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Hello. There is a question that I've been trying to understand for about a year. Why is it that the time dilation effect applies equally to all clocks as it does with "light clocks". For example, if John is moving on a space ship with respect to Alex who is on earth, and both are sitting next to a light clock where a beam of light is bouncing up and down, Alex will see John's beam of light moving upwards and horizontally in a diagonal line. Because the speed of light is constant for both, Alex will see John's beam of light taking longer to reach the top than the beam on his own clock, because he sees his own beam is going straight up and down, and he sees John's going up and sideways.

Time dilation with light clocks is affected by the constancy of the speed of light. Say we chose another type of clock, like a ball bouncing up and down instead of a beam of light. The speed of the ball is not constant for both of them. Therefore, if they both use clocks with a ball bouncing up and down, Alex will see John's ball going up a longer path just like in the light clock example. However, Alex will also observe the ball going up faster to compensate for the extra length. Therefore, although time dilation might occur here to, the amount of dilation should be different. Why then, does time dilate for all clocks on John's ship, in Alex's point of view, by the same factor of gamma? Shouldn't the factor be different depending on what type of clock is being used, and what the Lorenztian sum of the velocity is between the speed of John's ship and the speed of the clock?

If what I am saying is true, then the effects would be very strange: i.e. someone could go on a voyage at a very high speed with various clocks (including their biological clock) and come back with some of their clocks far behind those on earth, and other clocks only slightly out of sync.

Thank you.

Orodruin
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Hello. There is a question that I've been trying to understand for about a year. Why is it that the time dilation effect applies equally to all clocks as it does with "light clocks". For example, if John is moving on a space ship with respect to Alex who is on earth, and both are sitting next to a light clock where a beam of light is bouncing up and down, Alex will see John's beam of light moving upwards and horizontally in a diagonal line. Because the speed of light is constant for both, Alex will see John's beam of light taking longer to reach the top than the beam on his own clock, because he sees his own beam is going straight up and down, and he sees John's going up and sideways.

Time dilation with light clocks is affected by the constancy of the speed of light. Say we chose another type of clock, like a ball bouncing up and down instead of a beam of light. The speed of the ball is not constant for both of them. Therefore, if they both use clocks with a ball bouncing up and down, Alex will see John's ball going up a longer path just like in the light clock example. However, Alex will also observe the ball going up faster to compensate for the extra length. Therefore, although time dilation might occur here to, the amount of dilation should be different. Why then, does time dilate for all clocks on John's ship, in Alex's point of view, by the same factor of gamma? Shouldn't the factor be different depending on what type of clock is being used, and what the Lorenztian sum of the velocity is between the speed of John's ship and the speed of the clock?

If what I am saying is true, then the effects would be very strange: i.e. someone could go on a voyage at a very high speed with various clocks (including their biological clock) and come back with some of their clocks far behind those on earth, and other clocks only slightly out of sync.

Thank you.
Indeed, but this is precisely the point. The relation between the clocks would be observer dependent. If one observer sees clocks with the same motion tick at the same rate, all observers must.

Doc Al
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Why then, does time dilate for all clocks on John's ship, in Alex's point of view, by the same factor of gamma? Shouldn't the factor be different depending on what type of clock is being used, and what the Lorenztian sum of the velocity is between the speed of John's ship and the speed of the clock?
If time dilation affected different kinds of clocks differently, then we'd be able to tell our "absolute" speed by comparing two different clocks. Which would violate one of the basic premises of special relativity.

Realize that light clocks are chosen for analysis because they are easy to analyze! If you performed the correct analysis of a bouncing ball clock (or any kind of clock), you'd find the same time dilation factor at work. But unlike the light clock, that bouncing ball clock would be difficult to analyze.

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Indeed, but this is precisely the point. The relation between the clocks would be observer dependent. If one observer sees clocks with the same motion tick at the same rate, all observers must.
Thank you Orodruin. You said "if one observer sees clocks with the same motion tick at the same rate, all observers must". However, it doesn't seem like Alex would observe John's clocks ticking at the same rate? If both of them started out on earth stationary with respect to one another, and each had their own "light clock" and "ball clock" right next to them, both clocks would be ticking at the same rate. But once John starts moving, doesn't his light clock, according to Alex, start ticking slower than Alex's light clock? And, doesn't John's ball clock, according to Alex, start ticking slower, but only to a lessor degree, than Alex's. If John's light clock started ticking 50% slower than Alex's, for example, couldn't it be true that John's ball clock only started ticking 25% slower than Alex's?
You also said "the relation between the clocks would be observer dependent". Did you mean the relationship between John's clocks, or the relationship between John and Alex's clocks? Also, did you mean the relationship of how fast the clocks tick, or the relationship of the distances that the path of the clock takes (e.g. the path of the beam of light going up diagonally vs. straight up).
Just to clarify, is the answer to the question about whether the time on different clocks on a space ship, relative to those on earth, dilate differently depending on what type of clock is used, yes?

