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Acceleration doesn't cause the Twin Paradox?

  1. Jan 30, 2012 #1
    Acceleration doesn't "cause" the Twin Paradox?

    In a recent review of a physics textbook, the reviewer is critical of the author of the book because the the author doesn't correct the persistent notion of many students that it is the acceleration of one of the twins that "causes"[reviewer's quotes] the differential aging in the twin paradox.

    Suppose we have a set of twins, Eartha and Stella. Stella accelerates in a ship to nearly the speed of light and lands on a planet 30 light years away. Immediately upon landing,Stella sends a picture of herself to Eartha, the stay-at-home twin. What would Eartha say upon receiving the image of her twin? Eartha would say that her sister looks exactly like the day she left!

    All Stella did was accelerate to near the speed of light.There was no meet-up back on earth. There wasn't even a turn around or any change in direction. Just acceleration and deceleration

    My Question: Why is this persistent notion in error?
    If acceleration didn't cause the differential aging, what did?
    (Please no General Relativity)
  2. jcsd
  3. Jan 30, 2012 #2


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    Re: Acceleration doesn't "cause" the Twin Paradox?

    There's a few sources which just say "well one of the twins accelerated, so use GR for this". That's maybe what the reviewer was criticizing? In effect, it is the acceleration that breaks the symmetry of the problem and allows a non-ambiguous result to be reached. But one certainly does not need GR to analyze the twin paradox. Whichever twin switched reference frames will be the one who turns out to be older.
  4. Jan 30, 2012 #3
    Re: Acceleration doesn't "cause" the Twin Paradox?

    You mean younger?
  5. Jan 30, 2012 #4


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    Re: Acceleration doesn't "cause" the Twin Paradox?

    Right, younger. >.>

    I wrote that without thinking. <.<
  6. Jan 30, 2012 #5
    Re: Acceleration doesn't "cause" the Twin Paradox?

    Indeed acceleration is sufficient but not necessary - you need a way to break the symmetry, and acceleration is one way to do so. Consider the case where the the universe is compact and thus finite, say a torus, that allows its observers to travel in one direction and then arrives back where he started. See here for more details. This gives rise to the so-called "cosmological twin paradox", see this thread and this thread for detailed discussion. I just quote the scenario for clarity here:

  7. Jan 31, 2012 #6


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    Re: Acceleration doesn't "cause" the Twin Paradox?

    Note that you have two accelerations in your scenario: to reach near c, and to land on a planet. They are not essential to your interesting question (in fact they refute it if the goal is to show inessentiality of acceleration).

    Let's suppose stella passes right by a space station near earth, and stella and eartha look at each other, seeing they are about the same age (and they syncrhonize wristwatches). Then suppose stella sends eartha a self image as stella passes the distant planet. Further, let's suppose eartha sends stella a self image from a time such that it happens to reach stella at a time after stella passes the planet that is very slightly less than the time it took (per stella) to reach the planet. This means, stella would conclude eartha sent her image 'at the same time' per stella, that stella passed the planet. Eartha, meanwhile, by direct (delayed) communication with the planet verifies stella's image was sent when stella passed the planet.

    What do they see? Eartha sees a picture of stella that has aged only a month (for example), though sent after 30 years (per Eartha), and arriving after 60 years per Eartha. Stella gets a picture of Eartha after (for example) two months (planet passed after one month, per stella). But Eartha's picture (say, based on wristwatch in image) has aged less than an hour since stella left eartha! (It would be a good exercise for you to work out why this is so).

    Thus there is symmetrical time dilation, and no paradox, when all motion is inertial. Acceleration of at least one twin is necessary for a twin paradox in SR. Further, they have to get back together to have a mutually agreed on age difference. As to what part of difference in path through spacetime is responsible for the age difference, I hold that is a completely meaningless question. In that sense (only) I would say you can't say acceleration caused the age difference, but you can say it enabled it.
  8. Jan 31, 2012 #7


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    Re: Acceleration doesn't "cause" the Twin Paradox?

    Actually, what this cute example shows is that (if you allow flat but topologically nontrivial spacetime), acceleration is neither necessary nor sufficient. It is not sufficient, because if you symmetric acceleration, there is no differential aging (each accelerates away and back, with identical thrust profile). It is not necessary due to (and only due to) nontrivial topology.
  9. Jan 31, 2012 #8


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    Re: Acceleration doesn't "cause" the Twin Paradox?

    from an earlier post of mine
  10. Jan 31, 2012 #9
    Re: Acceleration doesn't "cause" the Twin Paradox?

