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Non-reciprocal Time Dilation

  1. Aug 9, 2010 #1
    This is a very simple scenario.

    Given:

    Earth (E) t=0, x=0 ,and a station (Stn.) at rest wrt (E) at 100 ls,.....t=0, x=100

    And a ship S which instantaneously accelerates to 0.8c at t=0, x=0 and t'=0.

    At t,t'=0 all three systems begin sending simple quantitative sequential time signals at one second proper time intervals. 1,2,3,4,5,...........

    Solely on velocity calculations we can assume that the ship arrives at Stn at t=125 and ship time t'=75
    At this time we can assume that the ship will have received 1 -->125 signals from the Stn and likewise the Stn will have received signals 1-->75 from the ship.
    In the Stn frame they will have received signals 1-->25 from E and the assumption would be that a further 100 signals would be spread out in space on the way.
    The ship will also have to have received the same number of of E signals at this point 1-->25.

    QUESTION #1): At what proper ship time did the ship receive signal number 1 from the Stn.???

    Back on Earth t= 125 ,,,they will also have received signals 1--> 25 from the Stn and I would assume signals 1-->15 from the ship.

    #2) Is there any calculation that would indicate E receiving a different number from the Ship???

    So it would appear that given intelligence on Earth they would calculate that at this time, if the Ship is on schedule, it is colocated with the Stn. This would infer that based on this distance in their frame, that they had received 20% of the signals sent by S, the other 80% being spread out in transit.
    From this it follows that 20% of their elapsed time = 25 and if they received 13 signals in this time the relationship is 25/15 =1.6666667 which is of course the dilation factor.
    On the Ship colocated with the Stn this same calculation could apply. That they had received 20% of the E signals = 25 and 20% of their elapsed proper time is 15 sec.

    SO it would appear that rationally, both frames could calculate the same non-reciprocal dilation factor.

    #3) Is there any principle that would indicate any of these premises would not apply??

    #4) Would any of the locations have received a different set of signals???
     
  2. jcsd
  3. Aug 9, 2010 #2
    This is not correct. When the ship leaves Earth at 0.8c (gamma factor=1.6666) the first second in the proper time of the ship occurs at time 1.6666 seconds in the Earth frame. In this time the ship will have travelled 1.6666*0.8c = 1.33333 and this is equal to the time it takes the light signal to get back to the Earth. The arrival time of each signal from the ship therefore corresponds to gamma+gamma*v/c = 1.6666+1.6666*0.8 = 3 seconds of Earth time. Therefore in 125 seconds of Earth time the Earth receives 125/3=41.6666 signals from the ship. The Earth first starts receiving sgnals from the station at time 100 and then receives 25 station signals between time 100 and time 125 Earth time. In the time it receives 25 signals from the station it therefore receives 25/3 = 8.3333 signals from the ship.

    Note that the equation gamma+gamma*v that I gave above, can be written as:

    [tex]\gamma+\gamma v/c = \gamma (1+v/c) = \frac{(1+v/c)} {\sqrt{1-v^2/c^2}} = \frac{\sqrt{1+v/c}\sqrt{1+v/c}} {\sqrt{1-v/c}\sqrt{1+v/c}} = \frac{\sqrt{1+v/c}} {\sqrt{1-v/c}} = \sqrt{\frac{1+v/c} {1-v/c}}[/tex]

    which is the standard relativistic Doppler factor given in text books.
     
    Last edited: Aug 10, 2010
  4. Aug 10, 2010 #3
    Hi kev thanks for your help. There remains some confusion in my mind.
    First off , can you answer question #1 ????

    QUESTION #1): At what proper ship time did the ship receive signal number 1 from the
    Stn.???



    Thanks
     
    Last edited: Aug 10, 2010
  5. Aug 10, 2010 #4
    Yes.

    When the ship meets the first signal from the station (sent at time t=0) the distance the ship has travelled (vt) plus the distance the signal has travelled (ct) is equal to 100 ls, so that:

    vt+ct = 100 ==> t = 100/(v+c) ==> t = 100/1.8 = 55.555 ls

    this is the time taken in the Earth frame, so the proper time of the rocket at this event is 55.555*0.6 = 33.333 seconds.

    This initial signal from the station passing the ship arrives at the Earth at time t =100 at the same time as the 33.333th signal from the ship.
     
