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Is this a paradox in Special Relativity?

  1. Jun 16, 2010 #1
    We covered SR last semester in Modern Physics I and also covered early Quantum Theory and started Modern Quantum Theory. I've been speculating a bit on SR recently since I started reading The Universe and Dr. Einstein a guy at work gave me.

    If we have the Earth as a stationary reference frame and a rocket ship as a moving frame at some appreciable value of c with respect to Earth, according to Earth's observations, humans on the rocket ship will age more slowly, the physiological processes will slow down and so forth.

    However, it's equally as valid to reverse the situation, i.e. the stationary frame is the rocket ship and the moving frame is Earth. The people on Earth from the rocket ship's observations will appear to age more slowly and the proper time will then be on the rocket ship.

    Which is actually physically true? Because both cannot simultaneously be true. They cannot both age more slowly and age more rapidly. I'm wondering if this is a conventionally-valid paradox I conjured up. I'm sure my textbook covered this situation, but I just haven't opened it up to refresh my memory on such an event.
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  3. Jun 16, 2010 #2


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    This is a variant of the so-called "twins paradox", which is only an apparent paradox due to a misunderstanding of the theory and poor definition of the case. The problem with your case is not understanding that you can't have a clear "who is older" unless both clocks are sitting next to each other, stationary, in the same reference frame. And in order to do that, you have to break the symetry of your scenario.
  4. Jun 16, 2010 #3
    Well, of course to show which is "older" you would have to explicitly compare the rates of time in the same reference frame, but that would destroy the apparent relativistic effect. However, you can still compare the situations and their logical conclusions according to the theory because they both happen actually physically concurrently. They are just viewed through different prisms and yield different results depending on your frame of reference.
  5. Jun 16, 2010 #4


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    No, it wouldn't. If you compare them in the same reference frame, one really would be ahead of the other*.
    If you make a truly mirror-image scenario, then either really could be considered older/younger, but making a truly mirror-image scenario is tougher than you may think and your scenario isn't specific enough to know if it is truly symmetrical.

    For example: where did the spaceship come from? If it came from earth, their situations are not symmetrical because the spaceship had to fire its engines to move away from earth. *So to define the situation explicitly and relate it to the above requirement:

    Start: Two clocks sitting next to each other on earth are synchronized.
    Next: One clock is put into a rocket and launched on a high-speed trip to Alpha Centuari and back.
    Finish: The two clocks are sitting next to each other again on earth.

    Which is ahead and which is behind at the finish?
  6. Jun 16, 2010 #5
    I remember reading in the textbook the situation you defined. The rocket ship undergoes acceleration leaving Earth, and when approaching Alpha Centauri, slowing down and turning back towards Earth. In this scenario, they mapped out the light-signal intervals and so forth over the span of years of the trip. The longest amount (interval) of time is always the proper time which is measured by a clock at rest with respect to the stationary frame. So, in this case the rocket ship would be slightly behind.

    I must admit I'm not familiar with the symmetry requirement in elucidating the application of SR. Why is it not valid to consider a rocket ship randomly passing Earth with some constant velocity?
  7. Jun 17, 2010 #6
    This problem usually comes about when people think of "time dilation" and "length contraction" as the defining results from which everything is derived. Instead, "time dilation" is actually referring to a very specific case of measurements.

    Consider two inertial frames with their spatial origins moving with respect to each other. If there are two events which occur at the same location according to frame1, then frame2 will claim there was less coordinate time between between the two events than frame1.

    Now let's look at it in terms of clocks. For the coordinate time in an inertial frame to match that of an actual clock, the clock has to be at rest according to the inertial frame. Therefore, in the previous experiment, you need three clocks. One in the frame1, and two in frame2 (at different spatial locations).

    The overarching point is: time dilation is NOT a relationship between two clocks moving with respect to each other. It is a relationship between coordinate time between two events according to two inertial frames, and only in a very specific case (when the events are at the same location according to one inertial frame).

    It is just hard to discuss SR accurately without using math. So many "popularist introductions" to SR unfortunatly cause a lot of confusion.
  8. Jun 17, 2010 #7


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    They're both equally true, and neither is objectively true (i.e. true in a coordinate independent way). These statements are just saying that if these guys assign coordinates (t,x,y,z) to events in spacetime using the standard procedure, the difference between the t coordinates that either of them is assigning to two arbitrary events where the other guy is present will be greater (by a factor of gamma) than the difference between his ages at the two events.

    There's no contradiction here. What you can learn from this is that the time coordinate assigned by the standard procedure isn't a good measure of a person's age. (The correct measure of his age, according to both SR and GR, is the proper time of the curve in spacetime that represents his motion).

