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Acceleration and the twin paradox

  1. Oct 30, 2014 #1
    After many years of agonizing over it, I have still failed to come to terms with the twin paradox. Here's a brief review of my understanding and a few questions:

    A standard story is as follows: Twin A leaves the earth for planet Zolan 10 light years away. Twin B stays on Earth. Let's say that, over a period of a week, twin A accelerates to 0.8c (relative to Earth and Zolan, which are at rest relative to each other) and then remains cruising at that velocity, decelerates for a week upon approaching Zolan, makes a sweep around the planet (waves at everyone), begins to accelerate once more towards Earth for a week, reaches a cruising speed of 0.8c, and then finally decelerates for a week before landing safely on the earth and having lunch with twin B.

    Now, I'm not worried about the exact figures, but we can all agree that twin A will have aged more slowly than twin B and will be "younger" than twin B.

    Why is this? The main problem most people seem to have is how do you get around the apparent symmetry of relative motion problem between the two twins.

    In order to address this, popular consensus among physicists/cosmologists that I've seen is that this apparent symmetry becomes broken once twin A accelerates from the earth and enters a non-inertial frame, and this is what is responsible for the slowing of twin A's aging relative to twin B. I have a few questions about this.

    1) "twin A accelerates from the earth and enters a non-inertial frame"--a non-inertial frame relative to what? To the earth/twin B? Some objective measure of spacetime displacement? To the CMB? I guess what I'm asking is that, is there a way to measure the acceleration of an object/frame without some measure of what to say it is accelerating in relationship to? I'm guessing there's a stock answer to this question, but I'm remiss to recall it at this moment.

    2) Lorentz transformations (LT): The equation determining the time dilation factor in twin A's slowing of aging is, AFAIK, contained completely with the LT equations. However, I see no term or provision at all for acceleration in the time dilation equation. So how does acceleration play a role quantitatively in this paradox when it seems as if all you need is a simple velocity difference to create the dilation effect? Are there some equations I am not aware of that address this, or some kind of formalism that can be used to quantify these measures? I'll give three examples of scenarios I'm having difficulty seeing how the standard formalism can address:

    2a) Say twin A were to accelerate (and decelerate) for 2 weeks instead of one week in both directions from earth to zolan, but compensates for the lost time by cruising at a speed slightly greater that 0.8c so that the two week acceleration round trip flight time was exactly the same as the one week. Would that make any difference in the age difference between twin A and twin B upon twin A's return?

    2b) Say twin A took the exact same trip as above (say the one week version), but now twin B also took a similar trip in the opposite direction to planet Xanadu 5 light years away, but accelerated at a different rate than twin A, and reached a different cruising speed that was, say, 0.5c. How would we calculate their age differences in this situation once they both returned to earth? Could we handle it with only the LT and the relativistic velocity addition formulas? If not, then how?

    2c) Finally, let's say we have twin A again going on his trip to Zolan (one week acceleration edition), but now we have twin B centrifuging himself in a 2001: Space odyssey style space station, undergoing rapid centripetal (or is it centrifugal?) acceleration. And then twin A and twin B go for lunch on earth when twin A comes back. Does that make a difference in their age? Or should twin B just as well stayed on earth during twin A's trip?

    I guess the more broader encompassing question is what it is specifically about twin A's undergoing an acceleration or transition into a non-inertial frame that sets his clock running slower than twin B's. It seems to be this act of accelerating that does it, not simply a relative difference in velocity. But this is the was the LT's lead you believe.
    Last edited: Oct 30, 2014
  2. jcsd
  3. Oct 30, 2014 #2


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    Have you ever discussed it here? Or read the other threads on it (see bottom of page)? Your questions seem like the usual ones, answered 1000 times already.

    Yes, with an accelerometer:
    Last edited: Oct 30, 2014
  4. Oct 30, 2014 #3


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    You should also work through the FAQ at http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html. Look especially at the Doppler and spacetime diagram sections.

    Part of your problem is that
    is not right. The acceleration is almost a red herring. It doesn't explain the differential aging; it does provide an objective basis (their accelerometers read different values) for saying that they have different experiences so it's at least possible for them to end up with different outcomes.

    Another misunderstanding you need to get rid of:
    Velocities are frame-dependent and relative, but whether a given frame is inertial an invariant property of that frame and independent of all other frames. If I am at rest relative to the origin of a frame, and I am experiencing no acceleration as measured by an accelerometer, then that frame is inertial.
  5. Oct 30, 2014 #4
    In fact it is possible to observe differential aging even in inertial travel between two points in space. See the second part of this link for an example: http://mathpages.com/rr/s6-05/6-05.htm
  6. Oct 30, 2014 #5
    Ugh, beating that dead horse again
  7. Oct 30, 2014 #6


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    This wording is incorrect. You do not enter or leave a reference frame. A reference frame is a mathematical construct, a tool for analysis. It is not a physical object which you can enter or leave.

    If being "in a reference frame" were to have any meaning then you would have to say that every object is always "in" every reference frame. They are just not at rest in most reference frames. So, the correct wording is that "twin A accelerates from the earth so A's rest frame is non-inertial".