    Last edited: Aug 10, 2010
  6. Aug 10, 2010 #5
    This is a diagram I have created to illustrate your problem, but I have changed the distance from the Earth to the Station to 12 light seconds to make things clearer. The velocity of the rocket is still 0.8c as in your original post.
     

    Attached Files:

  7. Aug 10, 2010 #6
    Hi kev
    SO it appears from this that only dilation is involved, that simultaneity plays no part.

    You did the calculations with the assumption of non-reciprocal dilation.

    What if we now have another ship S 2 that accelerates simultaneously (Identical proper acceleration) t,t' =0 with S from a location -100 ls , in Earth frame??

    It seems that the same result would obtain as in S and Stn regarding signals received in both S2 and earth upon arrival yes??

    Thanks and thanks for the drawing. I see you edited out your glitch so I edited out my snide comment.
     
    Last edited: Aug 10, 2010
  8. Aug 10, 2010 #7
    Hi kev see if this tracks:
    At t=25 the earth frqame has received 8.3333 SIGNALS from ship.
    Applying the 1.333 Doppler factor would mean 8.3333*1.3333=11.111 sec of that time was Doppler
    25-11.111 =13.899 s elasped earth time for 8.3333 signals
    13.899/8.3333= 1.6667 gamma factor for dilation

    In the earth frame at t=1 ,,,,ship is at x=0.8 when the signal is sent
    The signal would catch up to the ship at t=5 , x=4 at ship time t'=3
    On the ship the Doppler factor is 2.4 sec per signal received so at t'=75 having received 25 signals this is 2.4*25 = 60 ship seconds for Doppler leaving 15 proper seconds relative to 25 seconds earth time.
    So both frames could at any point after receiving a signal calculate the gamma factor and recognize a non-reciprocal dilation. Yes??? Or have my calculations run amok?
    Thanks again you have really helped :smile:
     
  9. Aug 11, 2010 #8
    Yep.

    Not sure where you get this Doppler factor from, but the maths seems to work using your own definitions.

    I would say the classical Doppler shift is given by:

    [tex]\frac{fe}{fo} = \left( \frac{1+Vs/c}{1+Vr/c} \right) [/tex]

    .. where fe is the frequency of the signal emmitted by the source moving with velocity Vs relative to the wave medium, fo is the frequency observed by the receiver moving with velocity Vr relative to the wave medium and c is the velocity of waves relative to the wave medium. In the case study you have presented the classical Doppler shift would be fe/fo = (1+0.8)/(1+0) = 1.8.

    In relativity, the frequency of the source is reduced by the gamma factor due to time dilation so the total relativistic Doppler factor becomes fe/fo = 1.8*1.6666 = 3 in your example. In other words if the ship sends signals at a frequency of 1 per second proper time of the ship, the Earth measures the signals to arrive with a frequency of one signal every 3 seconds.

    The relativistic Doppler factor in this context is given by:

    [tex]\frac{fe}{fo} = \sqrt{\frac{1+v/c} {1-v/c}}[/tex]

    Agree.

    The ship receives the first Earth signal at t'=3 (as you have calculated) and the second at t'=6 so the the ship is receiving Earth signals at the rate of 1 signal every 3 seconds of ship proper time. Not sure how you are defining the "Doppler factor". However, I agree that at t'=75 the ship will have received 25 Earth signals.

    possibly the latter :smile: They would both be receiving the other's signals at a rate of one signal every 3 seconds of their own time, so the relationship is reciprocal, so they both calculate the same gamma factor and there is no notion of who is "really" moving.
     
  10. Aug 11, 2010 #9
    I probably misused the term. I meant the percentage of the time interval between signals that was a result of system motion in contrast to the part that resulted from clock time and dilation. Editing out the former to get the latter.

    Yes this is relevant to calculating frequency shift in the signal itself.

    Uh Oh :confused: I made no suggestion of who was really moving but of course in this scenario everybody agrees that it is the ship I.e. This is just the twins sans one twin.
    WHere everyone seems satisfied that the ship having accelerated makes the motion real and the dilation non-reciprocal. I was simply interested to see if it was possible to determine this before actually reaching the station were it is then confirmed.
    Obviously as the ship approaches the station it is already clear before arrival, the relative relationship, as they have both received the majority of the signals, yes??
    In the beginning you started with the assumption that at t'=1 the earth time was t=1.66667,,,, if you instead make the assumption that at t'=1 , t=0.6 and therefore the ship has only moved 0.48 x ,,,then the first signal reaches earth at a different time ,,no??
     
