# Light clock calibration

1. Oct 30, 2014

### DAC

Hello PF.
Re. the light clock / train thought experiment. It seems to me that the clock calibration chosen, ( light's mirror to mirror path ), is determining the outcome of the experiment. And that a different calibration would give a different result. e.g. assuming the mirrors are one metre apart, calibrate the clock to tick over each time light travels one metre. One metre is the same measure in both frames as the distance between the mirrors can't change, and, the speed of light is constant. You still have the perpendicular and diagonal paths, but they are now measured in one metre lengths. The passage of time would therefore be the same in both frames.

2. Oct 30, 2014

### Staff: Mentor

Hi DAC,

First I need to make sure I understand which version of the light clock you are talking about. I assume you are talking about a light clock where the light, in the rest frame of the clock, moves perpendicular to the direction of relative motion between the clock and the observer. Is that correct?

Assuming that the above is correct, your analysis is not. It looks like the key error you are making is here:

The first part of this is true; one meter is the same in both frames, and the distance between the mirrors does not change (because that distance is perpendicular to the direction of relative motion). However, the two path lengths are different (because one path is perpendicular and the other is diagonal, as you say), so the time it takes the light to traverse them must also be different, since, as you agree, the speed of light is constant.

3. Oct 31, 2014

### DAC

The two path lengths are different, but if the clock is calibrated to each time light travels one metre,

4. Oct 31, 2014

### DAC

Hi PeterDonis
The fact that it takes light longer to travel a longer path is not surprising. Likewise if you calibrate a clock to that path's length its not surprising it takes longer to tick over. But that proves nothing regarding time dilation.
If instead you have a clock that ticks over each time light travels one metre, and that is the same in both frames, then the clock ticks at the same rate in both frames. Note that in the stationary frame the clock already ticks over each time light travels one metre, assuming the mirrors are one metre apart.

5. Oct 31, 2014

### Staff: Mentor

Ok, good.

But the clock isn't calibrated to that path's length in the frame in which the clock is moving; it's calibrated to that path's length in the frame in which the clock is at rest. You're leaving that out of your analysis.

You're describing this wrong. What you should be saying is this: in the frame in which the clock is moving, the light bouncing between the mirrors travels more than one meter from mirror to mirror, so it takes longer than one second to travel from mirror to mirror in that frame. That's true. And if you calibrate a clock to tick once each time light travels one meter, then that clock will tick more than once each time light goes from one mirror to the other.

But the definition of a light clock is a clock that ticks once each time light goes from one mirror to the other. And the whole point of the light clock is that one "tick" of the clock corresponds to the time it takes light to travel one meter in the clock's rest frame, but corresponds to a longer time--the time it takes light to travel the diagonal path between the mirrors--in the frame in which the clock is moving.

The question you should be asking is, why do we define one "tick" of the clock as the time it takes light to go between the mirrors? The answer is, because the light going between the mirrors is a way of marking out invariant events in spacetime, events which stay the same when we change frames. Think of each event of the light beam bouncing off a mirror as a marker in spacetime: different frames may assign different coordinates to the marker, but the marker itself does not change (just as, if we drive a stake into the ground on the Earth, the stake stays the same even if we describe it in different coordinates). And any consistent definition of "ticks" of a clock has to use invariant markers like these. Your proposal, to define one "tick" of the clock as the time it takes light to travel one meter in whatever frame you choose, amounts to changing the markers when you change frames; but that means you're changing clocks--physically changing the definition of what the "clock" is. The point of the light clock as illustrating time dilation is to keep the same clock--the same markers in spacetime that define its ticks--when you change frames.

6. Oct 31, 2014

### Staff: Mentor

Consider two identical light clocks. By "identical" I mean that if I put them side by side while they are at rest relative to me and each other, the distance between the mirrors is the same. It doesn't matter what that distance is, just that it is the same for both.

Now we can define our unit of time to be the "tick", the amount of time it takes for a flash of light to travel from one mirror to the other and back again when the clocks are at rest. When the two clocks are sitting side by side, they always agree; if we start them running at the same time they will have recorded the same number of ticks whenever we look at them later.

Now we set one of the two clocks in motion. It doesn't matter which one; we can say that clock A is traveling to the right at speed v or we can say that clock B is traveling to the right at speed v. It's the same physical situation either way.

