Time Dilation with a Tennis Ball

[Sturk200]

So I just finished reading my textbook on special relativity and am a bit confused by the discussion of time dilation. We equip an observer with a light source, a mirror, and a clock and herd him onto a train moving at constant velocity relative to a second observer located at the station. If D is the distance between the source and the mirror, then the time (measured by the observer on the train) that it takes for a pulse of light to strike the mirror and reflect back to the source is given by

Δt₀ = 2D/c.

We compare this to the time observed by the stationary observer, which will be:

Δt = Δt₀/√1-(v/c)².

The measured times are different and related by the Lorentz factor. Now the reason these two times are different is a consequence of the constancy of the speed of light. For the stationary observer, the light beam has to travel a longer path (a triangle of height D and base vΔt), and since it travels at the same speed it takes longer. Now here comes my question.

How would this result be applied to a tennis ball rather than a light beam? As I understand it, the idea of time dilation is supposed to apply to all phenomena, not just electromagnetic phenomena. So suppose we fire a tennis ball from a tennis-ball-souce and reflect it from a tennis-ball-mirror (i.e., wall). There is no principle of the constancy of the speed of sports equipment, so wouldn't the time measured by the observer on the train be equal to that measured by the stationary observer (I mean the time it takes for the tennis ball to leave its source and be reflected back). Because though the stationary observer watches the ball travel a longer path, the ball is moving with a greater speed due to the speed imparted to it by the movement of the train, and that greater speed is precisely enough to compensate for the extra distance, since both are caused by the movement of the train. So I guess my question is first of all, am I right that there would be no time difference measured if we used a tennis ball instead of a light beam? And second, how do we justify the application of special relativity to tennis balls if the effects are dependent upon the constancy of the speed of light?

 

[PeterDonis]

If D is the distance between the source and the mirror

Just to clarify, this distance is perpendicular to the direction of motion of the train, correct? It seems so from the rest of your description, but it's always good to be as explicit as possible in scenarios like this. In this particular case, it's not an idle question, because distances in the direction of motion of the train are length contracted, while distances perpendicular to the direction of motion are not.

For the stationary observer, the light beam has to travel a longer path (a triangle of height D and base vΔt), and since it travels at the same speed it takes longer.

Yes; but note that there is also the constraint that the light pulse has to hit the mirror and return to the source in both frames (the train frame and the station frame). This constraint will be there even if we use something besides light in the setup. See below.

There is no principle of the constancy of the speed of sports equipment, so wouldn't the time measured by the observer on the train be equal to that measured by the stationary observer (I mean the time it takes for the tennis ball to leave its source and be reflected back).

No. The increase in speed in the station frame is not enough to compensate for the longer path, because in relativity, velocities do not add linearly. You have to use the relativistic velocity addition formula:

https://en.wikipedia.org/wiki/Velocity-addition_formula#Vector_notation

If you work it out, you will see that the "tennis ball clock" still looks time dilated in the station frame, by the same factor as the light clock. A key factor in the solution is to remember the constraint I mentioned above, that the tennis ball has to hit the reflector and return to the source in both frames.

 

[Dale]

that greater speed is precisely enough to compensate for the extra distance,

Not precisely. As Peter mentioned, the relativistic velocity addition formula would yield a time dilated tennis ball clock also.

 

[Ibix]

Set up a train half a light second wide. In the train's rest frame the light pulse will take one second to cross the train and bounce back. Next to one end of this light clock set up a (much shorter) tennis ball clock that also takes one second to complete an out-and-back bounce. Rig a bomb to explode if the two clocks don't tick synchronously.

What happens in the platform frame if the tennis ball doesn't obey relativity but the light does? What happens in the train frame?

For a more formal argument, see (for example) Pal, who shows that the principle of relativity leads to either Galilean relativity or Einsteinian relativity without once mentioning light. You can differentiate between a Newtonian/Galilean universe and an Einsteinian universe with regular pendulum clocks on a train, if the train is fast enough or the clock precise enough. No light needed.

 

[Sturk200]

Thanks for the answers.

Just to clarify, this distance is perpendicular to the direction of motion of the train, correct?

