# Train experiment in special relativity: a problem?

Dear All,
In the http://galileoandeinstein.physics.virginia.edu/lectures/srelwhat.html" [Broken], there is one assumption I don't understand.

Why do we assume that the distance w between the mirrors is constant for both the observer in the train and the observer outside the train? We could also let this distance shrink for the observer outside the train as soon as the train starts moving, and keep the time between two reflections constant.

What am I missing?

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bcrowell
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This is an interesting question. The light-clock thought experiment has some advantages and disadvantages as a way of introducing special relativity. An advantage is that it's pretty simple. The disadvantages, IMO, are: (1) it makes it sound as though light has a special role in relativity and c should be thought of as the speed of light, and (2) it inherently requires at least three dimensions (2 space+1 time), whereas it would be cleaner to do a derivation in only two dimensions (1 space+1 time). (Here is an approach that avoids both these problems: http://www.lightandmatter.com/html_books/0sn/ch07/ch07.html [Broken] ) I think this issue is basically coming from disadvantage #2.

If you've already established the effect of motion in the x direction on x and t, then it's fairly straightforward to show that there can't be any Lorentz contraction in the y and z directions. The reason is that both area in the x-t plane and volume in an x-y-t space have to stay the same under a Lorentz transformation (proof for x-t: http://www.lightandmatter.com/html_books/0sn/ch07/ch07.html#Section7.4 [Broken] ). This requires that there be no stretching or contraction of y.

I think it's also fairly straightforward to show that you can't just get rid of time dilation by assuming transverse Lorentz contraction. If there was no time dilation, then it wouldn't be possible for observer A to see observer B's rulers as contracted and at the same time for B to see A's rulers as contracted. They would be able to synchronize clocks and use radar to agree on lengths. Either A or B would be wrong, and this would violate the principle that all frames of reference are equally valid.

But to make the light-clock a really logically satisfying intro to SR, you really would have to prove that there is no transverse length effect *before* you do anything else, and I don't see any easy way to do that.

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In the http://galileoandeinstein.physics.virginia.edu/lectures/srelwhat.html" [Broken], there is one assumption I don't understand.

Why do we assume that the distance w between the mirrors is constant for both the observer in the train and the observer outside the train? ... What am I missing?
Well, the scenario is designed for simplicity. The mirrors are defined to be "at rest" in the embankment frame, which means their separation cannot change over time ... it's constant. If their separation does not change in any one inertial frame, it cannot possibly change in any other inertial frame. The embankment POV sees their separation constant, with mirrors stationary. Other frames (that move relatively, eg the train) see their separation constant, while moving. The separation between mirrors can only change if the mirrors begin to move wrt one another, which requires either one or both mirrors "to move" along the embankment. However by scenario design, they do not.

We could also let this distance shrink for the observer outside the train as soon as the train starts moving, and keep the time between two reflections constant.
The mirrors are at rest in the embankment, and so their separation can never change per any inertial POV including the embankment. Here though, you are discussing what happens as the train begins to move, and thus accelerate. Einstein's scenario included no acceleration, as it related to the description of his all-inertial theory, special relativity. However if the train did accelerate from rest, there would still be no change in mirror separation per any inertial POV including the embankment POV. However train observers are non-inertial when undergoing proper acceleration, and so per them the mirror separation would indeed contract more and more with increased acceleration ... not because either mirror accelerated in its own right, but rather because of a relativistic effect caused by a change in one's own state of motion. Once the train steadies out at some specific inertial speed, the mirror separation then remains constant at whatever it was when the acceleration ceased.

GrayGhost

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bcrowell
Staff Emeritus
Gold Member
The mirrors are defined to be "at rest" in the embankment frame, which means their separation cannot change over time ... it's constant.
If I'm understanding the OP correctly, he's not talking about the possibility that the separation changes over time, he's talking about the possibility that observers in different frames disagree on it.

If I'm understanding the OP correctly, he's not talking about the possibility that the separation changes over time, he's talking about the possibility that observers in different frames disagree on it.
Ahh, yes. Duely noted, thanx.

So if I understand it correctly, this problem is resolved as follows:

- If we do not allow time dilation to account for the compensation in the train/clock experiment to keep the light speed constant in both frames, the distance w between the mirrors of the moving frame must shrink for the observer at rest, and also vice versa: the distance w of the mirror of the frame at rest must also shrink for the moving observer, since both observers see the same happening to each others light ray bouncing between the mirrors: /\/\/\/\/ etc.
- We place a ruler in the moving frame and in the frame at rest.
- We attach a sharp object on the zero meter and one meter mark of the moving ruler.
- We hold the rulers in such way that the zero meter marks of both rulers coincide, that is: both rulers scratch each other at their zero meter marks.
- According to the observer in the frame at rest, the moving ruler must shrink if no time dilation takes place, thus meaning that the ruler in the frame at rest is scratched at less than a meter.
- According to the observer in the moving frame, the ruler in the frame at rest must shrink relative to his/her own ruler, since there is no time dilation.
- Therefore the ruler in the frame at rest is scratched at more than a meter.
- Contradiction, measurement of the location of the scratch must give one well-defined value.
- Conclusion: length contraction perpendicular to the direction of motion does not take place.