thought experiment about time dilation

nonnonenon

My knowledge toward physics stay on high school level, and all i know about relativity are from wiki, so please make the explanation simple~

In wiki, it said it is possible and logical that B's time runs slower relative to A while A's time runs slower to B.

But i still can't understand how this is possible without causing contradiction.

Let's say A and B are relatively stationary and separated apart by distance d, this makes their clocks run at the same rate and display the same time.

Then they start traveling toward each other at the same high speed when both clocks show 3 p.m. They will stop and meet each other at the mid-point between them since both speed are the same.

Since B's time runs slower relative to A while A's time runs slower to B, if A's clock turns 4 p.m. when they meet, what time will B's clock show? after or before 4 p.m.?

PeterDonis

Both clocks will show 4 pm when they meet. The scenario is completely symmetric as you state it, so whatever applies to one of them also applies to the other. That's the short answer, and the correct one, but of course it's not very satisfying as it stands.

So if you want a longer answer, read on, but be warned: it won't be as simple as the above.

You don't give a reference, so I don't know which "wiki" you are reading (Wikipedia? If so, which specific article?). But just saying that "A's clock runs slower relative to B while B's clock runs slower relative to A" is not really a good way to describe what's going on. For one thing, it leads people to ask obvious questions like the one you are asking.

The thing that is missing from the description you got from the wiki is that it's not just apparent clock rates that change with relative motion; simultaneity changes too. That is, which events seem simultaneous to a given observer (which events appear to him to have happened "at the same time") depends on the observer's state of motion. This change in simultaneity has to be *combined* with the apparent change in clock rates, if you want to predict what someone else's clock is going to read when you meet them.

In your scenario, when A and B are at rest relative to each other, their sense of simultaneity is the same. So the two events "A's clock reads 3 pm" and "B's clock reads 3 pm" will seem simultaneous to both A and B.

But once A and B start moving towards each other, their senses of simultaneity are different. So, for example, the event "A's clock reads 3:30 pm" will *not* be simultaneous with the event "B's clock reads 3:30 pm", to either A or B. In fact, A will find that when his clock reads 3:30 pm, B's clock reads something *later* than 3:30 pm "at the same time", according to his sense of simultaneity while he is moving. That means that, according to A, when the two meet, A's clock will have ticked off half an hour from 3:30 pm, but B's clock will have ticked off *less* than half an hour "in the same time"--meaning, from what B's clock read "at the same time" as A's clock read 3:30 pm, according to A.

But how can B's clock read *later* than A's, according to A, when A's clock reads 3:30 pm? Because A's sense of simultaneity changes as soon as he starts moving--that is, right after his clock reads 3:00 pm. So, for example, when A's clock reads 3:01 pm, according to A's sense of simultaneity while he is moving, B's clock will read something *later* than 3:01 pm. But B was supposed to start moving when *his* clock read 3:00 pm, just like A; so A will have to conclude that, according to his sense of simultaneity while he is moving, B started moving *earlier* than 3:00 pm. (That is, earlier by A's clock--the event "B's clock reads 3:00 pm", according to A's sense of simultaneity while he is moving, happens "at the same time" as A's clock reading something *earlier* than 3:00 pm.)

That's how B's clock can "run slower" according to A, but still read 4:00 pm when A's clock reads 4:00 pm; B's clock runs slower, but B started moving *earlier*, according to A's sense of simultaneity while he is moving. So if A wants to predict what B's clock will read when he meets, using the frame of reference he is in while he is moving (and in which B's clock "runs slower"), he has to take into account this change in simultaneity. So A will find that B was moving for *longer* than 1 hour, by A's clock--just enough longer that B's clock, which "runs slow" according to A while he is moving, ticks off exactly 1 hour while B is moving. (And of course, B comes to similar conclusions with regard to A, just with everything reversed.)

This post has gotten pretty long, and I should stop and let you digest it.

nonnonenon

Thank you for the post, it is nice of you to make it as simple as possible and i think i understand you answer

So it means that A will see B start moving then he takes a look at his clock, he will read 2:59 pm, but according to his knowledge of relativity, he knows that at that time B's clock shows 3:00 pm at that instant. When A starts to move at 3 pm according to his clock, he knows B's clock has already passed 3 pm. Since B's clock has less time to pass to reach 4 pm, in order for both clocks to reach 4 pm at the same time, from A's pov, B's clock runs slower so that A's clock can catch up to B's clock bit by bit until they both reach 4 pm at the same time.

