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Special Relativity Clocks

  1. Apr 17, 2010 #1

    JM

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    Consider the following properties of SR clocks. The coordinate systems are the usual K( X,Y,Z,T) and k( x,y,z,t) where k is the coordinate whose origin moves in the positive direction of K.
    1.The clocks of K are synchronized with each other using the procedure given in the 1905 paper. Also the clocks of k are synchronized with each other similarly.
    2.When the origins of K and k are located at the same position the clocks are set to zero.
    3. A fundamental principle of physics requires all the terms of an equation to be expressed in the same units. Thus in the equation T=t/gamma, an example of time dilation, T and t must have the same units, and thus tick at the same rate.

    Don't these properties lead to the conclusion that the clocks of K and k must always be synchronized?
     
  2. jcsd
  3. Apr 17, 2010 #2
    There's two problems here. First, in point 2 you said that K and k are synchronized when they are in the same position at the same time. However, the meaning of "at the same time" depends on what coordinate system you're using. K may believe that the origins were at position X at the same time, but k will disagree. So you are presupposing that simultaneity is absolute, and you're using that assumption to try to disprove time dilation. But in relativity the question of whether two events are simultaneous depends on what inertial frame you're in.

    Also, in response to point 3, just because two variables have the same units doesn't mean they're equal. Think about it: if T=5s and t=10s, then they both have the same unit, which is the second, but they have different magnitudes.
     
  4. Apr 17, 2010 #3

    Dale

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    By this logic, since the perimeter, p, of a square is related to the length of a side, s, by the formula s=p/4, s and p must have the same units (true) and thus be the same length (false).
     
  5. Apr 17, 2010 #4

    JM

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    I gather that you agree that T and t must be expressed in the same units. What that means is that the clocks used to measure T and t must tick at the same rate, not that T and t are equal. So, are the clocks ( not T and t ) synchronous?
     
  6. Apr 17, 2010 #5

    JesseM

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    "Tick at the same rate" is meaningless unless you pick a coordinate system to measure the coordinate time between successive ticks of each clock so you can compare their rates. And there is no inertial frame where they tick at the same rate.

    Consider two Cartesian coordinate systems on a 2D plane, with the x-y axes of one coordinate system rotated at an angle relative to the x-y axes of the other. Each coordinate system can measure distance in meters, but if you draw two dots on the plane and ask for the difference between the y-coordinates of the dots in each coordinate system, they can disagree even though they both use the same units to measure distance along the y-axis. Similarly, if you pick two events in spacetime and ask for the difference between the t-coordinates of the events in two different inertial frames, they can disagree, because their time axes are rotated relative to one another.
     
  7. Apr 17, 2010 #6

    Dale

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    No, they are not.
     
  8. Apr 18, 2010 #7
    From the perspective of K, more time elapses between ticks of the k clocks than between ticks of the K clocks. From the perspective of k, more time elapses between ticks of the K clocks than between ticks of the k clocks. That is the general meaning of time dilation, and time dilation is completely symmetric among inertial frames. Similar remarks apply to the synchronization of distributed clocks. In the opinion of K, all of the clocks at rest in K are synchronized, and the clocks at rest in k are out of synch. It's all implicit in the Lorentz transformations or, if you prefer, the Lorentz transformations can be derived on the assumption of length contraction, time dilation, and the relativity of simultaneity.
     
  9. Apr 18, 2010 #8
    All clocks tick at the same rate in their proper frame. That doesn't mean they all measure the same time interval between the same two events.

    According to your logic, if the sun casts a shadow of a stake in the ground, then all rulers should measure the same width of the shadow at different locations because their units of measurement are identical.
     
  10. Apr 18, 2010 #9

    JM

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    Jason, can we examine this statement a bit? Do you mean that the rate of ticking of the clocks of K with respect to the K frame is the same as the rate of ticking of the clocks of k with respect to the k frame? If so, that is the point I am trying to make.

    This property doesn't make T and t equal, perhaps I should have said that the relation T = t/gamma implies that the clocks measuring T and t are ticking at the same rate.
    JM
     
  11. Apr 18, 2010 #10

    JM

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    Would you mind explaining in what respect you disagree with the properties of clocks?
     
  12. Apr 18, 2010 #11

    JM

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    Jesse, Think about it this way: If we want to compare the duration of an event as seen by two observers isn't it necessary that both observer use clocks that tick at the same rate? If the clocks tick at different rates how can we tell whether the disagreement is due to axis rotation or tick rate?
     
  13. Apr 18, 2010 #12

    Dale

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    You asked if the clocks in the different reference frames were synchronized. The answer is "no". This is called the relativity of simultaneity. A system of clocks which is synchronized in one frame will not be synchronized in any other frame.
     