    Yes. You are right. I should have been more careful :tongue:
  11. Jan 31, 2012 #10
    Re: Acceleration doesn't "cause" the Twin Paradox?

    What about the claim by Dr. Mendel Sachs that the twin paradox is itself not a valid interpretation?


    I should note that a post asking for clarification on how, given two paths relative to a given inertial frame, the integral of time along the path integral does not correspond to aging was deleted. I asserted that the path with the most acceleration relative to that frame should age more.

    I guess Mendel Sachs did not like my definition of aging as a biological measure of the rate of time, and thus subject to SR.
  12. Jan 31, 2012 #11


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    Re: Acceleration doesn't "cause" the Twin Paradox?

    I was actually in college when Sachs first proposed his thesis on the twin paradox (circa 1971). It caused a lot discussion and snickers.

    Sachs is/was a serious scientist, but on this, his position is crank, and in decades of writing has not swayed any to his side.

    The integral of proper time along world line (with any amount or lack of proper acceleration) is, by definition, the time experience by any physical process following that world line. In both SR and GR, this a definition leading to predictions. Any observation counter to this would be disproof of relativity. There are no such observations, so far as I know.
  13. Jan 31, 2012 #12
    Re: Acceleration doesn't "cause" the Twin Paradox?

    Thank you for confirming my thoughts. This appears to make his sites a haven for SR doubters. It seems the flaw in logic is outlined by TataKai in the following forum post.


    Here it is argued that Einsteins postulates are inconsistent, thus SR is in error. It would seem to me that these postulates are not well defined, leaving the status of SR inconclusive based on the invalid arguments deriving from these postulates.

    Unfortunately, it appears to me that those who refute SR based on this line of argument fail to observe that SR can also be derived based on the following, clearly consistent, postulates:

    1) An inertial reference frame is Euclidean.
    2) Any trajectory moving with a constant velocity in any reference frame defines the origin of another reference frame.
    3) The transformations between frames form a mathematical group.

    This then limits the form of the transformation to a single undetermined parameter, 1/c^2, that has been determined to be a precise constant using many methods beyond the optical methods implied by Einsteins postulates. In fact, this can be used to clarify what is intended by Einstein's postulates as theorems.

    Electromagnetic/optical phenomenon were the first system where we reached an accuracy able to distinguish this value from 0, and refute the Galilean transformations.

    Returning to the twin problem in SR, this derivation would require us to first fix a reference frame. Further, the twins trajectories must have identical initial and final coordinates for a comparison to be meaningful. Each twin will have their age reduced based on how much time dilation is observed in the given frame along the trajectory they followed.

    This delta in age must be Lorentz invariant. Otherwise the theory WOULD be inconsistent.
  14. Jan 31, 2012 #13
    Re: Acceleration doesn't "cause" the Twin Paradox?

    Hello starfish99:
    The Twin Paradox outcome is not dependent on having an acceleration phase. An age difference can also be demonstrated by other means of symmetry breakage.

    Let’s say that observers A and B are separated and are approaching each other on a collision course with constant relative velocity. Observer A takes the initiative of synchronizing their clocks as follows. When A’s clock reads T1 he sends a signal (at light speed) to B. B’s receipt of that signal resets his clock to zero and immediately sends a return signal to A, who receives it at his time T2. With the assumption that both light signals traveled at the same speed, A concludes that from his point of view, B reset his clock to zero when A’s clock was at (T1+T2)/2, the midpoint of the T1-to-T2 interval. That being so, A resets his clock at T2 to the time (T2-T1)/2. As far as A is concerned the two clocks have been synchronized.

    Nevertheless, when A and B finally collide (or, hopefully, pass one another) they compare clocks and find that A’s clock has advanced more than B’s clock (of course by an amount predicted by SR for their relative velocity.)

    As you might expect, if B initiates the synchronization process, the aging outcome would be inverted. And all this without any acceleration.

    It does no good to ask why time behaves this way. Such questioning falls in the same category as “Why are there only three spatial dimensions?” (if indeed that’s all there are.)