Orodruin
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However, it doesn't seem like Alex would observe John's clocks ticking at the same rate?
This is only true if you use classical addition of velocities instead of relativistic. The entire point is that you get into this problem only if you presuppose that classical addition of velocities is true. The resolution is of course to introduce relativistic velocity addition. There really is nothing more to it.
Just to clarify, is the answer to the question about whether the time on different clocks on a space ship, relative to those on earth, dilate differently depending on what type of clock is used, yes?
No.

Gold Member
Thank you Doc Al. "If you performed the correct analysis of a bouncing ball clock (or any kind of clock), you'd find the same time dilation factor at work."
It seems logically impossible, though. I must be making some mistake with the math or something? Take the hypotenuse of a right triangle which is the path that Alex sees John's beam of light moving up. That light moves at the same speed for Alex and Bob, but travels a further distance than Alex because he is stationary. In the ball example, the speed of the ball is not constant. Alex can add the velocities of John's ship and of John's ball (using the Lorentz addition). Therefore, even though the hypotenuse will be the same length, Alex will see the ball traveling up faster than John will see it. Am I making some fundamental error in the way I'm thinking about it?

Also, Orodruin, even if you use Lorenz addition, you are still adding velocities in the ball example, which you can't do at all in the light example which would seem that time should dilate less. I agree that with regular addition velocity, time wouldn't dilate at all.

Doc Al
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It seems logically impossible, though. I must be making some mistake with the math or something? Take the hypotenuse of a right triangle which is the path that Alex sees John's beam of light moving up. That light moves at the same speed for Alex and Bob, but travels a further distance than Alex because he is stationary. In the ball example, the speed of the ball is not constant. Alex can add the velocities of John's ship and of John's ball (using the Lorentz addition). Therefore, even though the hypotenuse will be the same length, Alex will see the ball traveling up faster than John will see it. Am I making some fundamental error in the way I'm thinking about it?
Why don't you actually do the calculation? It's not that difficult. Use the Lorentz transformation for velocity and figure out the time it takes for the ball to make a vertical bounce as seen by another observer.

(The reason that elementary treatments use light clocks --and not bouncing balls-- is that they're simple enough that you don't need the LT to figure them out, just the basic assumption that the speed of light is constant.)

Orodruin
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which you can't do at all in the light example
This is just wrong. If you put the velocity of the object to c in one frame, relativistic velocity addition will just give you c in any frame. It is just a special case of relativistic velocity addition.

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That makes sense- I wanted to do that. Physics has never been my major, I've just been really interested in the nature of time for a while.

I will use concrete numbers: Say John has traveled a distance of 4 light years from Earth while Alex has remained. Say both have a light clock that is 3 light years high. Using the Pythagorean theorem, Alex will see John's beam of light traveling a distance of 5 light years. He would also measure the beam of light to take five light years to reach the top when John measured it to take only 3. John's light clock is working 60% as fast as Alex's.

Now same example except John is traveling at half the speed of light. Instead of a light clock, there is a ball traveling upwards a distance of three light years, and the speed of the ball is half of the speed of light from John's perspective. Say John's ship is also traveling at half the speed of light. Using the Lorentz addition velocity, Alex would calculate the ball to be traveling at .8 the speed of light. He would also see the ball traveling a distance of 5 light years. At .8 the speed of light, this would take 4 light years from Alex's point of view. In this example, John's ball clock is working 75% as fast as Alex's.

Did I miss a step or miscalculate along the way?

Doc Al
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Now same example except John is traveling at half the speed of light. Instead of a light clock, there is a ball traveling upwards a distance of three light years, and the speed of the ball is half of the speed of light from John's perspective. Say John's ship is also traveling at half the speed of light. Using the Lorentz addition velocity, Alex would calculate the ball to be traveling at .8 the speed of light.
You are using the wrong velocity addition formula. The velocities are not parallel. (Treat the ball as moving in the y-direction with respect to the ship, and the ship as moving in the x-direction with respect to the other observer.)