You guessed it. Sorry for the vagueness.

the relativistic velocity addition formula would yield a time dilated tennis ball clock also.

You have to use the relativistic velocity addition formula

Now I see. For some reason I had it in my head that the velocity transformation was only for velocities in the direction of motion, but this would fail to take into account the time dilation effect. Thanks.

I was also wondering if there is some explanation of this phenomenon that doesn't assume special relativity theory. What I mean is that I can see how time dilation with a light beam is a direct consequence of the two postulates of relativity and the constancy of light speed, even before busting out the formulas. The distance the light has to travel is made larger by the movement of the train, yet due to the constancy of light speed the train cannot impart a velocity to the light beam, and therefore the beam traverses the path in a slower time. With a tennis ball, on the other hand, it doesn't seem to fall out of these two principles in quite the same way. I guess I'm having a hard time imagining why Einstein generalized his theory from light to non-electromagnetic phenomena. Am I being at all clear?

Maybe the rationale would be this, that by the principle of relativity, if it is true for electromagnetic phenomena, then it must be true for all phenomena?

 

[Dale]

guess I'm having a hard time imagining why Einstein generalized his theory from light to non-electromagnetic phenomena.

That is a direct consequence of the first postulate. If you could build a non-EM clock that did not time dilate then by comparing such clocks to EM clocks you could detect a frame-dependent variation in the laws of physics.

Maybe the rationale would be this, that by the principle of relativity, if it is true for electromagnetic phenomena, then it must be true for all phenomena?

Yes. Exactly.

 

[Sturk200]

That is a direct consequence of the first postulate. If you could build a non-EM clock that did not time dilate then by comparing such clocks to EM clocks you could detect a frame-dependent variation in the laws of physics.

Pretty amazing idea, that relativity. Do you know if there have been tests of special relativity on tennis ball clocks? I know it's been tested with atomic clocks which use microwaves I think. Are mechanical clocks too imprecise?

 

[Dale]

Pretty amazing idea, that relativity. Do you know if there have been tests of special relativity on tennis ball clocks? I know it's been tested with atomic clocks which use microwaves I think. Are mechanical clocks too imprecise?

Yes. Not only are mechanical clocks too imprecise, they are fundamentally based on EM also.

However, fundamentally non EM clocks can be made using the strong and weak nuclear forces both of which show SR time dilation. And gravity also shows time dilation.

 

[Sturk200]

they are fundamentally based on EM also.

How are mechanical clocks based on EM? I don't doubt the claim, I just have no idea how to even begin explaining it to myself.

 

[Dale]

All of the interactions that hold solids together are EM interactions between nearby molecules and atoms.

 

[Stephanus]

Set up a train half a light second wide[..]What happens in the platform frame if the tennis ball doesn't obey relativity but the light does? What happens in the train frame?

Can I answer? The bomb does not explode because

Set up a train half a light second wide. In the train's rest frame the light pulse will take one second to cross the train and bounce back. Next to one end of this light clock set up a (much shorter) tennis ball clock that also takes one second to complete an out-and-back bounce. Rig a bomb to explode if the two clocks don't tick synchronously.

But, that's according to the train frame. What about at the platform frame? Let say the tennis balls travels 5000m/s compared to the light 300,000,000. And supposed the train travels at 0.6c, what would happen?
I draw a picture here, to make it clear.

Fo the train travels from A to B. It tooks t time from A to B.
So AB (r) distance is Vt.
The light travels to A->D->B. It takes t time. So, AD(p) (and AB) distance is 0.5t
We know DF(q) is 0.5c for in train frame, it tooks 1 second bounce back. So here it its
p=0.5t, distance in a factor of c
r=Vt, V in a factor of c
q=0.5
If we pythagoras r, we'll have. $\frac{1}{2}r^2 = p^2 - q^2$
$\frac{1}{2}^2v^2t^2 = 0.5t^2 - 0.5^2$
$t=\frac{1}{\sqrt{1-v^2}}$ Okay..., so $t = \gamma$
$t=1.25 \text{ seconds}$ So, 1.25 seconds for the platform is 1 seconds for the train.
$r = 0.75t$
The tennis ball travels at,...
Let $V_{ball} = 5000m/s = 0.00005c$
$S_{ball} = 2 * \sqrt{V_{ball}^2+\frac{1}{2}^2r^2}$
$S_{ball} = 2 * \sqrt{0.0000000025+0.140625}$
$S_{ball} = 0.750000006667t$
Train frame: t = 1; S_Ball = 0.00005t
Platform frame: t = 1.25; S_Ball = 0.750000006667t
I hope this is true.