    The current answer to such questions can only be: “That’s the structure of spacetime.”
  15. Jan 31, 2012 #14


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    Re: Acceleration doesn't "cause" the Twin Paradox?

    Clocks that are moving with respect to one another cannot be synchronized since they tick at different rates. They can only be reset to the same time when they are colocated. By the time A responds to the signal from B, B's clock will no longer be at the same time. I don't know what you think this accomplishes.
  16. Jan 31, 2012 #15
    Re: Acceleration doesn't "cause" the Twin Paradox?

    Yes, clocks in motion relative to each other will not remain synchronized. That’s the gist of SR. That’s what accounts for the age difference, both in my example and in the standard twin paradox problem. Both have the same result for this same reason.

    In one case (the standard twin paradox problem) they momentarily synchronize their clocks and mutually agree that they had done so because they were together at the synch moment. In my example only A can claim the synch moment , and only by inference.

    For A to conclude that B’s clock read zero when A’s own clock read (T1+T2)/2 is most reasonable. And for A to reset his clock to (T2-T1)/2, so that it would have read zero at what he considered to be a simultaneous event with B’s clock-zero, gives him an easy way to compare elapsed times when they meet. In either case A’s synch moment is fleeting, and serves only to make it easier to compare the clocks’ elapsed times between events.

    The “mechanism” that brings about the aging difference is the same in both the standard twin problem and the one I showed here.
    My purpose was only to put to rest the idea that acceleration is a necessary ingredient.
  17. Jan 31, 2012 #16
    Re: Acceleration doesn't "cause" the Twin Paradox?

    Actually, we could have wrapped up the discussion with PAllen's post, because it tells the story in a nutshell. I'll just go ahead and add the space-time diagram for what PAllen has just said (but leaving out acceleration details). I've included a couple of hyperbolic calibration curves to help keep track of the proper times for the twins. The traveling twin takes 10 years going out and 10 years returning, while the stay-at-home twin sits there and ages 40 years.

    I think the reviewer of the text book referred to in post #1 was frustrated because he felt that it is the path taken through space-time that should be the point of focus when talking about the twin paradox.
  18. Jan 31, 2012 #17


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    Re: Acceleration doesn't "cause" the Twin Paradox?

    I have a question.

    Why are there no textbooks that actually DO the Lorentz Transformation?

    Why do we never actually take the event-mapping one-to-one, and see what the space-time diagram looks like from the traveling twin's point of view on the OUTBOUND trip?

    Why do we never actually take the event-mapping one-to-one, and see what the space-time diagram looks like on the RETURN trip?

    Is there some kind of conspiracy, or is it just considered "wrong" to do it, for some reason?
  19. Jan 31, 2012 #18
    Re: Acceleration doesn't "cause" the Twin Paradox?

    Is this what you are looking for? Or did you want to see space-time diagrams for the traveling twin's rest system?
  20. Jan 31, 2012 #19


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    Re: Acceleration doesn't "cause" the Twin Paradox?

    Since you can analyze any scenario from any Frame of Reference, why bother with doing a Lorentz Transformation?

    The Twin Paradox is so easy to analyze from a frame in which the one twin remains at rest. You just apply Einstein's super simple time dilation formula for a moving clock, τ=t√(1-β2), you plug in the speed, β, as a fraction of the speed of light that the traveler goes at and the time, t, in the rest frame that he is gone and you get his age, τ, when he returns. So if he's traveling at 0.8c and he's gone for 10 years (5 years out and 5 years back), he'll be only 6 years older when he gets back compared to the 10 years of his brother:


    Now if you want to use the Lorentz transformation to analyze this from a frame in which the traveling twin is at rest during the outbound portion of the trip, you will have to first assign events to the first frame. I prefer to only include t and x in the form [t,x] and use units of t in years and x in light-years.

    So we start with both twins at the origin of our frame, [0,0].

    Next we calculate where the traveling twin will be after 5 years at 0.8c which is 4 light-years away, [5,4].

    Meanwhile the other twin is at [5,0].

    Then 5 years later, both twins are reunited at [10,0].