  11. Aug 11, 2010 #10
    You didn't mean to make a suggestion of absolute motion, but if there is any way of measuring non-reciprocal time dilation, then that certainly implies a way of measuring absolute motion. How do you know that "the ship having accelerated makes the motion real"? Let us say we DID have a notion of "real" absolute motion, then we could say that the Earth/station system initially has a "real" velocity of -0.8c, and the rocket ship by accelerating, has inadvertently accelerated itself to a standstill. This situation and your assumed situation of a stationary Earth, both yield exactly the same predictions for the mutual velocities and time dilations measured.

    O.K, t'=1 and t=1.66667 is what is measured simultaneously in the Earth frame. t'=1 and t=0.6 is what is measured simultaneously in the ship frame. In the ship frame, the Earth has moved a distance of -0.8 ls in the first second. The ship sends its signal and the signal takes 4 seconds as measured by the ship to catch up with the Earth, so the signal arrives at the Earth after 5 seconds as measured by the ship. The Earth's clocks are time dilated according to the ship by a factor of 0.6, so the ship calculates the first signal it sends, arrives at Earth at t = 5*0.6 = 3 seconds Earth time. The relationship is completely reciprocal.
     
    Last edited: Aug 11, 2010
  12. Aug 11, 2010 #11
    You seem do disagree the presence of absolute motion. This has always confused me.
    Do you disagree absolute acceleration too?
    If not, then while you are accelerating (absolute) then aren't you in absolute motion?
     
  13. Aug 11, 2010 #12
    I do not dismiss the notion of absolute motion completely. It is just that as far as I am aware, no one has found a way (in practice or in principle) to measure the absolute velocity of anything.

    I do not disagree with the notion of absolute acceleration, because clearly there is such a thing.

    If you have absolute acceleration, then there is an implication that your absolute velocity is changing, but there is no way of knowing what that absolute velocity is and as I said before, you can never be sure if you have accelerated from a state of absolute rest to a state of absolute motion or if you have accelerated from a state of absolute motion to a state of absolute rest.
     
  14. Aug 11, 2010 #13
    Yeah, thats quite what I wanted to hear. But Why do you use the term absolute velocity at all? Can't it be worded like this:

    If you have absolute acceleration, then there is an implication that your velocity relative to every inertial frame is changing.



    We accelerated, we did absolute motion, and we changed our inertial frame. Thats all we did. Both our initial and final states are non-absolute motion states. I think, Its nonsense (not in offensive sense!) to think in terms like the above quoted text.
     
  15. Aug 13, 2010 #14
    This is not my individual idea but a concept that has been asserted many times by well known members of this forum. Certainly you must have read enough twins threads to be familiar with this??
    What is your explanation for the non-reciprocal dilation between ship and station on arrival??? Or if the ship reverses direction and returns to earth we assume 150s' to 250s earth time right?? NON-reciprocal dilation , yes???

    I think you are missing the point. In this scenario, you are describing, the assumption is still that the ship is the system that is accelerating and therefore the system that experiences non-reciprocal dilation.
    As it is an empirical truth that no real quantitative velocity measurement can be attached to any system, this still means that it is the ship that is really moving in your picture.
    Your -0.8 c velocity for the earth system has no meaning, yes?

    Can it not be said that the returning twin being younger is demonstrating non-reciprocal time dilation??
    And therefore the intermediate motion was in some sense real even if the vast majority of the actual time was inertial ???

    I certainly had no thought of any quantitative meaning to velocity of any system.

    You did not respond to this before. Specifically for e.g. when 80% of the trip to the station is complete and the ship has received a large number of signals from the station while the station has received a comparably lesser number from the ship would it not be obvious in both systems that the dilation was not going to be reciprocal?
    In the twins question, simultaneity is offered by some as a different possible foundation that doesn't require acceleration as a causative factor.

    Analysing the situation with reciprocal dilation and relative simultaneity. So why don't we apply that perspective to this simplified scenario and see where it goes???
     