Am i interpreting your answer correctly?

ghwellsjr

I think it helps in scenarios like this to provide all the specifics, not just that each clock started moving at 3pm and ended at 4pm. So let's say that in their initial mutual rest frame, the only one I'm going to consider, their speeds were 0.6c towards each other. This gives a time dilation factor of 0.8 which means that while each of their clocks advanced by 1 hour, the coordinate time advanced by 1.25 hours. Since their speed is 0.6c and they each traveled for 1.25 hours, the distance they each traveled is 0.75 light-hours. Now we know the value of d, their initial separation is double that or 1.5 light-hours.

This means that prior to 3pm, they each viewed the other ones clock as being 1.5 hours earlier than their own because it takes 1.5 hours for the image of the other clock to traverse the distance of 1.5 light-hours to get to them.

A 3pm they both instantly accelerate to 0.6c. This does not cause any jump in the reading they see on each others clock but it does instantly change the rate at which they see each others clock tick. The Relativistic Doppler equation tells us that at 0.6c, they will each instantly see the other ones clock tick exactly twice as fast as their own, at least for awhile, because they will not see the other one accelerating until some time later.

When the image of the other one accelerating finally reaches them, the Velocity Addition formula tells us that their relative speed will then be 0.882353c and from that the Relativistic Doppler formula tells us that they will see each others clock tick at four times their own. The point at which they see this happening during their trip is at 45 minutes.

So while each of them sees their own clock advance by 60 minutes during the trip, they have to see the other ones clock advance by 150 minutes (the initial 90 minutes plus the 60 minutes that the trip took) in order to both arrive at 4pm at the same time when they meet. During the first 45 minutes of their own clock advancing, they see the other ones clock advance by 90 minutes (2x) and during the last 15 minutes of their own clock advancing, they see the other ones clock advance by 60 minutes (4x).

To put times on this:
3:00 pm: each clock accelerates to 0.6c and they see 1:30 pm on the other clock and they see the other clock running twice as fast as their own.
3:15 pm: they each see 2:00 pm on the other clock
3:30 pm: they each see 2:30 pm on the other clock
3:45 pm: they each see 3:00 pm on the other clock and they each see the other clock running four times faster than their own.
4:00 pm: they each see 4:00 pm on the other clock as they arrive together at the same point and stop.

This is exactly what each observer actually sees of their own clock versus the other ones clock when they travel at 0.6c. This analysis was done in their initial mutual rest frame. You can pick any other frame(s) to do a similar analysis but it won't change what they each actually see.

*GrammawSally*

Quote by PeterDonis

[...]
The thing that is missing from the description you got from the wiki is that it's not just apparent clock rates that change with relative motion; simultaneity changes too. That is, which events seem simultaneous to a given observer (which events appear to him to have happened "at the same time") depends on the observer's state of motion.
[...]
But once A and B start moving towards each other, their senses of simultaneity are different. So, for example, the event "A's clock reads 3:30 pm" will *not* be simultaneous with the event "B's clock reads 3:30 pm", to either A or B.
[...]

Good explanation, Peter.

To try to further crystallize it for the OP, I'll just emphasize that, as soon each person accelerates, and suddenly starts to move toward the other person, they will each conclude that the other person's clock suddenly jumps ahead in time.

nonnonenon

Quote by ghwellsjr

I think it helps in scenarios like this to provide all the specifics, not just that each clock started moving at 3pm and ended at 4pm. So let's say that in their initial mutual rest frame, the only one I'm going to consider, their speeds were 0.6c towards each other. This gives a time dilation factor of 0.8 which means that while each of their clocks advanced by 1 hour, the coordinate time advanced by 1.25 hours. Since their speed is 0.6c and they each traveled for 1.25 hours, the distance they each traveled is 0.75 light-hours. Now we know the value of d, their initial separation is double that or 1.5 light-hours.

This means that prior to 3pm, they each viewed the other ones clock as being 1.5 hours earlier than their own because it takes 1.5 hours for the image of the other clock to traverse the distance of 1.5 light-hours to get to them.

A 3pm they both instantly accelerate to 0.6c. This does not cause any jump in the reading they see on each others clock but it does instantly change the rate at which they see each others clock tick. The Relativistic Doppler equation tells us that at 0.6c, they will each instantly see the other ones clock tick exactly twice as fast as their own, at least for awhile, because they will not see the other one accelerating until some time later.

When the image of the other one accelerating finally reaches them, the Velocity Addition formula tells us that their relative speed will then be 0.882353c and from that the Relativistic Doppler formula tells us that they will see each others clock tick at four times their own. The point at which they see this happening during their trip is at 45 minutes.

