# Light clock with something else than light

1. Sep 29, 2010

### homer5439

I think I understand how the classical thought experiment of the light clock (light bouncing between two mirrors work). The typical explanation that is given is "the stationary observer sees the light travel a longer distance (the diagonal), so since light can't go faster than the speed of light, it must be time that slows down".
That's all right.
But let's imagine that instaed of the light there is a ball bouncing up and down, at a non-light speed, in fact, quite slowly. But a constant speed.

So now the moving observer carrying the clock sees the ball taking time x to do a vertical movement between the walls. The stationary observer sees the ball taking time x to do a slightly longer trip (the diagonal).
Does this also mean that time slows down for the moving observer? Or that the ball moves faster for the stationary observer? How can the ball move at two different speeds at the same time?

2. Sep 29, 2010

### DaveC426913

Each observer observes the other observer as moving slower.
No.

Because time (and distance and speed) is dependent on the frame of reference.

3. Sep 29, 2010

### Janus

Staff Emeritus
The stationary observer will not see the ball travel the diagonal in time x, but in a longer time. The ball is subject to the Relativistic addition of velocities theorem. What this means is the the velocity of the ball with respect to the stationary observer will be slightly less than if you used Newtonian addition of velocities.

For example, with Newtonian addition, the speed of the ball relative to the observer would be

$$w=\sqrt{u^2+v^2}$$

Where v is the velocity of the clock and u the up and down velocity of the ball as measured by the moving observer.

In Relativity you would use

$$w=\sqrt{u^2+v^2-u^2v^2}$$
(assuming u, v, and w are measured in units of c)

Thus if u and v both equaled 0.5c, the Newtonian equation give an an answer of 0.707c and the Relativistic equation give 0.661c

Applying the Pythagorean theorem to both, the Newtonian approach will say that the stationary observer will measure the up and down motion of the ball with as being 0.5c, (the same as measured by the moving observer), while the Relativistic approach says that it will be 0.432c.

0.432c is 0.865 of .5c, The time dilation factor at 0.5c is 0.866, which when allowing for rounding errors, is the same.

The rate of the ball bouncing up and down matches the same time dilation as you get with the light.

4. Sep 30, 2010

### Saw

Yes, that is true. But I suppose that you don't mean that the ball is subject to time dilation because it is subject to the Relativistic addition of velocities theorem. I say so, because the latter (velocity addition formula) is derived *assuming* precisely time dilation.

I think it is not out of question to clarify to the OP that time dilation is not a consequence that you can draw from a mere thought experiment. It doesn't flow out of pure abstract logic. To hit upon it, you have to either (i) rely directly on physical experiments or (ii) make some sort of physical assumption about what happens when a ball is accelerated... If you don't like to go into the complexities of route (ii) (and for some reason, I don't know why, that seems to be the general preference), then there is only route (i), i.e. experiments like those with muons.

5. Oct 1, 2010

### homer5439

Thank you all. So I seem to understand that the same effects produced by the hypothetical light-clock are also produced if we use my slower ball-clock.
So why all the descriptions focus on the light-clock? There must be something special about it that makes it different from the ball-clock.

6. Oct 1, 2010

### Janus

Staff Emeritus
The light clock is used because it can be inferred directly from the two postulates of Relativity. It is obvious that if light travels at the same speed relative to both frames then in the "stationary" frame the light will take longer to travel the diagonal. It is a direct example of time dilation.

It isn't as obvious that the ball will without already understanding the concepts of time dilation, length contraction and the relativity of simultaneity beforehand.

7. Oct 1, 2010

### homer5439

Right. So is it correct to say that the same exact reasoning that is done for the light clock can be applied to the ball clock?

Also, I have another, a bit more "philosophical", question. Is "speed" (and other properties) something intrinsic to an object? I mean, whatever the object is doing, it can't depend on whether someone is there to measure it or not (ok, not at subatomic scales). So everything both frames see as "moving", must be moving at the same speed for both frames? (be it light, the ball, or whatever they both see as moving)

8. Oct 1, 2010

### curiousphoton

Impossible. A ball bouncing up and down cannot travel at a constant speed.

9. Oct 1, 2010

### JesseM

The idea is that it's bouncing in zero gravity, not in a gravitational field (of course even in a gravitational field we can get zero gravity if the walls the ball is bouncing between are in freefall). Of course its speed will change when it bounces against each wall, but we can idealize the collisions as perfectly elastic so the speed between walls remains constant with each bounce, and we can also idealize the period of contact with each wall as arbitrarily brief.

10. Oct 1, 2010

### Ich

No. You can't apply any reasoning at all to the balls, as you don't know what they'll do (hmmm - no pun intended). There's no ball speed postulate.
No.
No.

11. Oct 2, 2010

### Saw

If in the end the ball clock should be also affected by time dilation and by the same rate as the light clock…, yes, you are right in that a *similar* reasoning should apply to it, but I would not say that it is the *same* reasoning. The proof is that in the end, just as there are similarities, there are differences between the light and the ball. Unlike in Galilean relativity, in Special Reativity the speed of a light beam is the same in all frames. Instead the speed of a ball isn’t: it varies from frame to frame and you relate the ball speed in one frame to the speed of the same ball in another frame through the law of addition of velocities. This was also so in Galilean relativity. In Special Relativity it just happens that the equation is different. And the difference between the two formulas lies precisely in the fact that the second assumes time dilation. As I said before, it is derived by asuming time dilation.