However, the clocks will no longer be synchronized. An observer who remains at rest relative to clock A will see that A is ticking away at the same rate as always because nothing has changes for clock A when clock B starts moving. He will also say that the time between ticks of clock B increased when it started moving because the light in clock B is traveling a longer path than when it was at rest. However, an observer who remains at rest relative to clock B will see the exact opposite: nothing about clock B changes when clock A is moving, and the light has a longer distance to travel in clock A so A ticks more slowly.

That's time dilation, and the only calibration needed to demonstrate it was to ensure that the two clocks were constructed identically.

7. Oct 31, 2014

### DAC

8. Oct 31, 2014

### DAC

Thanks PeterDonis.
The calibration I proposed is the same in both frames. How does that constitute changing markers / clocks?

9. Oct 31, 2014

### Staff: Mentor

Because the markers are the events at which the light hits each mirror. In the light clock's rest frame, the calibration you proposed leads to exactly one tick of the clock between two such events. In the frame in which the light clock is moving, the calibration you proposed leads to more than one tick of the clock between two such events. A clock cannot be "the same" if it registers a different number of ticks between the same two events in spacetime, depending on what frame you choose.

10. Oct 31, 2014

### DAC

Thanks Nugatory.
If a clock is calibrated to each time light travels one metre, it seems to me it will tick at the same rate in either frame. Clearly this would not be in agreement with SR. Can you tell me why?

11. Oct 31, 2014

### Staff: Mentor

We make a light clock that ticks once (light flash out and back again is a tick) in the time that light travels one meter by placing the two mirrors exactly one-half meter apart, right? If you and the clock are at rest relative to one another, then the light will travel exactly one meter per tick and because the speed of light is exactly 299,792,458 meters per second, exactly one second will have passed after 299,792,458 ticks. That's the way that observer A describes clock A which is at rest relative to himself but not B; and it is the way that observer B describes clock B which is at rest relative to himself but not A.

A, looking at his clock, sees a light flash leave the near mirror, travel 1/2 meter to the far mirror where it is reflected, and travel 1/2 meter back to the near mirror to complete one tick. But what does A see when he looks at B's clock once it has been set in motion? He sees a flash of light leave B's near mirror, travel on a slantwise path that is longer than 1/2 meters to reach B's far mirror where it is reflected, and then travel on a slantwise path that is longer than 1/2 meter to get back to B's first mirror. Therefore, A will see that light travels a longer distance to complete a single tick of B's clock than to complete a single tick of his own clock; in A's frame B's clock is ticking more slowly.

In B's frame, his clock is at rest while A's clock is moving, so from B's point of view it is the flash of light in A's clock that is taking a slantwise path, and hence A's clock that is ticking more slowly.

12. Nov 1, 2014

### DAC

Agreed there is more than one tick of the clock in the moving frame which is longer, but the rate at which the clock ticks doesn't change.

13. Nov 1, 2014

### Staff: Mentor

I may not be understanding what you mean by "the rate at which the clock ticks".

When the two identically constructed clcks are side by side, clock B ticks five times during the same amount of time that clock A ticks five times. After clock B is set into motion at a speed of .6c, observer A observes that during that same amount of time, clock B now ticks only four times.

If that's not a change in the rate at which clock B is ticking, what is it?

14. Nov 1, 2014

### zbe

If I understand OP's reasoning, an analogous example would be comparing heights of two persons. Say person A is 2.00 m tall and person B is 1.50 m, so clearly person A is taller. But now we re-calibrate person B's meter so 1 meter = 1.33 meters and we conclude that, in fact, both persons are of equal heights.

15. Nov 1, 2014

### DrGreg

This diagram, which shows two views of the same scene, is a more relevant analogy.

16. Nov 1, 2014

### Staff: Mentor

Time dilation means "there is more than one tick of the clock in the moving frame which is longer". Whatever you mean by "the rate at which the clock ticks", it has nothing to do with time dilation.

(Like Nugatory, I'm not getting what you mean by "the rate at which the clock ticks", but whatever it is, it can't have anything to do with time dilation, because you have now said that your "rate at which the clock ticks" being "the same" doesn't affect how many ticks of the clock there are in the moving frame.)

17. Nov 2, 2014

### Staff: Mentor

This is pretty straightforward.