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Cool thanks! I will check it out.

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The formula for addition of non parallel velocities, as its stated on the website, is a little tricky for me.

The formula for addition of parallel velocities is: S = (V + U)/ UV/Csquared

What is the formula for addition of non parallel velocities using the symbols S, V, U, and C?

Thank you,

Joe

Mister T
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Now same example except John is traveling at half the speed of light. Instead of a light clock, there is a ball traveling upwards a distance of three light years, and the speed of the ball is half of the speed of light from John's perspective. Say John's ship is also traveling at half the speed of light. Using the Lorentz addition velocity, Alex would calculate the ball to be traveling at .8 the speed of light.

No, not 0.8 c because the two speeds of 0.5 c are not parallel. Take the component of the ball's speed that's parallel to John's motion and use the relativistic formula for velocity addition. The math is probably more than you want to do, but I assure you that when you do it, it works out that the clock is ticking at the same rate as the light clock.

The basic issue is this. Light clocks tick at the same rate as all other clocks, so whatever light clocks do, that's what other clocks do, too. The only room left for objection is that clocks don't really measure time. That's a philosophical objection, not a physical objection. As long as you accept that what clocks measure is time, then you must conclude that time dilation is real. Moreover, tests of time dilation have shown that it's real. Moving objects really do live longer than they would if they weren't moving. Or to say that more precisely, when we measure the lifetime of objects at rest relative to us, and then measure that lifetime when those same objects are moving relative to us, we find that the lifetime is greater when the objects move.

Doc Al
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The formula for addition of non parallel velocities, as its stated on the website, is a little tricky for me.

The formula for addition of parallel velocities is: S = (V + U)/ UV/Csquared

What is the formula for addition of non parallel velocities using the symbols S, V, U, and C?
Well, you could do it that way, but it's easier just to stick to the components of velocity. So the only transformation you really need is for the y-component of the velocity:
$$u_y = \frac{u'_y}{(1 + vu'_x/c^2)\gamma}$$
Where ##u'_x## and ##u'_y## are the components of the ball's velocity in the ship frame and ##u_y## is the y component of the ball's velocity in the other frame.

Since, in the ship frame, the ball moves vertically we can say that ##u'_x = 0##, which simplifies the above transformation to:
$$u_y = \frac{u'_y}{\gamma}$$

So, given that, imagine that the ball bounces through a vertical distance L. Figure out the time it takes according to both observers and compare.

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Thank guys! After playing with the math, I am starting to understand it more! I will need to go through more examples, but I feel like I'm making progress.

Mister T you wrote: "The only room left for objection is that clocks don't really measure time. That's a philosophical objection, not a physical objection." That's a very interesting point! One of the interesting things I've learned from SR is that time from the point of view of physics is, I think, just repetitive motion- e.g. the rotation of the earth around the sun, the ticking of a clock, etc, and that time can be measure by anything that has repetitive motion.

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I realized I'm still confused. Just when I think I'm starting to understand special relativity, I realize I'm far off.

Doc Al, where does this formula which you've posted come from?: The one where you divided by a number with C squared in the denominator? Are we looking for the Y component of John's velocity from Alex's point of view? Once we get that do we need to plug it in to the velocity addition formula?

Mister T, you said "Take the component of the ball's speed that's parallel to John's motion and use the relativistic formula for velocity addition." Isn't the ball traveling at speed zero in the X direction for John?

To make matters worse, I'm not even sure if my example makes sense. I used a 3,4,5 triangle for convenience, and I based the length of the sides based on the height of the light clock and on the distance that John traveled with respect to Alex. It seems like if the triangle is going to illustrate time dilation, the sides should be based on the speed that John is traveling with respect to Alex and on the ball's speed. The 3,4,5 triangle was arbitrary, and the lengths would come out differently based on the speeds of John and the ball.

Ibix
There's a fairly straightforward semi-mindless way of working out what happens in any special relativity problem.

1. Write down the problem specification in one frame.
2. Work out the coordinates of interesting events.
3. Lorentz transform the coordinates into the other frame
4. Think about what that is telling you.

So, step 1. I am at rest in a frame called S, which uses coordinates (x,y,z,t). I have a light clock which is two light seconds long pointed along the y axis, and a ball clock which is one light second long, whose ball does 0.5c in my frame, and which is also aligned along the y axis. One end of both clocks is at the origin. Both the light pulse and the ball bounce off this end at t=0. You fly past at speed v=0.8c in the +x direction. What do you see?

