# Confused About the Length of Time Between Ticks in Time Dilation

I am confused about time dilation. I understand that a common pedagogical device used is a light clock, in which a pulse of light flashes back and forth between two objects. I understand that when an observer in a vehicle carrying such a contraption is moving, to an external observer, as the light has to traverse a longer, diagonal path, and it cannot exceed speed c, it must take more time to make a tick, and each tick of the clock would take a longer time. I understand all of that.

The issue I'm having trouble comprehending is that, somehow, taking more time between ticks would result in less time elapsing for the moving observer than for the stationary observer. And this has nothing to do with the principle of relativity or the twins paradox or anything like that, as I know that the acceleration of the moving twin furnishes the answer to that particular paradox.

What I am confused about is that, since the moving observer's clock has more time between ticks, wouldn't MORE time, not less, elapse for the moving observer than for the stationary one? I cannot wrap my mind around how taking a *longer time* to tick could possibly translate into less time, overall, elapsed. Is the number of ticks so drastically reduced that it overcompensates, perhaps? Or is ticking in one direction a lot more time-consuming than ticking in the opposite direction, and the shorter half of the tick overcompensates? Or something else? What is it that I am missing here?

Basically, to sum up, I feel like it logically seems to me that, in time dilation, the reverse of what is said to happen ought to happen; the stationary observer ought to have less time elapse for them, and the moving observer ought to have more time elapse for them. Any explanations? Thank You.

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phyzguy
First of all, you need to understand that there is no "moving observer" and "stationary observer". If two observers are moving relative to one another, each one sees themselves as stationary and the other one as moving, and each one sees the other one's clock as ticking slower than their own. Second, imagine that you see your own clock going tick-tock-tick-tock.., and I go whizzing by you and you see my clock going tick.......tock.......tick.......tock.......tick..., won't you conclude that my clock is running more slowly than yours?

When an observer in frame A view's the light clock in frame B which is in relative motion to A, it takes more time between ticks as you mention. Thus, in the amount of time it takes for 10 ticks to happen on A's light clock, A will only observe, say 9 ticks on B's clock. Thus, less time has passed in B's frame relative to A's frame, from A's perspective. B doesn't notice anything, time just passes as normal for him. And in fact, it will look as if A's time is slower to him.

That's not the issue I have, though. I totally get it that each will perceive the other's clock as running slower. My issue is that I feel that someone's clock running slower would result in more, rather than less, time passing for them, but, in reality, somehow, it is the reverse.

For example, they say someone moving at 99.99% of the speed of light experiences less time than someone standing still because their clock runs slower, but it seems to me that, since there is a greater time between ticks of the clock, if their clock is running slower, they should be experiencing MORE time than someone standing still.

How does a beam of light taking a LONGER time to traverse a distance in one "tick-tock" possibly translate into LESS time passing in comparison to light taking a SHORTER time to traverse the distance of one "tick-tock"? That is really the issue I have here.

• nitsuj
Mister T
Gold Member
The issue I'm having trouble comprehending is that, somehow, taking more time between ticks would result in less time elapsing for the moving observer than for the stationary observer.
You are correct. But here's why you're confused. The phrase "moving clocks run slow" can result in the same type of confusion. To sort it out you need to focus on events. And on proper time, which is the time that elapses between two events that occur at the same place.

So, let's take an example where the speed ##v=\frac{\sqrt{3}}{2}\ c##, or about ##0.866\ c##. At this speed ##\gamma=\frac{1}{\sqrt{1-(v/c)^2}}=2##.

Now choose two events that occur in the same place. Let's have one event be the start of a one-hour physics class and the other event be the end of that physics class. These two events occur at the same place, the physics classroom, so the one hour is a proper time. We would write ##\Delta \tau=1##. If an alien could somehow observe these two events as he whizzes past the classroom at a speed of ##0.866\ c## he would measure an elapsed time ##\Delta t=\gamma \Delta \tau## of two hours.

The alien would conclude that the clock on the classroom wall must be running slow because during the class that lasted two hours only one hour elapsed on that classroom clock. Thus you see that in this example the clock that was running slow had less time elapse on it than the amount of time that elapsed on the moving clock. Thus it's not the moving clock that was running slow, it was the stationary clock in the classroom. Of course, the alien could claim that he had the stationary clocks (see below why he would need two clocks) and to him the clock on the classroom wall was moving, and so to him it was the moving clock that ran slow.