    To do any Lorentz Transforms, we start by calculating gamma for beta of 0.8:


    Now it's fairly easy to transform the first two events into the rest frame of the traveler using the simplified Lorentz Transform. In fact, the first one is the origin which is also the origin of any other frame. But the second event for the traveler is:

    t'=γ(t-xβ)=1.667(5-4*0.8)=1.667(5-3.2)=1.667(1.8)=3 years

    x'=γ(x-tβ)=1.667(4-5*0.8)=1.667(4-4)=1.667(0)=0 light-years

    But the event for the stay-at-home twin is:

    t'=γ(t-xβ)=1.667(5-0*0.8)=1.667(5-0)=1.667(5)=8.333 years

    x'=γ(x-tβ)=1.667(0-5*0.8)=1.667(0-4)=1.667(-4)=-6.667 light-years

    Now the event for the traveler looks good because since he is at rest in this frame, his position remains at 0 and his time is 3 years, half of the accumulated age that we calculated earlier (since he is half-way through his 6 year trip, according to him).

    But what about the event for the stay-at-home twin? Those numbers don't make any sense at all, do they? But they do if we remember that these are coordinates in a different frame. If we want to know how much the stay-at-home twin aged up to this point, we have to use the time dilation formula on the coordinate time to get his proper time which is 8.333 times 0.6 which is 5 years.

    Now we want to bring the twins back together using the last of the events:

    t'=γ(t-xβ)=1.667(10-0*0.8)=1.667(10-0)=1.667(10)=16.667 years

    x'=γ(x-tβ)=1.667(0-10*0.8)=1.667(0-8)=1.667(-8)=-13.333 light-years

    Again, the stay-at-home twin has a coordinate time of 16.667 years but if we multiply this by 0.6 we get 10 years.

    The traveling twin is a little more complicated because we don't know off hand what his speed is but it's not too hard to calculate if we just take the difference in the last two events for him, [16.667,-13.333] and [3,0] which is [13.667,-13.333]. This means he has traveled 13.333 light-years in 13.667 years for a speed of 0.9756c. Plugging this into the time dilation formula, we get an accumulated age of:

    τ=13.667√(1-0.97562)=13.667√(1-0.9518)=13.667√(0.0482)=13.667(0.2195)=3 years

    Wasn't that fun?
  21. Feb 1, 2012 #20


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    Re: Acceleration doesn't "cause" the Twin Paradox?

    Hi ghwellsjr.

    Yes, that was fun. I like your problem set-up. We have three major events
    e1: departure (x,t)=(0,0)
    e2: turnaround (x,t)=(4,5)
    e3: return (x,t)=(0,10)

    You also have defined another event (x,t)=(0,5), for which (x',t')= (-6.667,8.333). Then you said: "Those numbers don't make any sense at all, do they? But they do if we remember that these are coordinates in a different frame." I totally agree; but there's a lot more to say about that. Why doesn't it make sense. Why does it make sense? How does it make sense? How long is the outbound twin's reference frame relevant to the outbound twin? For only three years. But this event happens at t=8.333 years! How far away is it going to happen? 6.667 light-years from the origin.

    We should also figure out when and where this event happens according to the inbound twin's reference frame.

    Here is my calculation of the coordinates of e1, e2, and e3 in the outbound and inbound frames:


    e1': departure (x,t)=(0,0)
    e2': turnaround (x,t)=(0,3)
    e3': return (x,t)=(-13.333, 16.667)

    We can calculate the necessary change in velocity by figuring Δx/Δt between event 2 and event 3.


    And in the return-frame, we have
    e1'': departure (x,t)=(-13.333,-10.667)
    e2'': turnaround (x,t)=(0,3)
    e3'': return (x,t)=(0,6)

    My point is, those numbers DO make sense if you show the space-time diagram in the other frames. However, none of the textbooks on relativity actually SHOW the space-time diagrams in the other frames, so the reader is always left with these loose ends, wondering if it really makes sense, or just accepting the authority of the author, who claims it makes sense.

    I know it does make sense, but that's because I've gone through the effort of actually looking at it from the different frames. But I've never seen ANY relativity texts actually go through the effort of transforming the coordinates of the events to other reference frames, and showing how those coordinates DO make sense.

    When an author says "it doesn't make sense" does he mean
    • "it doesn't make sense to me," or
    • "it doesn't make sense to most people" or
    • "it really makes no sense, i.e. it is really wrong"
    • It makes no sense, i.e. it is meaningless;
    • "it does not make sense to our primitive human brains, but it is mathematically correct."
    • other?

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    Last edited: Feb 1, 2012
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