# Standard derivation?

1. Aug 11, 2007

### bernhard.rothenstein

standard derivation??

I read in a paper that the "light clock" approach is a standard derivation of the time dilation formula.
Please let me know when do we consider that a derivation is a standard one?

2. Aug 11, 2007

### robphy

- Widely recognized or employed as a model of authority or excellence: a standard reference work.
- Normal, familiar, or usual: the standard excuse.
- Commonly used or supplied: standard car equipment.

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3. Aug 11, 2007

### neutrino

A related question: In one book I came across a derivation of time-dilation from the LT by assuming the two events take place at the same point in frame S, so that while calculating delta-t' in S', delta-x is zero - Is this "general enough"? I mean, in the case of the light clock, there are three events - emission, reflection and reception, at least two of which happen at different points in both frames.

4. Aug 11, 2007

### robphy

To answer this, one really needs to see the details of the proof... probably including a clear spacetime diagram.

Without those details and given the possibility of implicit assumptions, I'm not sure how useful it is to merely count explicit events.

5. Aug 11, 2007

### neutrino

Okay, these were the best scans I could get. Please tell me if something isn't clear to you.

http://img260.imageshack.us/img260/6127/tdltdi6.jpg [Broken] [From LT, eq(4.8) is the inverse transformation for time coordinates]

http://img183.imageshack.us/img183/4783/tdstda9.jpg [Broken] [From the invariance of space-time intervals]

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6. Aug 11, 2007

### bernhard.rothenstein

light clock problem

Please let me know which are the two events that happen at different points in both frames,
I am interested in the problem of the "light clock" which involves in its stationary frame two "wrist watches" located on the two mirrors respectively whereas in the reference frame relative to which it moves three of them (located on the mirror where from the light signal starts, located where the upper mirror arrives and located where the first mirror arrive when the reflected light signal returns). So the learner does not know which are the clocks related by the time dilation effect.

7. Aug 11, 2007

### neutrino

Emission and reflection certainly happen at two different points in both frames. In the rest frame of the mirrors, emission and reception happen at the same point.

8. Aug 11, 2007

### robphy

Note that one pedagogical advantage of the "light clock" is that it can be used to derive time dilation (more simply) [and other effects] without explicit use of the (more complicated) Lorentz Transformation.

9. Aug 11, 2007

### robphy

As I think I might have commented in the past, at least for inertial motions, the light-clock (local mirror and distant mirror and reflecting light signals) is the clock-mechanism [for the local mirror]. It does not need additional wristwatches... but one can certainly include them (possibly at the expense of unnecessary complication).

10. Aug 11, 2007

### country boy

To derive time dilation you need two stationary clocks and one moving clock. In the rest frame the two time readings (emission and reception) happen at the same point, but there is a second clock, that has been synchronized with the first one, that is needed to compare with the moving clock at the later time. We say that the measurements are made at the same point in the moving frame, but at different points in the rest frame.

So even though it seems like emission and reception are at the same point in the rest frame, a supposedly simpler situation, there had to have been a previous measurement involving two separated points. When you derive time dilation using light clocks, you usually look at the light traveling in one moving clock and one stationary clock, but you are allowed to do this because the two separated stationary clocks have already been synchronized.

Einstein makes this this statement in his 1905 paper: "We assume that this definition of synchronism is free from contradictions ..." If he can, I can.

11. Aug 11, 2007

### bernhard.rothenstein

light clock and clock syncronization

Thanks. The important facts you mention are obscured by the light clock approach. In the rest frame of the light clock we have to consider the events E'(1)[0.0.0.0] and E'(2)[0,0.0.2d/c) separated by a proper time interval 2d/c measured as a difference between the readings of the same "wrist watch" located on the mirror where from the light signal starts.
In the reference frame relative to which the light clock moves we have to compare the events E(1)(0,0,0,0) and E(2)(x,0,0,t) associated with the fact that the light signal starts and returns after reflection respectively. In order to realise events E'(1) and E(1) we have to ensure that the clocks of the two frames located at the overlapped origins of the two frames read zero (initialization) whereas in order to realise events E(1) and E(2) we should synchronise the clocks of the stationary frame using a clock synchronization procedure say a la Einstein. The time interval (t-0) is a non-proper time interval. Please tell me if I am correct. Without telling all that to the learner (probably in a better English) the teaching via "light clocks" is lacunary.
In what concerns standard derivation I think that De gustibus non est disputandum

12. Aug 11, 2007

### pmb_phy

Hi Rob

The derivation I gave on my website at

http://www.geocities.com/physics_world/sr/light_clock.htm

made no use of a spacetime diagram and when I was learning SR I found this derivation the easiest to understand .... hence my usagage of it.

Best regards

Pete

13. Aug 11, 2007

### robphy

Yes, that is more or less the standard proof. I use it myself (for that "transverse light clock").
For the "longitudinal light clock", I think a spacetime diagram is more enlightening than a spatial diagram.

The point of mentioning the possible use of the spacetime diagram is to make the presentation of an attempted proof [e.g. the one being proposed] as unambiguous as possible. When not accompanied by some kind of unambiguous diagram, my eyes glaze over when I see attempted algebraic proofs with x's, t's, v's, and their primed counterparts.

Last edited: Aug 11, 2007
14. Aug 12, 2007

### bernhard.rothenstein

light clock

Hi Pete. Thank you for your help. I must confess that I have seen many presentations of the light clock but the way in which you present it is the single one on which the clocks of the the two involved reference frames are marked and the essential events are mentioned.
Please have a look at a thread of mine some lines above in which I have tried to express my point of view concerning the light clock and the other clocks it involves. Your oppinion is appreciated. It is essential in a discussion I have with the Auhor of
arXiv:0708.0988v 1 [physics.ed-ph] 7 Aug 2007
Kind regards

15. Aug 12, 2007

### neutrino

Anyone...?

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16. Aug 12, 2007

### robphy

Once you have the Lorentz Transformations and know how to use them, you have practically all you need to know. So, it seems general enough to me. (For teaching purposes, the issue is then "How do you get the Lorentz Transformation?")

In the [easy] "standard" derivation of time dilation using a light clock, you haven't established the [difficult] Lorentz transformation yet... that's the pedagogical value of the light clock derivation. In the light clock derivation, note that the distant reflection event is used to obtain the expression $\gamma$ that will be called the time-dilation factor...and will appear in the Lorentz Transformations.

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17. Aug 12, 2007

### country boy

Thank you for the very clear derivation and diagrams. But I have a couple of questions.

When you say "Let t equal the time between clock ticks as measured in S" and "t is called the proper time," don't you mean "tau"? If the clock is moving with the mirror it will measure tau.

Also, the right-hand diagram leaves out the important stationary clocks that measure the times at the three (or two) positions as the moving clock passes. These previously synchronized clocks measure t at separated points (thus t is not the proper time). The stationary clocks can be light clocks using the same L, so that the comparison is easy. The relation between tau and t is then simply derived from the geometry in your diagram and the result is the one you give.

The distance L is the key here. L is the invariant that does not change between the stationary and moving frames. The invariant proper time tau is given by L in your derivation. The role of L is similar to that of the invariant ds in the Minkowski diagram (for motion in the x-direction dy and dz, like ds, do not change).

The description of light clocks in often confusing. The whole apparatus is called a "light clock," but then it is descibed a two mirrors with a clock attached to one of them. The attached "clock" is actually just a counter keeping track of the number of cycles of the light pulse. In a conventional clock the counter keeps track of cycles of a pendulum or something else that moves with repeated motion. The clock is the combination of the cycler and the counter.