    A reference frame is inertial or not inertial without reference to anything else. Similarly, a given object is inertial or non-inertial in an absolute sense, not relative to any reference frame. You do not need to specify "relative to what" because it is not relative.
  8. Oct 30, 2014 #7
    Well, this is a good example of where my trouble lies because the "acceleration solution" is what I keep coming across in my perusals on the matter. Here are a couple of instances:

    Fast forward to 1:29:00 to listen to Lenny Susskind's comment on the legitimacy of using the acceleration of the traveling twin to explain the twin paradox.

    Also, fast forward here to 4:46 where this guy from sixty symbols starts off with, "The resolution to this paradox..."

    Yes I have discussed it here, I've followed many of the related threads and have even started one or two. And I'm still not comfortable with my understanding of it or I would have not made the post. Case in point, I'm not getting from the two physicists' videos I posted that "acceleration is a red herring," I'm getting from them that it's an explanation (of sorts). And the sixty symbols guy goes on to say that you need to invoke general relativity to address the problem. But I've also heard that you don't need to invoke GR to explain the twin paradox. So...

    From what I can see this horse is not quite dead yet. At least from the level of understanding I am able to acquire, but's why I'm asking here from those "in the know." Hopefully I can glean some insight from the links provided and/or any further discussion in this thread.
  9. Oct 30, 2014 #8


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    It is part of the explanation in this particular scenario because, as Nugatory said, it is a physical difference between the two twins: the traveling twin feels acceleration for at least some portion of his journey, while the stay-at-home twin never does--he's in free fall, feeling no acceleration, the whole time.

    However, this difference is not the complete explanation: it can't be, because, as others have commented, there are other scenarios in which both "twins" are in free fall the whole time, but still have different ages when they meet up again. So just saying "acceleration explains the twin paradox" is not sufficient, because that "explanation" doesn't generalize.

    Here's an explanation that does generalize: two twins that take different paths through spacetime between the same two events (where they separate, and where they meet up again) can experience different elapsed times between those two events. The role acceleration plays in the standard twin paradox is to explain how both twins can pass through the same pair of events--the traveling twin has to turn around, and he has to feel acceleration when he does, because the scenario is set in flat spacetime. In the scenarios where both twins are in free fall the whole time, spacetime is curved, so there can be multiple free-fall worldlines between the same pair of events, and they can have different elapsed times along them.

    Many discussions of the twin paradox (including, apparently, the videos you linked to) don't go to this level of generality, which is a shame, because it makes it harder for people to understand the general rule. One of the reasons I like the Usenet Physics FAQ article on the twin paradox is that it explicitly talks about the general rule (in the Spacetime Diagram section).
  10. Oct 30, 2014 #9


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    The sixty symbols guy is flat out wrong on that point if that is what he said.
  11. Oct 30, 2014 #10


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    That's why I pointed at you at two specific sections of that FAQ. Read them.
  12. Oct 30, 2014 #11


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    DiracPool, suppose that you had a piece of paper with two points on it, one labeled "start" and one labeled "finish". Suppose further that there were two paths connecting the start and finish, one is a straight line and the other has a bend in it.

    Do you understand that the bent path is longer? Could you explain why the lengths are not the same?
  13. Oct 30, 2014 #12
    \rhetorical: Which part of the bent path contributes the extra length?
  14. Oct 30, 2014 #13
    I plan on it, thanks.

    Because one is curved and one is straight, and the shortest distance between 2 points is a straight line? Is that a trick question or am I missing something? Lol

    Is that the same one Nugatory posted? Yes, I plan to read it, thanks.
  15. Oct 30, 2014 #14


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    It is not a trick question. The analogy to the twin paradox is almost exact, for reasons that hopefully will become clear to you later. If you really understand the path lengths then you will understand the twin paradox also.

    Why is the shortest distance between two points a straight line? If you were speaking with someone who was unfamiliar with geometry on a plane, how could you explain to them why a straight line is the shortest distance or how to determine the length of a path? In particular, what would you do to explain the asymmetry in the lengths?
  16. Oct 31, 2014 #15
    Here's an explanation not involving curvature.

    I will assume you know what a space-time diagram is... now think about what is "now" for a certain event on such a diagram. This is what is called a simultaneity plane in literature, and is represented as a simultaneity line on the diagram and is actually a simultaneity space in reality :p Hint, it is the horizontal line though the event on the diagram.

    For the two diagrams representing the two different reference frames centered on our traveller right before he turns around and right after he turns around (assume instantaneous acceleration, for simplicity), the two simultaneity lines will be different.