So while each of them sees their own clock advance by 60 minutes during the trip, they have to see the other ones clock advance by 150 minutes (the initial 90 minutes plus the 60 minutes that the trip took) in order to both arrive at 4pm at the same time when they meet. During the first 45 minutes of their own clock advancing, they see the other ones clock advance by 90 minutes (2x) and during the last 15 minutes of their own clock advancing, they see the other ones clock advance by 60 minutes (4x).

To put times on this:
3:00 pm: each clock accelerates to 0.6c and they see 1:30 pm on the other clock and they see the other clock running twice as fast as their own.
3:15 pm: they each see 2:00 pm on the other clock
3:30 pm: they each see 2:30 pm on the other clock
3:45 pm: they each see 3:00 pm on the other clock and they each see the other clock running four times faster than their own.
4:00 pm: they each see 4:00 pm on the other clock as they arrive together at the same point and stop.

This is exactly what each observer actually sees of their own clock versus the other ones clock when they travel at 0.6c. This analysis was done in their initial mutual rest frame. You can pick any other frame(s) to do a similar analysis but it won't change what they each actually see.

then the explanation you gave is opposite to peter's one.
Peter said they both see other's clock runs slower
but you said they see other clock runs faster
peter said if i were A, i would see B starts earlier than me
you said if i were A, i would see B starts later than me ( from what you said, it means that i would think B starts when my time is 3:45 pm?)

So which one is right? or you two are just saying the something i am the one who gets it wrong

PeterDonis

Quote by nonnonenon

Peter said they both see other's clock runs slower
but you said they see other clock runs faster

The word "see" can mean two different things. I was using it to mean "each one *calculates* that the other's clock is running slow", meaning after taking into account, and correcting for, the time delay for light signals traveling between the two observers. ghwellsjr is using it to mean "each one actually *sees* the other's clock running faster", meaning the actual images they see, without making any corrections for light travel time. Both of our descriptions are correct, given our different usages of the word "see".

nonnonenon

So if A and B have installed a special CCTV that can transfer signal instantly on each other which allows them to see the time of other's clocks, and with this condition, my interpretation of your explanation on post #3 will be correct?

harrylin

Quote by nonnonenon

So if A and B have installed a special CCTV that can transfer signal instantly on each other which allows them to see the time of other's clocks [..]

Sorry, with such a scenario SR cannot work - SR relies on the fact that no signal can be transferred faster than the speed of light, otherwise contradictions occur. With a special CCTV that can transfer signal instantly, relativity breaks down.

And perhaps that's the clue to the answer that you need. Mutual time dilation works because the two reference systems define distant time differently; and the "measured" moving clock rate depends on that definition. We cannot know "true" distant time.

Addendum: I now see that the others already explained this more elaborately; but in case you got lost in their long explanations, my summary may still be helpful.

*Vandam*

Quote by PeterDonis

Both clocks will show 4 pm when they meet. The scenario is completely symmetric as you state it, so whatever applies to one of them also applies to the other. That's the short answer, and the correct one, but of course it's not very satisfying as it stands.

So if you want a longer answer, read on, but be warned: it won't be as simple as the above.

You don't give a reference, so I don't know which "wiki" you are reading (Wikipedia? If so, which specific article?). But just saying that "A's clock runs slower relative to B while B's clock runs slower relative to A" is not really a good way to describe what's going on. For one thing, it leads people to ask obvious questions like the one you are asking.

The thing that is missing from the description you got from the wiki is that it's not just apparent clock rates that change with relative motion; simultaneity changes too. That is, which events seem simultaneous to a given observer (which events appear to him to have happened "at the same time") depends on the observer's state of motion. This change in simultaneity has to be *combined* with the apparent change in clock rates, if you want to predict what someone else's clock is going to read when you meet them.

In your scenario, when A and B are at rest relative to each other, their sense of simultaneity is the same. So the two events "A's clock reads 3 pm" and "B's clock reads 3 pm" will seem simultaneous to both A and B.

But once A and B start moving towards each other, their senses of simultaneity are different. So, for example, the event "A's clock reads 3:30 pm" will *not* be simultaneous with the event "B's clock reads 3:30 pm", to either A or B. In fact, A will find that when his clock reads 3:30 pm, B's clock reads something *later* than 3:30 pm "at the same time", according to his sense of simultaneity while he is moving. That means that, according to A, when the two meet, A's clock will have ticked off half an hour from 3:30 pm, but B's clock will have ticked off *less* than half an hour "in the same time"--meaning, from what B's clock read "at the same time" as A's clock read 3:30 pm, according to A.