  16. Aug 20, 2010 #15
    Can you analyse this same scenario from the perspective of relative simultnaeity???
     
  17. Aug 20, 2010 #16

    mysearch

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    Austin,
    I do not mean to cut across the direction of your thread, but I saw the subject of absolute motion has been raised in a few of the posts. Therefore, the link below may be of some interest as its raises the following question:

    http://au.answers.yahoo.com/question/index?qid=20100415182125AA5qBrv"

    Somebody has provided an answer, which you may (or may not) feel is a point of discussion in this thread. However, if you accept the arguments of an expanding homogeneous universe along the lines of a simplified FRW metric, e.g. [tex]ds^2 = dt^2 + a^2(t)dr^2[/tex], then an absolute frame of reference seems problematic. In this context, a stationary frame with respect to CMB would seem to imply that the rest of the universe is accelerating away from this point. Equally, it might even question the notion of absolute acceleration based on [F=ma] because it is not clear to me whether you can detect the force of acceleration with respect to the accelerating expansion of the universe. Anyway, apologises if this is not the intended focus of your thread.
     
    Last edited by a moderator: Apr 25, 2017
  18. Aug 20, 2010 #17

    JesseM

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    The acceleration does imply that the traveling twin will have aged less when they reunite, and that the traveling twin changed velocity relative to every inertial frame, but that's not quite the same as "makes the motion real"--for any given point on the traveling twin's worldline, there is always going to be a valid inertial frame where the twin is at rest at the point, i.e. not in a state of motion. For example, as I'm sure you probably know, we can analyze the twin paradox from the perspective of a frame where the Earth twin was in motion the whole time, while the traveling twin was at rest during one of the two phases of the trip (either the outward trip before the acceleration, or the inward trip after).

    Also note that initial acceleration when the twin first leaves the Earth is completely irrelevant to the twin paradox scenario--after all, instead of taking off from the Earth you could imagine the traveling twin's ship just passed by the Earth moving at constant velocity. You could even replace "Earth" with a space station, and it might even be that the initial difference in velocity between the ship and the station was caused by the station firing its rockets and accelerating away from the ship. In any of these scenarios, it would still remain true that if the ship accelerated two turn around and travel back towards the Earth/station twin after they'd been traveling apart for a while, when they finally reunited they would find the twin on the ship had aged less (and if the accelerations are assumed to be quasi-instantaneous, the amount that each twin had aged when they reunited would be exactly the same regardless of whether the ship initially accelerated away from the station, the station initially accelerated away from the ship, or they simply passed next to each each other moving inertially)
     
    Last edited: Aug 20, 2010
  19. Aug 20, 2010 #18

    JesseM

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    Simultaneity is only relative between two different inertial frames. Do you want to analyze kev's scenario from the perspective of a different frame? If so which one?

    If you're interested, in post #63 here I gave my own example where I analyzed the aging of two twins from the perspective of two separate frames, one where the inertial twin Terence was at rest, the other where the non-inertial twin Stella was at rest during the outbound phase of her trip (and had a speed even greater than Terence during the inbound phase).
     
  20. Aug 21, 2010 #19
    Hi JesseM For some reason I cant access your thread link but unless my memeory is even worse than I think (which is bad enough) I am quite familiar with your analysis of the twins. It was you specifically, I was referring to above. Unless I am much mistaken you viewed the twins from the perspective of the fundamental Lorentz math. In your workup you maintained reciprocal dilation thoughout and the final cumulative proper time differential was attributed to the shift in relative simultaneity as a consequence of the turn around. Is this more or less accurate???
    I found the perspective both consistent and provocative as it suggested a literal interpretation of spacetime i.e. that the space twin by translating through space was also literally moving through time into the future of the earth twin. I have since thought about it quite a bit and hoped at some point to discuss it with you.
    In this current thread the point was not the question of real motion but actually the possibility of detection of non-reciprocal time dilation before actual colocation and comparison. Addtionally a context to examine simultaneity also.
    I am sure you can absorb the conditions here in a moment, so if you care to help in implementing simultaneity it would be great. We can assume the acceleration occured instantly just prior to reaching the earth.
    As far as I see At t=0,x-0 , ship t'=0 colocated with earth ,according to the ships simultaneity the station t=80.
    Based on this, the first station signal #1 received by the ship would not occur at the proper time or location according to kev's calculations .

