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Thought experiment about time dilation

  1. Aug 18, 2012 #1
    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.?
  2. jcsd
  3. Aug 19, 2012 #2


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    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. :wink: 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.
  4. Aug 19, 2012 #3
    Thank you for the post, it is nice of you to make it as simple as possible and i think i understand you answer:smile:

    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?
  5. Aug 19, 2012 #4


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    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.
    Last edited: Aug 19, 2012
  6. Aug 19, 2012 #5
    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.
  7. Aug 19, 2012 #6
    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:confused:
  8. Aug 19, 2012 #7


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    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".
  9. Aug 20, 2012 #8
    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?
  10. Aug 20, 2012 #9
    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.
    Last edited: Aug 20, 2012
  11. Aug 20, 2012 #10
    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 :)
  12. Aug 20, 2012 #11
    ?? A positivists is someone who hardly uses 'seem' and 'apparent' - positivists tend to take things at face value. :wink:
    - https://en.wikipedia.org/wiki/Positivism
  13. Aug 20, 2012 #12
    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.
  14. Aug 20, 2012 #13
    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:
    Last edited: Aug 20, 2012
  15. Aug 20, 2012 #14
    ?? Definitely not more than a 'seems' and 'apparent' and 'sense' terminology!
    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
  16. Aug 20, 2012 #15


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    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.
  17. Aug 20, 2012 #16


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    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.
  18. Aug 20, 2012 #17


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    No, because it helps to have *some* standard phrasing, even if it's clumsy. Do you know of a non-clumsy one?
  19. Aug 20, 2012 #18
    In theory, their clocks tick slower to their partners because of time dilation.
    But in reality, you have to take Relativistic Doppler effect into consideration if you want to simulate how they see other's clocks. And this makes their clocks tick faster to their partners.

    Is this version correct?
  20. Aug 20, 2012 #19


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    I would put it the other way around. What each observer actually sees--meaning, the images that actually reach each observer from the other, containing the actual images of clock readings--is what is directly observed. (It is not "simulated".) The relativistic doppler effect tells you about that without having to bring in time dilation at all; all you need to know is the relative velocity of the two observers. And, as you say (and as ghwellsjr explained in detail), each observer in this scenario actually sees images of the other observer's clock running faster than his own.

    However, each observer knows that the images he is seeing at a given instant took some time to get to him (at least, that's true until the two observers meet up at 4:00 pm, by both their clocks, at which point they are at the same spatial location). When each observer corrects for that, he finds that the other observer's clock is running slower than his own while they are in relative motion, by a factor equal to the time dilation factor based on their relative velocity. This is not "in theory"--it's what each observer will actually conclude when he corrects for light travel time.

    The above is, I believe, a sufficient answer to your question; but I feel compelled to say some more, though it might only add confusion. If you find that the above is enough, feel free to ignore the rest of this post. :redface:

    Which answer is correct "in reality"? IMO, if you feel you have to ask that question, you're thinking about it wrong. If two observers are spatially separated, there is no single answer to the question of whether either one's clock is "running slower" or "running faster"; it depends on how you want to define your terms. The only questions that have definite, unique answers are questions about invariants.

    Things that are directly observed are invariants: for example, "what clock reading on B's clock is contained in the image of B that observer A receives, at the instant when A's clock reads 3:30 pm?" Also, when two observers are *not* spatially separated (as A and B are not at 4:00 pm), they can directly observe each other's clock with no light travel time delay, so those observations have to be invariant--for example, the fact that A's and B's clocks both read 4:00 pm when they meet.

    But for example, if A asks himself, at the instant when his clock reads 3:30 pm, "what time is B's clock reading *right now*?" the answer will depend on how he wants to define "right now". The image he is receiving from B at that instant will show a clock reading; that, as above, is a direct observable. A can adjust that clock reading for the light travel time if he wants, and the result of that computation will be "B's clock reading right now" for A, according to the usual convention for constructing A's frame of reference while he is moving. But there is no principle of physics that *requires* A to adopt the result of that computation as "B's clock reading right now".

    Does that mean that time dilation is "not real"? Of course not. Consider the standard "twin paradox" scenario: observer T sits at rest in his spaceship, floating freely, while observer S flies away at relativistic speed, then turns around and comes back. S's clock will have less time elapsed than T's; that's an invariant physical fact, and the usual explanation is time dilation. (There's a lot lurking beneath the "usual explanation", of course; but it's still essentially correct.) So yes, time dilation is "real". But notice that, to show that it was "real" here, I gave a scenario where it's a direct observable--T and S both pass through the same pair of events, at the start and end of the trip, so the difference in elapsed times on their clocks is directly observed by both of them.