But how can B's clock read *later* than A's, according to A, when A's clock reads 3:30 pm? Because A's sense of simultaneity changes as soon as he starts moving--that is, right after his clock reads 3:00 pm. So, for example, when A's clock reads 3:01 pm, according to A's sense of simultaneity while he is moving, B's clock will read something *later* than 3:01 pm. But B was supposed to start moving when *his* clock read 3:00 pm, just like A; so A will have to conclude that, according to his sense of simultaneity while he is moving, B started moving *earlier* than 3:00 pm. (That is, earlier by A's clock--the event "B's clock reads 3:00 pm", according to A's sense of simultaneity while he is moving, happens "at the same time" as A's clock reading something *earlier* than 3:00 pm.)

That's how B's clock can "run slower" according to A, but still read 4:00 pm when A's clock reads 4:00 pm; B's clock runs slower, but B started moving *earlier*, according to A's sense of simultaneity while he is moving. So if A wants to predict what B's clock will read when he meets, using the frame of reference he is in while he is moving (and in which B's clock "runs slower"), he has to take into account this change in simultaneity. So A will find that B was moving for *longer* than 1 hour, by A's clock--just enough longer that B's clock, which "runs slow" according to A while he is moving, ticks off exactly 1 hour while B is moving. (And of course, B comes to similar conclusions with regard to A, just with everything reversed.)

This post has gotten pretty long, and I should stop and let you digest it.

Your 'seem' and 'apparent' do not make sense to me (because I'm not a positivist), so be it, but your 'sense of simultaneity'... Waaw!, that really blows my socks off! Never read that one before :)

harrylin

Quote by Vandam

Your 'seem' and 'apparent' do not make sense to me (because I'm not a positivist), so be it, but your 'sense of simultaneity'... Waaw!, that really blows my socks off! Never read that one before :)

?? A positivists is someone who hardly uses 'seem' and 'apparent' - positivists tend to take things at face value.

- https://en.wikipedia.org/wiki/Positivism

*Vandam*

Quote by harrylin

?? A positivists is someone who hardly uses 'seem' and 'apparent' - positivists tend to take things at face value.

- https://en.wikipedia.org/wiki/Positivism

Well, why use that type of phrasing then? Because they are optical illusions? That's even worse.

I should not have reacted to Peter's post. You'll make a philosophical issue of it. That's off topic. But I think his way of phrasing is rather off topic.

harrylin

Quote by Vandam

Well, why use that type of phrasing then? Because they are optical illusions? That's even worse.

I should not have reacted to Peter's post. You'll make a philosophical issue of it. That's off topic. But I think his way of phrasing is rather off topic.

There is no claim about "optical illusions" but with disagreeing existential "is" statements one creates self contradictions - that's why. However, such a discussion about rather standard* phrasings is indeed off topic here; if you like you can start it as a topic - but please first search this site, it has been discussed already and perhaps one of the old threads is still open.

* you can even find "appear" (instead of "is") here:
http://www.fourmilab.ch/etexts/einstein/specrel/www/

*Vandam*

Quote by harrylin

There is no claim about "optical illusions" but with disagreeing existential "is" statements one creates self contradictions -

?? Definitely not more than a 'seems' and 'apparent' and 'sense' terminology!

That's why. However, such a discussion about rather standard phrasings

You mean standard phrasings that everybody copy from everybody, without thinking about it because that's philosophy... I see what you mean.

O.K. topic closed

is indeed off topic here; if you like you can start it as a topic - but please first search this site, it has been discussed already and perhaps one of the old threads is still open.

PeterDonis

Quote by Vandam

'sense of simultaneity'

There's nothing mysterious about it; it's just a way of referring to the time coordinate a given observer assigns to events, in the inertial frame in which he is at rest. I agree it's a clumsy expression, but there isn't really a non-clumsy way of referring to relativistic concepts in English. I could have stated it mathematically, but I wasn't sure how much mathematical background the OP has.

PeterDonis

Quote by Vandam

Well, why use that type of phrasing then? Because they are optical illusions?

No, of course not. Again, those terms were ways of referring to the coordinates that observers assign to events, in the inertial frames in which they are at rest. If there is a non-clumsy way of doing that in English, I'm not aware of it. If you know of one, please enlighten us all.

PeterDonis

Quote by Vandam

You mean standard phrasings that everybody copy from everybody, without thinking about it because that's philosophy

No, because it helps to have *some* standard phrasing, even if it's clumsy. Do you know of a non-clumsy one?

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nonnonenon	So if A and B have installed a special CCTV that can transfer signal instantly on each other which allows them to see the time of other's clocks, and with this condition, my interpretation of your explanation on post #3 will be correct?