Now, you can do the same thing with a tick of the light clock. A beam of light leaves one mirror, bounces off the other mirror, and returns to the original mirror. One event is the leaving of the light beam, and the other event is the return of the light beam to that same mirror. Thus the time it took for the beam to make a round trip is a proper time ##\Delta \tau##. That's in a frame of reference where the clock is not moving!

Now, if you want to understand how the student in the classroom can observe the alien's clock running slow, you have to realize that the alien needs two clocks. One clock passes the classroom when class begins, but the other one passes the classroom when the class ends. The alien synchronizes the two clocks, but the student doesn't agree that he did it correctly. Thus when the second clock is passing by the classroom just as the class ends, it shows that two hours have elapsed since class began, but to the student it was not synchronized correctly. The student would say that had it been synchronized correctly it would show an elapsed time of only one-half hour. So the student sees the alien's clocks as not only running slow, but out of sync.

russ_watters
Mentor
...since there is a greater time between ticks of the clock....
That's a contradiction. Clocks tell time. The time between ticks of a clock is 1 second. Period.

• Grinkle
You are correct. But here's why you're confused. The phrase "moving clocks run slow" can result in the same type of confusion. To sort it out you need to focus on events. And on proper time, which is the time that elapses between two events that occur at the same place.

So, let's take an example where the speed ##v=\frac{\sqrt{3}}{2}\ c##, or about ##0.866\ c##. At this speed ##\gamma=\frac{1}{\sqrt{1-(v/c)^2}}=2##.

Now choose two events that occur in the same place. Let's have one event be the start of a one-hour physics class and the other event be the end of that physics class. These two events occur at the same place, the physics classroom, so the one hour is a proper time. We would write ##\Delta \tau=1##. If an alien could somehow observe these two events as he whizzes past the classroom at a speed of ##0.866\ c## he would measure an elapsed time ##\Delta t=\gamma \Delta \tau## of two hours.

The alien would conclude that the clock on the classroom wall must be running slow because during the class that lasted two hours only one hour elapsed on that classroom clock. Thus you see that in this example the clock that was running slow had less time elapse on it than the amount of time that elapsed on the moving clock. Thus it's not the moving clock that was running slow, it was the stationary clock in the classroom. Of course, the alien could claim that he had the stationary clocks (see below why he would need two clocks) and to him the clock on the classroom wall was moving, and so to him it was the moving clock that ran slow.

Now, you can do the same thing with a tick of the light clock. A beam of light leaves one mirror, bounces off the other mirror, and returns to the original mirror. One event is the leaving of the light beam, and the other event is the return of the light beam to that same mirror. Thus the time it took for the beam to make a round trip is a proper time ##\Delta \tau##. That's in a frame of reference where the clock is not moving!

Now, if you want to understand how the student in the classroom can observe the alien's clock running slow, you have to realize that the alien needs two clocks. One clock passes the classroom when class begins, but the other one passes the classroom when the class ends. The alien synchronizes the two clocks, but the student doesn't agree that he did it correctly. Thus when the second clock is passing by the classroom just as the class ends, it shows that two hours have elapsed since class began, but to the student it was not synchronized correctly. The student would say that had it been synchronized correctly it would show an elapsed time of only one-half hour. So the student sees the alien's clocks as not only running slow, but out of sync.
Alright, interesting, but why would the alien need two clocks, and why would the student see the alien's clocks as being out of sync?

Janus
Staff Emeritus
Gold Member
That's not the issue I have, though. I totally get it that each will perceive the other's clock as running slower. My issue is that I feel that someone's clock running slower would result in more, rather than less, time passing for them, but, in reality, somehow, it is the reverse.

For example, they say someone moving at 99.99% of the speed of light experiences less time than someone standing still because their clock runs slower, but it seems to me that, since there is a greater time between ticks of the clock, if their clock is running slower, they should be experiencing MORE time than someone standing still.

How does a beam of light taking a LONGER time to traverse a distance in one "tick-tock" possibly translate into LESS time passing in comparison to light taking a SHORTER time to traverse the distance of one "tick-tock"? That is really the issue I have here.
According to the person "standing still" the "moving" clock takes longer to complete one tick than his own clock. So let's say his clock take one sec to tick, then he will measure the moving clock as taking longer than one second. A person traveling with the "moving clock" measures that clock as taking 1 sec to tick.
So let's say that our "standing still" observer notes that the moving clock takes 1.5 secs per tick as compared to 1 sec per tick for his own. Then after his clock has ticked off 15 times, he see the moving clock only tick off 10 time. And while he aged 15 sec, the moving observer only aged 10 sec. If someone has aged 10 sec while you aged 15 sec, he aged slower than you did.