 

[Ibix]

You're working way too hard for this.

We know that the light clock will tick every $\gamma$ seconds in a moving frame. The OP accepted that. If relativity somehow did not apply to tennis balls then the tennis ball clock would obey Newtonian physics and tick every second. But then we end up with a contradiction - the bomb would go off according to any frame except the $\gamma=1$ rest frame of the train, where it wouldn't. Which is nonsensical. Either it goes off or it doesn't.

The only self-consistent solution is for the tennis ball clock to tick at whatever rate the light clock ticks at, whatever frame is in use.

 

[Janus]

Thanks for the answers.
You guessed it. Sorry for the vagueness.Now I see. For some reason I had it in my head that the velocity transformation was only for velocities in the direction of motion, but this would fail to take into account the time dilation effect. Thanks.

I was also wondering if there is some explanation of this phenomenon that doesn't assume special relativity theory. What I mean is that I can see how time dilation with a light beam is a direct consequence of the two postulates of relativity and the constancy of light speed, even before busting out the formulas. The distance the light has to travel is made larger by the movement of the train, yet due to the constancy of light speed the train cannot impart a velocity to the light beam, and therefore the beam traverses the path in a slower time. With a tennis ball, on the other hand, it doesn't seem to fall out of these two principles in quite the same way. I guess I'm having a hard time imagining why Einstein generalized his theory from light to non-electromagnetic phenomena. Am I being at all clear?

Maybe the rationale would be this, that by the principle of relativity, if it is true for electromagnetic phenomena, then it must be true for all phenomena?

Basically yes. Look at it this way. If you are in the same frame at the light clock and the "tennis ball" clock. the tennis ball bounces back and forth at 0.0000001 c. This means for you, the light in the light clock bounces back and forth 10 million times for every one time for the ball.

Now assume I am in another frame in which you and your clocks are moving at 0.866 c. Compared to my own light clock, your light clock is ticking 1/2 as fast. If the same did not apply to your tennis ball clock, i would see it "tick" twice for every 10,000,000 ticks of your light clock, or put another way, I would see your light clock tick 5,000,000 times for every tick of the tennis ball clock . This will lead to contradicting results when we get together again after the experiment ( let's assume that you have a simple device that records the number of ticks for each clock, I will see it record a different ratio of ticks between the two clocks than you do. When we get back together would we be looking at the same physical record and see different numbers?. This would make no sense. The only way things cam make sense is if I measure your tennis ball clock as ticking slower by the same rate as your light clock does. Then there is no disagreement between us as to the ratio of ticks between the clocks.

 

[Stephanus]

You're working way too hard for this.

We know that the light clock will tick every $\gamma$ seconds in a moving frame.

"We"? Whom do you mean by "We", I didn't know that $t = \gamma$, before I did the calculation. .

[..]then the time that it takes for a pulse of light to strike the mirror and reflect back to the source is given by

Δt₀ = 2D/c.

[..]the time observed by the stationary observer, which will be:

Δt = Δt₀/√1-(v/c)².

I should have made it sense before!

[..]Which is nonsensical. Either it goes off or it doesn't.

The only self-consistent solution is for the tennis ball clock to tick at whatever rate the light clock ticks at, whatever frame is in use.

"Either it goes off or it doesn't", I like that, reminds me of Schrodinger cat. Of course, just like Pavlov Dog, the tennis ball clock ticks when the light clock ticks.

 

[Ibix]

"We"? Whom do you mean by "We", I didn't know that $t = \gamma$, before I did the calculation. .

The first post states this. It's also a standard result for time dilation - if two things happen at the same place a time t apart (in some frame) then in any other frame they happen $\gamma t$ apart. Note all the restrictions around when this simple form of the Lorentz transforms can be used.