for a thought-experiment that just isn't correct according to current theories of physics. The description says:

Will you be able to see your reflection in the mirror? Yes,

Not according to relativity. You'd need an alternate theory to justify this, and who knows what this theory would say about clock rates and speeds in different frames.

the reflection will be seen because, according to the principle of relativity, it would not be possible for the person in the train to do anything to detect the constant motion with which he or she is travelling.

The principle of relativity says nothing of the sort, since it only applies to inertial frames moving at sublight speeds (think about it, if you have two photons moving alonside each other, this would imply that in each photon's 'rest frame' the other photon would move away at c, so each photon would predict the other photon would reach their destination first, a contradiction). I would ask your teacher if you could modify it so the person with the mirror was moving at some relativistic speed less than c, like 0.6c. In the case of sublight speeds, the fact that all observers measure the light beam to move at c can be understood with a combination of length contraction, time dilation, and the "relativity of simultaneity" (which says that if two clocks are synchronized in their own rest frame, they are out-of-sync in a frame where they're moving along the axis between them...if x is the distance between them in their own rest frame, and v is their speed in the frame where they're moving, then in the frame where they're moving they'll be out-of-sync by a factor of vx/c^2). You may not want to go into this much detail, but here's an example I came up with on another thread to show how it works out:

Say there's a ruler that's 50 light-seconds long in its own rest frame, moving at 0.6c in my frame. In this case the relativistic gamma-factor (which determines the amount of length contraction and time dilation) is 1.25, so in my frame its length is 50/1.25 = 40 light seconds long. At the front and back of the ruler are clocks which are synchronized in the ruler's rest frame; because of the relativity of simultaneity, this means that in my frame they are out-of-sync, with the front clock's time being behind the back clock's time by vx/c^2 = (0.6c)(50 light-seconds)/c^2 = 30 seconds.

Now, when the back end of the moving ruler is lined up with the 0-light-seconds mark of my own ruler (with my own ruler at rest relative to me), I set up a light flash at that position. Let's say at this moment the clock at the back of the moving ruler reads a time of 0 seconds, and since the clock at the front is always behind it by 30 seconds in my frame, then in my frame the clock at the front must read -30 seconds at that moment. 100 seconds later in my frame, the back end will have moved (100 seconds)*(0.6c) = 60 light-seconds along my ruler, and since the ruler is 40 light-seconds long in my frame, this means the front end will be lined up with the 100-light-seconds mark on my ruler. Since 100 seconds have passed, if the light beam is moving at c in my frame it must have moved 100 light-seconds in that time, so it will also be at the 100-light-seconds mark on my ruler, just having caught up with the front end of the moving ruler.

Since 100 seconds passed in my frame, this means 100/1.25 = 80 seconds have passed on the clocks at the front and back of the moving ruler. Since the clock at the back read 0 seconds when the flash was set off, it now reads 80 seconds; and since the clock at the front read -30 seconds, it now reads 50 seconds. And remember, the ruler was 50 light-seconds long in its own rest frame! So in its frame, where the clock at the front is synchronized with the clock at the back, the light flash was set off at the back when the clock there read 0 seconds, and the light beam passed the clock at the front when its time read 50 seconds, so since the ruler is 50-light-seconds long, the beam must have been moving at 50 light-seconds/50 seconds = c as well! So you can see that everything works out--if I measure distances and times with rulers and clocks at rest in my frame, I conclude the light beam moved at 1 c, and if a moving observer measures distance and times with rulers and clocks at rest in his frame, he also concludes the same light beam moved at 1 c.

If you really wanted to go all-out you could "demonstrate" something like this by having students hold up a shrunken "moving ruler" next to a normal-length "resting ruler" with little signs showing the times on clocks at either end of the moving ruler, first at 0 seconds in the resting ruler's frame and then again at 100 seconds in the resting ruler's frame, showing how far the moving ruler had moved along the resting ruler at that time, and how far some little ball representing a light beam had moved in that time too.

And you could also simplify this scenario by avoiding worrying about the relativity of simultaneity, and instead of putting a clock at the front end of the moving ruler, just put a mirror there so when the light catches up to the front it immediately turns around and moves back towards the back end. In this case the light will catch up with the mirror at the front end of the ruler at position 100 light-seconds and time 100 seconds in the resting ruler's frame, after which the light will bounce back, and will reach the back end of the moving ruler when it's at position 75 light-seconds and time 125 seconds in the resting ruler's frame, at which time the clock at the back end will read 100 seconds since it started at 0 seconds and was slowed down by a factor of 1.25 (and naturally if the back end of the ruler is moving at 0.6c, it will have moved 75 light-seconds in 125 seconds since 125*0.6 = 75). And remember that in the moving ruler's own frame its length is 50 light-seconds, so the distance for the light to go from one end to the other and back would be 100 light-seconds in its frame, so if the clock at the back end measures a time of 100 seconds this means the light is measured to move at c in its frame (naturally it also moved at c in the resting ruler's frame too, since it took 100 seconds to reach a position of 100 light-seconds and then was reflected back, taking another 25 seconds to move a distance of 25 light-seconds back to a position of 100 - 25 = 75 light-seconds).