Ibix
If I see your clock ticking slowly I also see your brain processes running slowly. Therefore, to you, one second elapses betwen ticks of you clock, always.

So if my clock ticks ten times while yours only ticks eight then you think eight seconds have passed while I think ten seconds have passed. You see nothing odd about your clock, so you experienced less time than I did.

Mister T
Gold Member
Alright, interesting, but why would the alien need two clocks,
One for each event.

and why would the student see the alien's clocks as being out of sync?
Relativity of simultaneity.

• SiennaTheGr8
Grinkle
Gold Member
As @russ_watters pointed out, this is your problem.

taking a *longer time* to tick
This line of reasoning invokes a preferred frame that is true time against which moving clocks can to be compared to resolve questions like 'why do we call this slower and we don't call it faster'. There is no such frame in relativity.

robphy
Homework Helper
Gold Member
That's not the issue I have, though. I totally get it that each will perceive the other's clock as running slower. My issue is that I feel that someone's clock running slower would result in more, rather than less, time passing for them, but, in reality, somehow, it is the reverse.

For example, they say someone moving at 99.99% of the speed of light experiences less time than someone standing still because their clock runs slower, but it seems to me that, since there is a greater time between ticks of the clock, if their clock is running slower, they should be experiencing MORE time than someone standing still.

How does a beam of light taking a LONGER time to traverse a distance in one "tick-tock" possibly translate into LESS time passing in comparison to light taking a SHORTER time to traverse the distance of one "tick-tock"? That is really the issue I have here.
A spacetime diagram on rotated graph paper might help.
Light signals travel along the rotated grid lines.
The light-clock's light-signal reflects off the light-clock worldlines [not shown].
When that reflection is received by an observer's worldline,
that marks "one tick of that clock [elapsed since the last transmission from that worldline]".
Along the lines of what others have been saying, at this instant Alice has aged 5 years, she will say Bob has aged 3 years [because Alice says that ##P_{simA}## on Bob's worldline is simultaneous the event ##P## on her worldline].

You might ask how I chose this "scale" for Bob's light-clock diamonds...
• math answer: Bob's diamonds occupy the same area since the first reception events from O lie on a hyperbola centered at O (the unit-hyperbola in Minkowski spacetime).
• physics answer: this scaling satisfies the principle of relativity because if each observer agreed to send a light-signal "1 year after they separated", each observer will report the same time stamp on their clock ["3", the doppler factor for this case of (4/5)c] when the other observer's signal is received. Mister T
Gold Member
That's not the issue I have, though. I totally get it that each will perceive the other's clock as running slower. My issue is that I feel that someone's clock running slower would result in more, rather than less, time passing for them, but, in reality, somehow, it is the reverse.
After digesting some of the other replies I think I now better understand your question. Let's say that as I sit here my friend sits next to me as we attend a physics class. The class lasts for one hour and my watch verifies that between two events (start and end of class) one hour elapses. My friend's watch runs slow so that on his watch only one-half hour has elapsed between those same two events. Thus according to my friend less time has elapsed while we attended class, not more!

Your confusion arises if instead of focusing on the same two events (start and end of physics class) we focus on events that are different for each of us. For example 4 pm and 5 pm on our watches. For me one hour passes between those events. For my friend two hours elapses between those events because his watch is malfunctioning.

Note that these are different pairs of events. If the first event is the start of class, the second event can't be the end of class for my friend. But it can be for me.

But in relativity theory watches and clocks don't run slow because they are malfunctioning. So my analogy doesn't apply there. But hopefully it helps you resolve your paradox. In my opinion if you're not confronting paradoxes you're not learning this stuff.

That's not the issue I have, though. I totally get it that each will perceive the other's clock as running slower. My issue is that I feel that someone's clock running slower would result in more, rather than less, time passing for them, but, in reality, somehow, it is the reverse.
I'm presuming you're using the light clock as a visual and in turn seeing clearly how the photon travels a further distance when said clock is viewed in motion. Yes the photon travels a further distance....due to this further distance traveled it manages fewer tick tocks when returned at rest to a comparative clock, in turn has AGED less.

If you want to call that "more time" go ahead, but as you see that doesn't "feel" right. It was more length that the photon traveled, how much more? Equivalent to the differential in accumulated tick tocks. Clocks measure time, not length (though can do both with this "perfect" clock"). In relative motion this is the "trade off" because c is invariant.