Step 2. Interesting events are when the ball bounces off either end of its clock and when the light pulse bounces off either end of its clock. For the light pulse this happens at ##(x,y,z,t)## coordinates ##e_0=(0,0,0,0)##, ##e_1=(0,2,0,2)## and ##e_2=(0,0,0,4)##. For the ball clock it happens at ##e_3=(0,0,0,0)##, ##e_4=(0,1,0,2)## and ##e_5=(0,0,0,4)##.

Step 3. Use the Lorentz transforms to work out the ##(x',y',z',t')## coordinates that you would assign to the events.

Step 4. Now you know the positions and times of the bounces in your frame it's easy to work out the speeds in your frame.

I'll leave those last two steps to you. There's no substitute for doing it yourself.

You will find, if you break the velocity into x' and y' components, that the x velocity transforms according to the equation Doc Al gave (##u_x'=(u_x-v)/(1-u_xv/c^2)##). The y velocity transforms according to ##u_y'=u_y/\gamma##. In this particular case ##u_x=0##, which makes the first one easy. You can derive these formulae directly from the Lorentz transforms if you want.

Does that help?

Doc Al
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Doc Al, where does this formula which you've posted come from?: The one where you divided by a number with C squared in the denominator?
It's a relativistic velocity transformation: One of the Lorentz transformations for velocity. You can think of it as a version of the 'velocity addition formula' when the velocities are not parallel.

Are we looking for the Y component of John's velocity from Alex's point of view?
Yes.

Once we get that do we need to plug it in to the velocity addition formula?
No. The formula quoted is the velocity addition formula. (What you are calling the 'velocity addition formula' is only valid for the special case where the velocities are parallel.)

Nugatory
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don't we use trigonometry to break a speed into it's x and y components?
We can, but you have to know the angles to do that, and they're frame-dependent so it's not obvious what they will be. Ibix's approach is to calculate the lengths of the sides of the right triangle; from this you could calculate the angles if you needed them, but it's easier to go directly to the x and y components of the velocity. The x component of the velocity is the distance travelled in the x direction divided by the time, and similarly for the y component.

Janus
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Hello. There is a question that I've been trying to understand for about a year. Why is it that the time dilation effect applies equally to all clocks as it does with "light clocks". For example, if John is moving on a space ship with respect to Alex who is on earth, and both are sitting next to a light clock where a beam of light is bouncing up and down, Alex will see John's beam of light moving upwards and horizontally in a diagonal line. Because the speed of light is constant for both, Alex will see John's beam of light taking longer to reach the top than the beam on his own clock, because he sees his own beam is going straight up and down, and he sees John's going up and sideways.

Time dilation with light clocks is affected by the constancy of the speed of light. Say we chose another type of clock, like a ball bouncing up and down instead of a beam of light. The speed of the ball is not constant for both of them. Therefore, if they both use clocks with a ball bouncing up and down, Alex will see John's ball going up a longer path just like in the light clock example. However, Alex will also observe the ball going up faster to compensate for the extra length. Therefore, although time dilation might occur here to, the amount of dilation should be different. Why then, does time dilate for all clocks on John's ship, in Alex's point of view, by the same factor of gamma? Shouldn't the factor be different depending on what type of clock is being used, and what the Lorenztian sum of the velocity is between the speed of John's ship and the speed of the clock?

If what I am saying is true, then the effects would be very strange: i.e. someone could go on a voyage at a very high speed with various clocks (including their biological clock) and come back with some of their clocks far behind those on earth, and other clocks only slightly out of sync.

Thank you.

In a way, you've answered your own question. Imagine the following set up:
John has both a light clock and a "ball clock". He also has a means of physically recording the comparative "ticks" of the clock. Let's say that it is a strip of paper running between two spools. Each time the light hits it, it leaves a mark and the same with the ball. This leaves spaced out marks along the paper strip. If the ball is moving at 0.5 c, you get two light marks per ball mark.

Both Alex and John can watch the marks being made, and at the and of the experiment both Alex, John and the paper are brought back together. If Alex does not see the Ball clock time dilated by the same factor as the light clock, he will see it leaving marks on the paper that are spaced differently than the One per two light marks that John does. When John, Alex and Paper are brought together at the end, you can't have two people standing in the same room looking at the same strip of paper and seeing two different things. Nor can you have the marks of the paper suddenly change for one of the two. Trying to have the two disagree on the comparative tick rates of the two clocks produces physical contradictions. Time dilation occurs not only because light speed is consistent between observers, but also because, in the end the observers have to agree as to what end result occurred if we bring them back to together.