    One additional thought. I have noticed a strange asymmetry in the initial scenario. The perceived Doppler is reciprocal in both systems but the percentage of received signals is not.
    At the point of arrival at the station the earth has received over half of the sent ship signals while the ship has received only 20% of the earth's signals. There may be some obvious explanation but so far it has eluded me.
    Thanks
     
  21. Aug 21, 2010 #20

    JesseM

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    Maybe there was a temporary bug, but the link seems to be working fine now. If it still doesn't work for you, try going here and scrolling down to post #63.
    I'm not sure what you mean by "shift in relative simultaneity as a consequence of the turn around". I didn't analyze it from the point of view of a non-inertial frame where the age of the Earth twin suddenly jumped forward by a large amount at the turnaround, if that's what you mean. Instead I analyzed it from the point of view of two different inertial frames, one in which the Earth twin Terence was at rest, and one in which the traveling twin Stella was at rest during the outbound phase of her trip (but not the inbound phase). Either way, there are no sudden shifts in either twin's age at any point in the trip. The two frames do disagree about what event on Terence's worldline was simultaneous with the event of Stella turning around, though.
    Why do you say that? In the perspective of the second frame where the traveling twin Stella was at rest during the outbound phase, Terence was the one aging more slowly during the outbound phase, so in this frame you could say that the Earth twin Terence was "moving through time into the future of the traveling twin" during the outbound phase. Likewise you could analyze things from the perspective of a frame where Stella was at rest during the inbound phase, and in this frame Terence would be aging more slowly than Stella during the inbound phase. Since all three frames are equally valid, obviously there can be no "objective" truth about who was aging more slowly during any particular phase of the trip, although of course all frames agree that Stella has aged less in total by the time she reunites with Terence.
    What do you mean "implementing simultaneity"? Do you just want an analysis of which events were simultaneous in the rest frame of the Earth/station vs. which events were simultaneous in the rest frame of the ship?
    As I said above in the second paragraph of post #17, initial acceleration is irrelevant in this sort of problem. You can assume the ship was traveling at 0.8c relative to the Earth/station for all eternity, and just happened to pass by the Earth at t=0 in the Earth/station frame, it won't affect any of your conclusions.
    Yes, in the ship's frame the event of the station frame reading t=80 seconds is simultaneous with the event of the ship passing the Earth when the Earth clock reads t=0 seconds and the ship clock reads t'=0 seconds. And remember, the station clock is slowed down by a factor of 0.6, so in the ship's rest frame the station didn't send the first signal 80 seconds earlier, instead it sent the first signal 80/0.6 = 133.333... seconds earlier, at t'=-133.333... in the ship's rest frame (or if you want to imagine the ship accelerated at t'=0, we should say the event of the station sending the first signal happened at t'=-133.333... 'in the inertial frame where the ship was at rest after its acceleration away from Earth'). Since the station is moving at 0.8c in the ship rest frame, at the moment the station sent the first signal it was considerably further away from x'=0 then it was at t'=0, by an amount of 133.333...*0.8c = 106.666... light-seconds. And the station's distance from x'=0 at t'=0 must have been only 60 light-seconds due to length contraction, since the distance between Earth and station is 100 light-seconds in their rest frame and they are both traveling at 0.8c in the ship rest frame. So, at the moment the station sent the first signal (at t'=-133.333... seconds), it must have been at position x' = 60 + 106.666... = 166.666... light-seconds. Thus the first signal will reach x'=0 (the position of the ship) 166.666... seconds later, at time t' = -133.333... + 166.666... = 33.333... seconds. Naturally since the ship is at rest in this frame, its own proper time matches the coordinate time, so it receives the first signal at a proper time of 33.333... seconds, same as what kev found by analyzing things from the perspective of the Earth/station rest frame.
    But when you talk about how many signals the Earth has received "at the point of arrival at the station", you're talking about the point on the Earth's worldline that's simultaneous with the event of the ship reaching the station according to the definition of simultaneity in the Earth/station frame. If you instead considered the point on the Earth's worldline that's simultaneous with the event of the ship reaching the station according to the definition of simultaneity in the ship frame, then in this case the number of signals the ship has received from the Earth when it passes the station is greater than the number of signals the Earth has received from the ship at that point on the Earth's worldline. So by considering both definitions of simultaneity, you can see the ratio of (number of signals ship has received from Earth when it passes station):(number of signals Earth has received from ship at point on Earth's worldline that's simultaneous with ship passing station) is completely symmetrical between the two frames (i.e. if the ratio is 3:5 using the Earth frame's definition of simultaneity, then it's 5:3 using the ship frame's definition of simultaneity).