    I hope this didn't muddy the waters too much. :redface:
  21. Aug 21, 2012 #20

    Thanks for the link to Einstein's 1905 paper. The english translation. I hope you ever read the original german paper:
    http://nausikaa2.mpiwg-berlin.mpg.de/cgi-bin/toc/toc.x.cgi?dir=Einst_Zurel_de_1905&step=thumb [Broken]
    I did it for you. I speak flemish which is close to german. That helps a lot.

    Let me first show you the verbs in the english and german versions:

    It is known that Maxwell's electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena.

    Daß die Elektrodynamik Maxwells -- wie dieselbe gengen-
    wärtig aufgefaßt zu werden pflegt -- in ihrer Anwendung auf
    bewegte Körper zu Asymmetrien führt, welche den Phänomenen
    nicht anzuhaften scheinen, ist bekannt.

    It might appear possible to overcome all the difficulties attending the definition of “time” by substituting “the position of the small hand of my watch” for “time.”

    Es könnte scheinen, daß alle die Definition der ,,Zeit“ be-
    treffenden Schwierigkeiten dadurch überwunden werden könnten,

    Thus, whereas the Y and Z dimensions of the sphere (and therefore of every rigid body of no matter what form) do not appear modified by the motion, the X dimension appears shortened in the ratio ...

    Während also die Y - und Z-Dimension der Kugel (also
    auch jedes starren Körpers von beliebiger Gestalt) durch die Be-
    wegung nicht modifiziert erscheinen, erscheint die X-Dimension
    im Verhältnis ...

    We still have to find the amplitude of the waves, as it appears in the moving system. If we call the amplitude of the electric or magnetic force A or A' respectively, accordingly as it is measured in the stationary system or in the moving system, we obtain ...

    Wir haben nun noch die Amplitude der Wellen, wie
    dieselbe im bewegten System erscheint, zu suchen. Nennt
    man A bez. A' die Amplitude der elektrischen oder magne-
    tischen Kraft im ruhenden bez. im bewegten System gemessen,
    so erhält man ...

    It follows from these results that to an observer approaching a source of light with the velocity c, this source of light must appear of infinite intensity.

    Es folgt aus den entwickelten Gleichungen, daß für einen
    Beobachter, der sich mit der Geschwindigkeit V einer Licht-
    quelle näherte, diese Lichtquelle unendlich intensiv erscheinen


    You noticed that Einstein used two different verbs: 'scheinen' and 'erscheinen'. He doesn't mix these at random. They have different meanings:

    'Sheinen' means: illusion - an appearance that does not correspond to reality - it appears so, but it may not be true - what you see is mere appearance - only outward show, things are not what they seem to be, etc. (Anschein= farce, sham, make-believe, pretence etc...)

    'Erscheinen' is more: as it shows, come to light, as it is, etc.

    In the english version 'sheinen' and 'erscheinen' are translated by one verb only: 'appear'. Strictly speaking the translation is not wrong (ask google to translate the english words and somehow you will find 'appear'), but the very important difference in meaning in german disappears in the english translation. Or at least 'might very well' get lost. I suppose that in english one can use the verb 'appear' in both meanings as long as the context makes clear what the semantics are. In the english 1905 paper translation that's not so obvious as in the original german text. Prove is that in thousands of texts dealing with SR the english 'appears' is often replaced by 'seems', which is a synonym of 'appears', but not the correct one to match the german significance. 'Seems' refers to 'scheins' (= illusion). Dalespam's use of 'apparent' (= seeming, not proven real, illusive, illusory, likely, ostensible) is also prove of this, otherwise there would be no need to add that adjective. And his 'sense' of simultaneity is superb poetry, but no physics.
    (The same mistake occurs in other translations, because a lot of them are translations of the/an english text. I will not go into that.)
    Worse is that authors of those ambiguous texts (because of the use of 'appear' without proper explanation, or the word 'seem'), are probably not aware of the real significance of SR: trains ARE shorter, events ARE not simultaneous for one observer and ARE simultaneous for the other, meaning both observers ARE in different 3D worlds. etc. Those authors (not unlike many PF members) hide themselves in a type of Lorentz Ether Theory interpretation of the Lorentz Transformations as illusionary abstract calculations, because it matches perfectly the incorrect 'seems' interpretation of the german 'erscheins'. Unfortunately all those hundreds of thousands of people over the last 100 years are wrong. That's the most dreadfull and horrible scenario Einstein could ever imagine.
    I hope I made my point clear why I get extremely nervous, with a sense of (to say the least) acute desperation, when I am confronted with a text using 'appear' vocabulary. (There is a tree in front of you. Nobody says that a tree appears in front of you. And for me a moving train is shorter, not appear shorter. A blitz of 10.000 volt shivers through my body. And make it a 20.000 volt when I read that the train 'seems' shorter. And over the last 20 years it was (and still is) flabbergasting to read how people try to defend that false, erroneous approach of SR.
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