Also, it isn't so much that light speed is consistent as it is that there is a finite speed, c, that is invariant. Light just happens to travel at this speed so it makes it convenient to use in examples. It is the existence of this finite, invariant speed in the universe(Newtonian physics also has an invariant speed, but it is infinite) that causes us to measure time dilation in all processes which have a relative motion with respect to us.

Doc Al
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Thank you very much guys for the help.

Doc Al, I played with the numbers for while this evening, and I worked out a problem using the formula for converting U to U1 which you gave. Then I calculated the comparative time for Alex and John (who is traveling at .5c away from Alex) that it takes for a ball traveling upwards at .5c in John's frame (.4330 in Alex's frame) to reach the top. I then did the same calculation for a light clock and took the comparative times for the beam of light to reach the top, and I get the same answer like in the ball clock example. Its helpful to have worked it out, but I still have intuition problems, and need to work out the derivation for the formula U = U1/Gamma.

Janus that's a very interesting example you gave with the two clocks next to each other. It makes logical sense that if John has two times as many marks on the paper for the light clock as the ball clock, then Alex should see the same thing if they come together. There is still an intuitive problem for me though, and I am wondering there is a logical explanation: Going back to the light clock example using a right triangle, U and U1 are = since C is constant. The reason why there is time dilation is because for Alex the light is traveling up a longer path. As soon as we make U less than C, then Alex will see the ball (or whatever the object is) going up faster than John will see it since Alex can add V (the speed he sees John's ship going) to U and get the total speed S. If U and U1 were the same in the light clock example, shouldn't the comparative times be different when S is greater than U1? Is there some factor that I'm missing which counterbalances the increase in speed of the clock for Alex vs. John?

Doc Al that leads back to the formula of U = U1/gamma. Based on that formula, U is less than U1. But why aren't we adding the speed of John's ship to U to get the total speed? You said that for John, there is no horizontal velocity. However, Alex still perceives one.

Nuqatory, that makes sense. I still am wondering why we only use the y component of John's velocity though.

Ibix, I will try and plug the numbers into the Lorentz transformations.

Doc Al
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Doc Al that leads back to the formula of U = U1/gamma. Based on that formula, U is less than U1. But why aren't we adding the speed of John's ship to U to get the total speed?
We're just finding the y-component of the ball's velocity, which does depend on the speed of the ship. If you calculated the total speed of the ball in Alex's frame, it will indeed be greater that U1.

You said that for John, there is no horizontal velocity. However, Alex still perceives one.
That's true. Alex will measure the horizontal velocity of the ball to be equal to the ship's speed.

I still am wondering why we only use the y component of John's velocity though.
It just makes it easier to calculate the time for the ball's bounce. (The distance along the y-axis is the same in each frame.)

Gold Member
"We're just finding the y-component of the ball's velocity, which does depend on the speed of the ship. If you calculated the total speed of the ball in Alex's frame, it will indeed be greater that U1."

That makes sense. I thought of another problem with that this this morning though: When the clock is a light clock, U1 = C. Because C is the same for Alex and John, how can Alex divide C by gamma?

Thanks

Doc Al
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That makes sense. I thought of another problem with that this this morning though: When the clock is a light clock, U1 = C. Because C is the same for Alex and John, how can Alex divide C by gamma?
Recall that we are just calculating the y-component of the velocity when we use that formula. You are correct: The speed of light is the same for both. But while in John's frame the light moves vertically, in Alex's frame the light moves at an angle. Thus, the vertical component of the velocity is less than c.

Janus
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Thank you very much guys for the help.

Janus that's a very interesting example you gave with the two clocks next to each other. It makes logical sense that if John has two times as many marks on the paper for the light clock as the ball clock, then Alex should see the same thing if they come together. There is still an intuitive problem for me though, and I am wondering there is a logical explanation: Going back to the light clock example using a right triangle, U and U1 are = since C is constant. The reason why there is time dilation is because for Alex the light is traveling up a longer path. As soon as we make U less than C, then Alex will see the ball (or whatever the object is) going up faster than John will see it since Alex can add V (the speed he sees John's ship going) to U and get the total speed S. If U and U1 were the same in the light clock example, shouldn't the comparative times be different when S is greater than U1? Is there some factor that I'm missing which counterbalances the increase in speed of the clock for Alex vs. John?