In any given frame $S$ let $\Delta D_S$ be the distance that the light travels in one tick in frame $S$ and let $\Delta t_S$ be the time between ticks in $S$. In any frame $\Delta D_S/\Delta t_S = c$, by the second postulate.

Then comparing frames $S$ and $S'$ we have $\Delta D_S/\Delta t_S=\Delta D_{S'}/\Delta t_{S'}$. Since $\Delta D_S \ne \Delta D_{S'}$ then necessarily $\Delta t_S \ne \Delta t_{S'}$

18. Nov 2, 2014

### DAC

No because you have changed the distance

If you change the distance the light has to travel, you change the time it takes for the clock to tick over. The rate at which the clock ticks doesn't alter. It just takes longer in time.
For example if you are driving home from work, and a diversion lengthens the distance you must travel, it doesn't alter your speed.

19. Nov 2, 2014

### Staff: Mentor

That is time dilation.

20. Nov 2, 2014

### Staff: Mentor

Of course I've changed the distance - that's the while point of the exercise.

If the speed is constant in all frames and the distance is different in different frames, then the time must be different in the different frames. Pre-relativistic physics assumed that the time must be the same in all frames (without specifying a procedure for measuring time to see if this assumption was valid) while the distance was different (that's the part that we all agree about) so concluded that the speed must be different. Experiments such as Michelson-Morley that showed that the speed is not different came as a rude shock to this line of thinking.

The key to this entire discussion is to clearly specify how you are measuring the passing of time. The argument you're making, if you dig deeply enough, contains an assumption that there could be some giant clock in the sky that all observers can look up at and agree about what time it is. But that clock doesn't exist, so the only way that I have of measuring time is by counting the ticks of a light clock that is at rest relative to me.

21. Nov 2, 2014

### DAC

22. Nov 2, 2014

### Staff: Mentor

Only because we changed frames. The events in spacetime (the markers) have not changed. Only the distance and time between them has changed, because we changed frames.

Basically, you are thinking of the distance as the "constant", the thing we are supposed to keep the same in order to make an apples-to-apples comparison. In Newtonian physics, that would be correct; but in relativity, it's not. In relativity, the "constant", the thing you need to keep the same in order to make an apples-to-apples comparison, is the spacetime interval--the two "markers" in spacetime and the quantity $ds^2 = dt^2 - dx^2$ that gives the invariant "spacetime distance" (or interval--the latter word makes it clear that this is not the same thing as a spatial distance in Newtonian physics) between them.

The reason we do this in relativity is that it matches experiment: it turns out that the way the world actually works is not Newtonian.

23. Nov 2, 2014

### DAC

Thanks Nugatory.
If the clock is calibrated to tick over each time light goes mirror to mirror, then the longer path will mean the clock takes longer to tick over, and a longer interval between ticks means the clock runs slower. I agree.
But if you change that calibration to each time light travels one metre, the longer path, will be measured in one metre lengths which are constant in both frames. ( The mirrors are one metre apart in both frames, and light is a constant. )

A separate issue:- If there are two identical light clocks, on a moving train, and one is placed vertically, but the other horizontally, aligned in the train's direction of motion, what happens? Both should be time dilated, but the horizontal clock will also be length contracted which brings its mirrors closer, hence speeding up its tick rate. What prevails?

24. Nov 2, 2014

### Staff: Mentor

This is simply false, and it contradicts what you have said several times, that the distance the light travels is longer in the frame where the clock is moving.

As you said, the mirrors are 1 m apart in both frames, so the distance traveled in the frame where the clock is moving is necessarily greater than 1 m. The hypotenuse is longer than the length of either leg in a right triangle. That is simply basic Euclidean spatial geometry.

Also, the calibration thing is a red herring. You cannot calibrate the clock to tick every time the light goes one meter in all frames because different frames disagree on how many meters the light has gone between any two ticks.

You should work out the math on this one. The length contraction has the exact value needed to ensure that the two light clocks tick at the same rate as each other in all frames.

Last edited: Nov 2, 2014
25. Nov 2, 2014

### DAC

Yes the mirror to mirror distance is greater in the moving frame. I think you will find I said it could be measured in one metre lengths, which is not the same as saying its length is one metre. Sorry if that caused a misunderstanding.