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I'm presuming you're using the light clock as a visual and in turn seeing clearly how the photon travels a further distance when said clock is viewed in motion. Yes the photon travels a further distance....due to this further distance traveled it manages fewer tick tocks when returned at rest to a comparative clock, in turn has AGED less.

If you want to call that "more time" go ahead, but as you see that doesn't "feel" right. It was more length that the photon traveled, how much more? Equivalent to the differential in accumulated tick tocks. Clocks measure time, not length (though can do both with this "perfect" clock"). In relative motion this is the "trade off" because c is invariant.
But here's the thing, though: Theoretically, one could always shorten the distance between the mirrors, making more tick-tocks occur. This would mean that, even within one tick-tock, time would still elapse. Am I getting this wrong?

russ_watters
Mentor
But here's the thing, though: Theoretically, one could always shorten the distance between the mirrors, making more tick-tocks occur. This would mean that, even within one tick-tock, time would still elapse. Am I getting this wrong?
Clock tick length is arbitrarily chosen based on unit standard and measurement precision needed. You don't get taller by measuring your height in feet instead of meters. Nor are you required to only measure height in integer numbers of feet. Same thing here.

• nitsuj
Mister T
Gold Member
If you want to call that "more time" go ahead, but as you see that doesn't "feel" right.
When a light clock is in motion relative to an observer, it will take more time between ticks than if it were at rest.

To an observer at rest relative to a light clock, the time between ticks is a proper time ##\Delta \tau##. All observers in relative motion will measure a time between ticks that is larger than ##\Delta \tau##.

But here's the thing, though: Theoretically, one could always shorten the distance between the mirrors, making more tick-tocks occur. This would mean that, even within one tick-tock, time would still elapse. Am I getting this wrong?
No you're getting it right, but it doesn't matter! If you have ##n## ticks of the clock the proper time elapsed is just ##n\ \Delta \tau##, regardless of the distance between the mirrors.

But here's the thing, though: Theoretically, one could always shorten the distance between the mirrors, making more tick-tocks occur. This would mean that, even within one tick-tock, time would still elapse. Am I getting this wrong?
A "tick tock" is proper time, I thought this was understood already. With consideration to the COMPARATIVE length the photon travels, the AT REST length it travels is the proper time you experience. ( i should stop calling that distance in this context)

Ibix
But here's the thing, though: Theoretically, one could always shorten the distance between the mirrors, making more tick-tocks occur. This would mean that, even within one tick-tock, time would still elapse. Am I getting this wrong?
Have you realised that all clocks tick slowly if they are moving with respect to you? That includes a moving observer's heartbeat and brain processes. You would have to muck around with multiple cameras to do it, but you would be able to construct a movie of a moving observer that looks like a video running at half speed. Sure, the clocks tick slowly but someone moving along with it would react and move equally slowly. So I'm not surprised that they see their clocks as normal.

Certainly the moving observer could build a broken clock that ticks at a different rate. But, as Russ points out, you don't change reality by changing your measurement unit.

Someone moving at 99.99% of the speed of light experiences less time than someone standing still because their clock runs slower.
It doesn't matter how fast or slow the clocks tick. Even if both the moving and the standing still clocks are ticking at the correct rate, less time may pass for him than someone in another frame. Both clocks will be correct but will not agree with each other.

morrobay
Gold Member
The proper time between ticks in S frame and S' frame is equal. The reason for less elapsed time in S' clock : is related to length contraction. A lessor distance traveled in spacetime.
l = l0 √ 1-v2 . then as seen from S the elapsed time in S' , t' = l0√1-v2 / v
Recall that t' = γ( t - vx). Let x = l0 = distance traveled. and let t = l0/v
Then equating t' = l0/v - vl0 / √1-v2 = l0 √1-v2 / v

= l0(1-v2) = l0 (1-v2)

morrobay
Gold Member
The proper time between ticks in S frame and S' frame is equal. The reason for less elapsed time in S' clock : is related to length contraction. A lessor distance traveled in spacetime.
l = l0 √ 1-v2 . then as seen from S the elapsed time in S' , t' = l0√1-v2 / v
Recall that t' = γ( t - vx). Let x = l0 = distance traveled. and let t = l0/v
Then equating t' = l0/v - vl0 / √1-v2 = l0 √1-v2 / v

= l0(1-v2) = l0 (1-v2)
The last line at bottom is just proof of equality by cross multiplying