    Last edited: Apr 18, 2010
  14. Apr 18, 2010 #13
    That's incorrect. "At the same position at the same time" is a frame-invariant concept. Simultaneity is relative, but when two events are simultaneous in frame K, the time difference between the events in another frame k is proportional to the distance between the events. If two events occur at the same time and place in one frame, they do so in all frames; the theory would hardly make sense otherwise.
     
  15. Apr 23, 2010 #14
    I'm not so sure about that. Suppose you have a train of length L traveling speed v to the right. If two beams of light are emitted from either end of the train, at the same time according to the train's reference frame, then they will arrive at the center of the train at the same time according to the train's reference frame. But if I'm not mistaken, according to an observer on the ground they will have arrived at the center of the train at different times. Correct me if I'm wrong, though, since I haven't really studied special relativity in great detail. What I know comes mainly from popular books, which can be misleading.
     
  16. Apr 23, 2010 #15

    JesseM

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    I don't understand what you mean by "due to axis rotation or tick rate"--how are those two distinct possibilities? The fact that the axes of one frame appear rotated in a particular way from the perspective of another frame, combined with the fact that a physical clock at rest in a given frame always keeps pace with that frame's time coordinate, implies that a clock which is moving in some frame must tick at a different rate relative to the frame's time coordinate than a clock at rest in that frame. There's no way the first part could be true without the second part also being true! It's not meaningful to ask whether the difference in time intervals between two events is "due to axis rotation or tick rate" unless you can explain what it would mean to have a situation where it was due to axis rotation but not tick rate, or a situation where it was due to tick rate but not axis rotation.
     
  17. Apr 24, 2010 #16

    JM

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    I think you all are missing the point. Your comments don't seem to be related to the properties listed in the first post. Why jump to Relativity of Simultaneity? Does the idea that k sees Ks clocks as out of synch change the property that K sees his own clocks as in synch? Do you think that T and t can be expressed in different units in the equation t = T/m?
     
  18. Apr 24, 2010 #17

    JesseM

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    My analogy of spatial coordinate systems in 2D directly addressed your first post, can you tell me if you think it's not analogous to the situation you were discussing involving clocks?
    It's not something they "see", it's how they each define what it means for two clocks to be "synchronized" (i.e. how they assign time-coordinates to events, since two clocks are synchronized in a given frame if they both show the same reading at the same time coordinate). In much the same way, if you have two x-y coordinate systems on a 2D plane, with one system's y-axis rotated relative to the other system's y-axis, then they have a different definition of what it means for two points to have the "same y-coordinate".
    If that's supposed to be the time dilation equation where t is the time between events in the frame where they happen at the same location, T is the time between the same events in another frame moving relative to the first, and m is the relativistic gamma-factor, then in that case t and T must be in the same units. In terms of the spatial analogy, this is just like how if you have two points on a 2D plane, different coordinate systems can disagree on the difference between the y-coordinates of those points, with dy = A*dy' (where A is some constant), in spite of the fact that both dy and dy' are expressed in the same units. If you think there's some problem with the idea that two frames can use the same units yet disagree on the time between a pair of events, you need to explain whether you also have a problem with the idea that two 2D coordinate systems can use the same units yet disagree on the difference in y-coordinates between a pair of dots, and if you don't have a problem with the second you need to explain what the relevant difference is.
     
  19. Apr 24, 2010 #18

    JM

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    Doesn't this refer to point 3, discussed above?
     
    Last edited: Apr 24, 2010
  20. Apr 24, 2010 #19

    JesseM

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    Points 1 and 2 can also have analogues in the geometric analogy. In place of a bunch of clocks at rest in K, you can have a bunch of rulers which are aligned parallel to the y-axis of one of the 2D coordinate systems (we can call this coordinate system K as well). In this case, saying that the clocks at rest in K are synchronized is equivalent to saying that these parallel rulers are all arranged so that any given marking--say, the 3-meter mark on each ruler--is at the same level relative to the y-axis of K, so that identical markings on each ruler have the same y-coordinate in K. In this case the parallel rulers are analogous to the worldlines of the clocks, and the marks on each ruler are analogous to the tick-events on the worldline of each clock. And of course, you can have a different set of rulers which are arranged to be parallel to the second 2D coordinate system k.
     
    Last edited: Apr 25, 2010
  21. Apr 26, 2010 #20

    JM

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    So, where do we stand? Given my clarification of point 3, do you agree with the properties listed in post 1, and do you think that they lead to the conclusion that the clocks of both frames K and k are in synch?
     
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