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Relativistic Paradox

  1. Oct 29, 2007 #1
    Say 2 particles A and B.

    Particle A is stationary in this inertial reference frame.
    Particle B is moving with constant velocity v=0.8c

    Both is a similar isotope of the same radioactive element.(hence, supposedly same half-life)

    Q1: Which will decay faster in this reference frame?
    In theory, it's A(stationary), B will decay slower due to time dilation.

    However, if we change our perspective and take a different reference frame, B is stationary in this refernce frame. Hence, A is moving in the opposite direction with v=0.8c relative to B.

    Q2: Which will decay faster this time?

    Not sure if there a paradox in this 1. Just a problem that occurred to me that troubles me a lot.
  2. jcsd
  3. Oct 29, 2007 #2


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    Why does it bother you? The one that is at rest with respect to you will, on average, decay faster than the one moving with respect to you (on average since decay is a statistical phenomenon)
    Last edited: Oct 29, 2007
  4. Oct 29, 2007 #3
    hmm.. which means if B is moving with respect to me, B is decays slower.
    If A is moving with respect to me, it decays slower.

    So if there are simultaneously 2 observers, both is stationary with respect to A and B respectively.
    Each of them will observe a different outcome?

    Is there a paradox in this reasoning?
    Last edited: Oct 29, 2007
  5. Oct 29, 2007 #4

    Shooting Star

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    The other way round.
  6. Oct 29, 2007 #5


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    oops, you are right. I typed the wrong thing. I will edit that. Thank you.
  7. Oct 29, 2007 #6

    Not a paradox really, more so that one has to accept that even the simultaneity of events of relative to motion, so yes, different observers will record different outcomes (the simultaneous events being, say, a certain time on the observer's stopwatch and the decay of the particle, so will both would record the same decay times for their respective at rest and in motion particles, they would say that the other observer mistimed one particle, not recording it's decay until not only after it had decayed, but after the other had as well).
  8. Oct 29, 2007 #7


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    Most "paradoxes" in relativity are solved by understanding the relativity of simultaneity, and this one is no exception. Wildreamz, are you familiar with the idea that different reference frames define simultaneity differently? That is, if two events happen at the same time-coordinate in one frame, then the same two events can have different time-coordinates in another frame, and different frames can disagree about which event happened earlier and which happened later?
  9. Oct 29, 2007 #8
    in tis cse tthey will disagree on the outcome however using special relativity, they would agree with the other observer by transformining there coordinates into theres.
  10. Oct 29, 2007 #9
    hmm.. your explaination is profound, I find it difficult to visualise. But I believe there must be 1 particle who will decay first.. Both obervers obtaining different results(meaning: A observes that B decays first while A observes that B decays first) cannot be possible.

    In the Twin "Paradox", stella experienced acceleration, so there is a change in inertial reference frame, so it is obvious that time pass more slowly when she approaches the speed of light. So when she return, she would be the 1 who aged less than her twin.

    In this case, since both particle "exist" at the start of the question at different velocities, it is impossible to determine which would decay first, unless we know which of it has accelerated from rest. This is my understanding. If both of them "come into being" at different velocities. I think the only logical prediction is that they will decay at the same rate.

    These are my understanding.. correct me if im wrong.
  11. Oct 29, 2007 #10


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    The relativity of simultaneity is a basic part of the theory of relativity! If you think it cannot be possible than you are disagreeing with relativity itself.
    In the case of the twin paradox, both twins reunite at a single location, so there can be no disagreement about their ages. It's only for distant events that different frames will disagree; for example, if the trip lasted 8 years for Stella and 10 years for Terence, and they were both 30 years old when Stella departed, then all frames would agree that when they reunited Terence was 40 years old and Stella was 38, but they could disagree on what age Terence was when Stella was partway through the journey at age 33; in one frame Terence might be 34 at that moment, in another frame he might be 32.

    So, if you want to set things up so that both isotopes are at the same location when one of them decays, then there would be no disagreement about which decayed earlier. But if they'd both been moving towards each other at constant velocity, having been created at different locations, then there would be a disagreement about which was created first, so that the two frames would disagree about the total span of time between the moment each isotope was created and the moment they met and one decayed (one frame might say the one that decayed first did so because it was aging more rapidly, another frame might say it was decaying more slowly but it still decayed first because it was created at a much earlier time).
  12. Oct 29, 2007 #11
    Thank you for all your helpful replies. My course on Special Theory of Relativity did not touch on time coordinates, I shall do my own read-up on the topic. The relativity of simulataneity is also quite difficult to visualise but i think i get abit of the picture. Meaning, a certain event A happens before event B might be true to 1 observer but not necessarily true to another observer at a different reference frame.

    So it is possible that the 2 oberservers in my example above observed a different outcome.
  13. Oct 30, 2007 #12

    Shooting Star

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    At least, get your mis-understanding technically correct. Each frame observes the particle in the other frame to decay later (on the average).
  14. Oct 30, 2007 #13
    lol, ur right, i messed up again.
  15. Oct 31, 2007 #14
    Can i add on to the problem:
    Im an observer with the same reference frame as A, upon observing that A has decayed(on average) first, i accelerate instantaneously until i have the same velocity as B(same reference frame). What will i observe?

    Will i see that A has miraculously "gone back in time" and reappeared? Because SR of simultaneousity predicts that B is "supposed" to observe that A decayed slower. (paradox of causality?)


    Would i see that B has suddenly "gone forward in time" and dissappeared, because in B's reference frame, the sequence of events is: B decays-->A decays-->I accelerated. (maybe no paradox of causality?)

    I think case 2 is more likely but i would like to hear you guys opinions.

    ps: Since there is an upper limit to speed (c), is there an upper limit to acceleration?
    Last edited: Oct 31, 2007
  16. Nov 2, 2007 #15

    Shooting Star

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    Good question!

    The ordering in B’s frame is B decaying --> A decaying --> you starting to move.

    You starting to move and A decaying are causally connected, because you start to move after seeing that A has decayed. But B’s decay is not causally connected to either of the other two. That’s because a light signal sent from the location of A at the time of A’s decay cannot reach B before it decays. And you cannot reach faster than light, whatever your acceleration wrt to frame of A.
    Last edited: Nov 2, 2007
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