    It's certainly valid, but it's not useful if you're trying to find a logical inconsistency. There's nothing inconsistent about the fact that their coordinate assignments disagree with the other guy's age.
    Last edited: Jun 17, 2010
  9. Jun 17, 2010 #8
    Okay, so there's a difference between coordinate time and actual clock time, or rather rate of time. Feel free to use math. I'm not sure if you read the first sentence in my original post. Your explanations make sense. However, the overarching point seems to imply merely a difference in coordinate time, not necessarily a difference in rate of time, which is my primary concern - actual rate of time in different inertial frames. The textbook seemed to make the connection between time dilation and a slowing of biological processes and so forth in that moving inertial frame.

    I fully understand assigning arbitrary units and measures of time in different inertial frames and relating them using the Lorentz transformations. I'm concerned with the conclusions and experimental verifications of SR that claim the biological processes slow down in that moving inertial frame, i.e. the rate of time slows down. Let's assume we have a mirror-image, perfectly symmetrical situation involving two moving inertial frames. The statement was made that either could be considered older or younger. This of course cannot actually be true. So, as was stated, to determine the correct age of someone or something you must look at the object's proper time which is of course the longest interval of time. Is the claim that biological processes slow down misleading?
  10. Jun 17, 2010 #9
    There is no "actually." If you have two astronauts freely floating past each other, then each observes the other's clock to run slow. There is no answer to which clock is faster or slower in an absolute "actually" sense. There is no universal clock. There is no absolute rest frame with the "real," "actual" clock. There are only clocks that move with the observers. The answer depends on reference frame. That is why it is called "Relativity."

    No, all processes, biological or not, occurring in a moving laboratory will be observed to run slow. "Moving," that is, relative to the person doing the observing. Each astronaut observes the other's biology to run slow.
  11. Jun 17, 2010 #10
    So then if I observe someone's clock to be running more slowly, they're not really aging more slowly? There doesn't have to be an "absolute" sense but an actual sense, i.e. the person and their proper time, etc.
    Last edited: Jun 17, 2010
  12. Jun 17, 2010 #11
    Alright! Proper time. This concept makes things much easier.

    First though, your concept of proper time seems to be based on coordinate time in inertial frames. Let's make it as general as possible: the proper time is the length of the wordline of whatever object/observer/hypothetical we are talking about.

    So if two people want to compare their aging, using proper time, they need to meet up at two different times. Indeed, one's worldline can be longer than the others between these two "meeting events". So yes, one can actually age more (or less) than the other.

    Since proper time is coordinate system invariant, regardless of what coordinate system you use to describe the travels, you will get the same answer for who aged more.

    It depends on what you are extracting from that claim. Locally, neither astronaut will "feel" like their biology is going slow. In fact they can perform any experiment they want and they shouldn't be able to tell that they are moving in any absolute sense. Yet if they look out and try to describe the physics of another astronaut flying by, they will measure that astronaut's clocks running slow -- any clocks, biological or otherwise. And when they meet up, their ages will be different just as any inertial observer (an inertial frame used consistently for the entire experiment) would have expected from time dilation.
  13. Jun 17, 2010 #12
    What do you mean by "really" and "actual" other than "that which is observable?"
  14. Jun 17, 2010 #13
    I suspect, as you do, that Shackleford is still thinking of absolute time which of course does not exist in SR. This absolute time, or Newtonian time was thought to have an existence distinct from our measurement (observation) of it.

  15. Jun 17, 2010 #14
    Yes, well, the purpose of asking the question was to get him to think about that ;)
  16. Jun 17, 2010 #15
    Sorry for any misunderstanding. I was really aiming the comment at Skackleford but your question seemed appropriate to quote.

  17. Jun 17, 2010 #16


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    The question actually doesn't make sense, because "really" indicates that you're talking about something coordinate independent, and the word "slowly" is a reference to a coordinate system.
  18. Jun 17, 2010 #17
    Well, I'm not talking about an absolute time. I'm well aware of that historical notion being discarded in SR.

    The statement was made that if you have a mirror-image, perfectly symmetrical situation, you could compare the clocks of the reference frame and moving frame to see which one aged more or less. This of course is a relative difference, not absolute, because you can always find some arbitrary frame to modify the perspective and observations. This implies to me that it's not simply an observation based on coordinates and equations, but an actual change in the rate of time in space the object experiences. That has to be the case if you bring back together your inertial frames to compare the clocks at rest in the stationary frame and they yield different times as russ said.

  19. Jun 17, 2010 #18
    Realize that in order to measure this, you must have the two parties at the same place twice: once at the beginning and once at the end. In order to do this one must accelerate out of his initial inertial reference frame. This change not only breaks the symmetry of the situation but also destroys the synchronization of the clocks (relativity of simultaneity).