Let's draw out the right angle example with the two clocks John's velocity with respect to Alex is 0.5c and the ball's speed with respect to John is 0.5c.

On the left is Alex's view and on the right is John's ( with John I separated the paths so they don't overlap each other.) . With John, the light goes straight up and down at c while the ball travels upwards at 0.5c

In Alex's view, both John and his clocks are moving to the right at 0.5c. The light travels on the diagonal at c. Which means that if we break it down into vertical and horizontal components, we get a horizontal component of 0.5c and vertical component of 0.866c. For the ball, the horizontal component is 0.5c and the vertical component is 0.433c. Note that the base of the triangle for the ball is twice that of the triangle for one leg of the light path. This makes sense, since if the ball is traveling upwards at half the speed of the light, John is going to travel twice as far in the time it takes the ball to go from the bottom to the top as he did in the time it took the light to go from the bottom to the top. This gives the speed of the ball along its diagonal path as 0.6614c. This is faster than the 0.433c vertical component, but the longer base of the triangle requires this.

The trick is that in order to get S, the velocity along the blue line by adding V, the velocity of John to U1, the upward velocity of the ball as measured by John, you need to use the formula:

$$S = \sqrt{U1^2+V^2 - U1^2 V^2}$$

(where S, U1 and V are measured in units of c.)

Thus we get
$$S = \sqrt{0.5^2+0.5^2 - 0.5^2 0.5^2} = 0.6614c$$

which is also what you get if you add the horizontal component of the ball's velocity (0.5c) and its vertical component as measured by Alex (.433c) using the Pythagorean theorem:

$$\sqrt{0.5c^2+0.433c^2}= 0.6614c$$

Doc Al
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Doc Al and Janus I think I am finally starting to get this! (I hope!).

The horizontal distance or portion of the horizontal distance traveled by the clock and by John's ship is not what is causing time dilation and has no affect on time dilation. If John's ship traveled one light second in the X direction according to Alex, it traveled zero distance according to John. Alex sees the light and ball traveling infinitely further in the X direction that than John does because John sees zero movement in the X direction. To compensate for that, Alex sees the light/ball traveling infinitely faster in the X direction since for John, he has no movement in the X direction. So, time dilation happens because U is less than U1, and we only need to visualize the fact the the light and ball going upwards at U takes longer to travel up the clock than at U1. We can ignore the distance the ship traveled (which is the X component of the velocity) in doing the calculation.

Does that make sense?

Gold Member
Also, Janus thank you for creating that diagram which is very helpful.

I wanted to clarify my previous post:

Please let me know if my understandibg is now correct, or if there is still a flaw in it: According to Alex, all the clocks on John's ship go out of sync with his once John starts moving with respect to him. However, the ship itself is also a clock, and that does not go out of sync with Alex's clocks. Is that correct? The previous discussions we've had regarding only the need for the Y portion of the velocity led me to this idea.

Janus
Staff Emeritus
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Also, Janus thank you for creating that diagram which is very helpful.

I wanted to clarify my previous post:

Please let me know if my understandibg is now correct, or if there is still a flaw in it: According to Alex, all the clocks on John's ship go out of sync with his once John starts moving with respect to him. However, the ship itself is also a clock, and that does not go out of sync with Alex's clocks. Is that correct? The previous discussions we've had regarding only the need for the Y portion of the velocity led me to this idea.

How exactly would John use his ship as a clock? With the light and ball, he measures how long it takes for them to travel a fixed distance in the Y direction relative to him, but the ship has no relative motion with respect to John.
If he tries to use the relative motion between his ship and Alex, there is the problem the Alex and John don't measure distances in the X directions as being the same. This is why we work with the Y direction in the above examples as both John and Alex agree on that.

For example, let's assume that there is a marker a distance 1 light min(as measured by Alex) from Alex in the x direction. Thus according to Alex, it will take 2 min by his clock for John to travel from Alex's position to the marker at 0.5c. Since he measures John's clock running at a rate of 0.866, he will see it reading 1 min 43.92 sec when it reaches the marker.

For John it works out like this: Alex and the marker have a velocity of 0.5c relative to him, and due to this relative motion and the resulting length contraction, he measures the distance between the marker and Alex as being 51.96 light sec. It take 1 min and 43.92 sec to cross that distance at 0.5c. So he also agrees that 1 min, 43.92 sec passed on his clock between the time he was next to Alex and when he was next to the marker.