    You can plot each "now" on the other space time diagram to understand it better, but the specifics are not really important. It simply boils down to "relativity of simultaneity". When you accelerate, or in other words when you switch to a different reference frame, you change the definition of what you call "now". So you can say that the stay-home twin's clock goes slower before you accelerate, goes slower after you accelerate as well, and yet shows more time in the end when you return because of this jump in your "now" right when you accelerate.
  17. Oct 31, 2014 #16


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    Explanation of what
    exactly? Differential proper acceleration is the reason why there is no symmetry. And if you want to describe the whole thing from the non-inertial twin's frame, based on certain simultaneity conventions, then proper acceleration affects how space time coordinates look there.
  18. Oct 31, 2014 #17
    There are several standard answers for that, all stemming from classical mechanics. There are also answers from General Relativity that are incompatible with the former. Special relativity uses (or at least, used!) the definitions of classical mechanics. IMHO it's not helpful to mix them up, but regularly answers are given in the context of "flat" GR, and that commonly adds to confusion. More likely you are searching for a clear understanding in the context of pure SR, without GR jargon.
    If not, you can ignore what follows. :)

    Concerning SR and classical mechanics: Newton used the concept of the "fixed stars"; and despite the fact that far away stars are not really "fixed", that serves rather well up to a very high precision. As classical mechanics and SR consider free fall in gravitational fields to be non-inertial, another way to approximate a measure of inertial motion is to consider objects that are freely moving far away from heavy masses. And of course, one can use the observed laws of effects of heavy masses to make corrections to readings of accelerometers. This should have been clearly explained when you learned classical mechanics, but many textbooks fail in this respect.

    Your point is totally correct. SR does not account for any effect of acceleration itself on the speed of clocks; only indirectly, by means of a change of velocity, does acceleration have an effect. A change of velocity is required in order for the two to meet again, it's as simple as that! BTW, this was already pointed out by Langevin, before it became a "paradox"....

    Elaboration: the point is that as long as the two are in inertial motion (SR definition), a comparison of their clocks by means of coordinate systems in which each remains in rest must be symmetrical according to the LT equations. The only way to break this symmetry (according to SR) is that at least one of the two changes velocity. Only by the rather insignificant fact that a change of velocity is called "acceleration", does acceleration play an important role in this account according to SR. No change of velocity -> no asymmetry possible - and that's all there is to the essential role of "acceleration"!

    From there on it's all rather trivial, similar to, as mentioned in Susskind's video, a Pythagoras calculation. In the most simple variant you can pick an inertial coordinate system in which the one is at rest, and then you easily find how much the time of the other one will be delayed when they meet up again. And, as you noticed, quantitatively only the speed matters for that case.

    Also, just as in classical mechanics, you can switch and jump frames as and when you like for your analysis: that complicates the calculation but it won't change the answer (of course).

    On a sidenote, Susskind's remark about "experiencing acceleration" is a bit misleading; nothing about sensorial perception is in the SR equations or in SR theory, as you likely noticed. I doubt that he read the old literature, for in the first two examples of such calculations (by Einstein and by Langevin, neglecting the effect of gravitation on time keeping), locally no detectable acceleration is experienced by the traveling clock or by the traveler during the time under consideration.

    PS while writing this, the Susskind video was still playing; I spotted a number of inaccuracies, regretfully... but most of those are of no consequence for the topic here.
    Last edited: Oct 31, 2014
  19. Oct 31, 2014 #18


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    Used correctly, I think relating SR and GR clarifies the issues.
    Do you have a reference for Newton using fixed stars? My knowledge of history is that the distance to, and any concept of the nature of fixed stars came only with Herschel centuries after Newton's death. Einstein never referenced 'fixed stars'. Newton, I believe, discussed the idea that a spinning bucket was distinguishable in a universe with no stars at all.

    A correct, IMO, way to distinguish SR inertial frame from GR inertial frame is to exploit geometric understanding from GR rather than reject it. SR is a theory of flat Minkowskian space-time in which global inertial frames are well defined, and global comparison of velocities are well defined (due precisely to absence of curvature). Given that global comparison of velocities are well defined, acceleration is globally defined relative to global inertial frames, rather than being a local observation due to accelerometers. This means that in SR (irrespective of the fact that no satisfactory theory of gravity is possible), readings of accelerometers have no fundamental significance (because they are a complex dynamical instrument, rather than in expressing - in the ideal - a feature of geometry as in GR).
    It has also been pointed out by several posters in this thread, as well as in the referenced FAQ.

    [edit: A clarification on acceleration in SR. As I mentioned, in SR an accelerometer is not a 'fundamental' instrument as is a clock or or rod. However, acceleration is a fundamental invariant in SR. Thus, irrespective of what is 'felt', a planetary flyby in SR is proper acceleration and breaks symmetry between observers. In SR, there must be acceleration in a non-geodesic path, just as in Euclidean geometry, there must be a bend in a non-geodesic path. As a final note to pedants, I am discussing only standard topology of spacetime, not e.g. cylindrical flat Minkowski spacetime.]
    Last edited: Oct 31, 2014
  20. Oct 31, 2014 #19
    For using the concept of "fixed stars" as reference? Yes of course, nowadays the Principia is online (just text search the page for occurrences of "stars"):
    - https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1729)/Definitions

    And as usual, a discussion is provided in Wikipedia:

    No doubt so - but who wants to reject understanding of GR? Once more, there is no need for GR to clarify the standard (SR) "twin paradox" issues that the OP asked about.
    Yes indeed!
  21. Oct 31, 2014 #20


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