    To give exact numbers, if they are moving apart at 0.8c and each is sending signals at a rate of 1/second in their own frame, then using the relativistic Doppler shift formula, they are each receiving signals from the other at a rate of [tex]\sqrt{(1 - 0.8)/(1 + 0.8)}[/tex] = 0.333... per second, or one signal received every 3 seconds according to their own clock. The ship takes 60/0.8 = 75 seconds to reach the station in its own frame, so in that time it has received 75/3 = 25 signals from the Earth. In the Earth's frame, the Earth's clock has ticked 100/0.8 = 125 seconds at the moment the ship passes the station, so the Earth has received 125/3 = 41.666... signals from the ship at that point, so the ratio above is 25:41.666... = 3:5. But in the ship's frame, the Earth's clock has only ticked 75*0.6 = 45 seconds at the moment the ship reaches the station, so at that moment the Earth has received only 45/3 = 15 signals from the ship, so the ratio is 25:15 = 5:3.
     
    Last edited: Aug 21, 2010
  22. Aug 22, 2010 #21
    yes today I got it. A very complex analysis. I am going to have to give it some thought.
    Particularly the inclusion of the galilean velocity subtraction.
    In any case I must have mistaken somebody else's analysis for yours in past threads unless you have just changed your approach over time and have done so many twins threads you've lost track, :-)



    So here you spend a whole long paragraph raising objections I was already aware of, only to finish with a corroboration of exactly the view I was toying with. I.e. Stella traveled though more spacetime and ended up in Terrence's future.
    Not that I take it seriously. In fact I think an examination of a diagram of the classic situation shows that the lines of simultaneity are reciprocal. Into the past in one segment and into the future in the other as you have pointed out. And so are mutually cancelling .
    At the turn around, when for a moment they share simultaneity their proper times already do not agree.



    I am also going to have to think about all this. I had done a ballpark in my head from this perspective and as soon as it was clear that this was going to be the approximate situation I took it no further because it did not relate to any objective reality I could envision or any possible observers.
    I can understand simultaniety from the perspective of a system of clocks and rulers where local observers will always agree on the observed relationship. This to me is logically consistent and represents an objective reality, even if so far only hypothetical.
    But certain other inferences relative to spatially separated events leaves me seeking
    some greater understanding of what we are all really talking about.
    So I am going to think over your analysis here before trying to comment on it. But thanks



    OK this is fine. But it seems that from this perspective then the view would be that at t'=0
    the station t=80 and the elapsed station time for the total trip is also dt =45 s
    But this would also seem to mean that at t'=0 there were 80 station signals spread out through the intervening space , no??

    Beyond this, there still seems to be a fundamental asymmetry.
    If as you say both frames can be considered fully inertial then how is it that the signals from the ship are spatially more separated than the signals from the earth or the station??
    This would seem to be an objective reality. The signals exist independent of the emitting frames and yet all possible observing inertial frames would agree on this asymmetric spatial separation even if they did not agree on the quantitative distance between signals. They would also agree there were more signals, per any given distance, from the earth and station than from the ship.
    You took exception to my comment about acceleration making motion real , but actually that was a responce to kev's comment that any demonstration of non-reciprocal time dilation would be equivalent to showing real motion.
    Yet here seems to be a situation where two inertial frames seem to have non-reciprocal dilation as determined by their respective signals in space without acceleration or change of direction. SO more thought for sure.
    Thanks
     