    If you want a truly symmetric situation, then have BOTH parties accelerate toward one another in the same way. If you do that, then they will meet with the same biological age.
  20. Jun 17, 2010 #19
    Okay, if it's possible for someone to age more or less relative to someone else, and measure this accurately, why does the rate of time vary at appreciable values of c? What happens along that journey through the space-time continuum?
  21. Jun 17, 2010 #20
    I'm also a little confused as to why my use of words actually and physically are controversial. In reference to the muon decay experiment, the decay of muons traveling near c is physically less than that predicted by an Earth-frame observer because the muons actually experience less time and so lost less mass through nuclear decay. Right? Or am I just mistaken in thinking something actually, physically changes affecting the matter and the disparities are simply positional and observational?
  22. Jun 17, 2010 #21


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    But you're not comparing the clocks in that scenario. You're comparing the time displayed by clock B at some event on its world line, to the time coordinate assigned to that event by the inertial coordinate system associated with the motion of clock A.

    Suppose that you're moving with clock A and read the time it displays at some event P on its world line. How do you know which event on the world line of clock B that you should compare it with? We can imagine that you have an assistant moving with clock B and taking readings all the time, but which one of them should you compare to yours?

    Actual=coordinate independent
    Experiences=coordinate dependent
    ...so the statement doesn't quite make sense.

    Suppose you set both clocks to 0 at an event P where they're both present, and then have them move differently for a while, until they meet again at an event Q. They will usually display different "times" at Q because what they're displaying isn't a time coordinate, it's a coordinate independent property (proper time) of the curves that describe their motion.
    Last edited: Jun 17, 2010
  23. Jun 17, 2010 #22
    Sorry for all my confusion, guys. I think this is what I'm a bit fuzzy on. I'll have to go back and read the SR chapter in my textbook. The proper time, a coordinate independent property, accurately describes the instantaneous physical "experience" of matter. Is this correct?
  24. Jun 17, 2010 #23


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    Yes, they can both "simultaneously" be true. To see this better, instead of a single Earth and a single rocket ship, imagine two sets of rocket ships arranged along a line. One set is stationary (in the Earth's reference frame, say), and the other set is moving at the same speed in the same direction in a sort of "parade." Of course, in the frame in which the second set of ships is stationary, the first set is moving backwards with the same speed.

    The first set of ships synchronizes their clocks so they all read the same time in their "rest frame" (the frame in which they are stationary). The second set of ships synchronizes their clocks so they all read the same time in their rest frame.

    In the first set of ships' rest frame, the second set of ships' clocks run at a slower rate than the first set of ships' clocks. In the second set of ships' rest frame, the first set of ships' clocks run at a slower rate than the second set of ships' clocks.

    Yet, whenever two ships (one from each set) pass each other, everyone agrees on the time readings of those two ships' clocks at that moment! (We assume that they pass right next to each other, closely enough that they can read each other's clocks with a high-speed camera or something, without significant error caused by light's travel time between the two ships.)

    How is this possible? Because of relativity of simultaneity, the first set of ships' clocks are not synchronized in the second set's rest frame. Nor are the second set of ships' clocks synchronized in the first set's rest frame.

    I'm sure there are diagrams illustrating this setup somewhere on the web, but I can't find one at the moment. I first saw it many years ago in an article in the American Journal of Physics, I think by N. David Mermin. I'd make up a set of diagrams myself, and attach them, but I'm on the road and don't have the tools or the time to do this properly. If nobody's found something like this by the time I get home in about a week, I'll do it then.
    Last edited: Jun 17, 2010
  25. Jun 17, 2010 #24


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    When we talk about an inertial observer's "experience" we're really talking about a description in terms of the inertial coordinate system associated with his motion. His world line is taken as the time axis of that coordinate system. The time axis is labeled by the times displayed by his clock at each event on it. The assignment of coordinates to events not on his world line is done using a specific synchronization procedure. For example, if he emits a light pulse at (t=-T,x=0), and the light is reflected once and returns to him at (t=T,x=0), the reflection event is assigned coordinates (t=0,x=T).
  26. Jun 17, 2010 #25
    Okay, I busted out the textbook and perused the SR chapter a bit, particularly the muon experiment. This site also helped too by explicitly putting the situation in both reference frames.

    In the Earth Frame, I am at rest with respect to my ruler, so I observe no length contraction. My clock is not at rest with respect to the muon, so I do observe time dilation.

    In the Muon Frame, I am not at rest with respect to the ruler (Earth), so there is length contraction. My clock is at rest with respect to me, so I observe no time dilation.

    Now for the Twin Paradox, the conclusion is the Moving twin is younger because of the asymmetrical journey, i.e. the acceleration undergone and leaving the initial inertial frame. I also of course understand the shortest time-interval being at rest with respect to the inertial frame and that moving clocks undergo time dilation and yield a larger time interval (magnitude). However, it is said they run slowly and in the case of the Twin Paradox the Moving twin is younger. Why is it said it runs slowly when it yields a larger time interval? I guess you could say the proper time finishes more quickly and the moving frame takes more time to end.
    Last edited: Jun 17, 2010
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