  23. Aug 22, 2010 #22

    JesseM

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    There was no Galilean velocity subtraction in my analysis, since Galilean velocity subtraction involves the relation between velocities in two different frames. I did assume that if one single frame measures the velocities of two objects to be v1 and v2 in the same direction, then the "closing speed" in that frame (i.e. the rate at which the distance between the two objects was decreasing in that frame) would be (v2 - v1), but this only concerns speed and distances in one frame, so it's just as true in relativity as it is in Newtonian physics. For example, suppose in some frame object A's position as a function of time is x(t)=0.8c*t, while object B's position as a function of time is x(t)=0.5c*t + 3 light years. So, at t=0 the distance between them is 3 light years, how long will it take for object A moving at 0.8c to catch up with object B moving at 0.5c? Well, the "closing speed" is 0.8c-0.5c=0.3c, so (initial distance)/(closing speed) = 3/0.3 = 10 years. If you calculate more explicitly when the positions of the two will coincide by setting the right sides of the two equations equal, you have 0.8c*t = 0.5c*t + 3, subtracting 0.5c from both sides gives 0.3c*t = 3, so you conclude t=3/0.3c=10 years. "Closing speed" is just a shortcut for a full analysis where you write down x(t) for each object and use that to figure out at what t their positions will coincide.
    You didn't explain what you meant by the phrase "literally moving through time into the future of the earth twin", though. It seems to me that a "literal" interpretation of that phrase would suggest that the rate at which Stella was "moving through time" was objectively taking her "into the future of the earth twin", and rates concern what is going on over short time intervals, not over the course of an entire journey. But for any short interval, you can always find a frame where Terence is aging slower than Stella in that interval, i.e. Terence is "moving through time into the future of Stella" during that interval, the opposite of what you said. If you were only saying that Stella's overall path through spacetime ended up taking her into Terence's future, it seems to me that the phrase "moving through time" obscured your meaning.
    I would say simultaneity is a property of frames, not physical observers. The fact that Stella is instantaneously at rest relative to Terence at some point in the acceleration doesn't mean she "shares simultaneity" with Terence unless you impose the (arbitrary) rule that an accelerating observer should always judge simultaneity according to their instantaneous inertial rest frame.
    What is an "objective reality"? Would you say that any frame-dependent judgments can be called "objective realities", given that different frames disagree on such judgments and no frames are more "correct" than any others?
    Any frame-based judgment can be interpreted physically in terms of local measurements on a system of inertial rulers and synchronized clocks extending throughout space, like the ones I depicted in this thread. For example, we could imagine a long ruler moving inertially with synchronized clocks fixed at every marking (synchronized in the rest frame of the ruler of course, the clocks will be out-of-sync in other frames as illustrated in the thread above), such that the ship is at rest at position x'=0 on this ruler while it travels from Earth to the station, and the clock at x'=0 read a time of t'=0 at the moment the ship first started out from Earth. Then it would also be true that the event of the station sending out its first signal coincided with it passing the x'=166.666... mark on this same ruler, and that at that moment the clock sitting at that mark read a time of t'=-133.333... seconds.
    Yes, so the station clock reads t=80+45=125 at the moment the ship reaches it (this is frame-independent, although other frames disagree that the station's clock ticked forward by dt=45 during the ship's journey from Earth to station)
    Yes, in the ship's frame this is true. Now to make the symmetry more clear, imagine a second ship B traveling behind the first ship A, 60 light-seconds behind it in the Earth/station frame, 100 light-seconds behind it in the rest frame of the two ships. Suppose that the clocks of the two ships are synchronized, and that ship B also starts sending signals when its clock reads t'=0. In this case, in the Earth/station frame, ship B's clock already reads t'=80 at the moment ship A is passing Earth at t=0, so there are already 80 signals from ship B traveling through space at that moment.
    Where'd you get that idea? The Earth is receiving signals from the ship once every 3 seconds according to its clock, and likewise the ship is receiving signals from the Earth once every 3 seconds according to its clock. So, the separation between ship signals in the Earth frame must be 3 light-seconds, and the separation between Earth signals in the ship frame must be 3 light-seconds.
    No, the only asymmetry in your scenario comes from the fact that you have two objects at rest and sending signals in one frame, but only one in the other. If you add a second ship B as I suggested, the symmetry is absolutely complete; any statement about what's true in one frame can be matched with an equivalent statement (involving the same numbers) in the other frame.
    There is absolutely nothing non-reciprocal in the scenario, at least not if we add the second ship B behind the first ship A.
     
    Last edited: Aug 22, 2010
  24. Aug 22, 2010 #23
    Last edited: Jul 30, 2012
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