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Temporal distortion along train DURING Acceleration

  1. Nov 6, 2006 #1
    During acceleration,

    will the train's rear clock appear to tick faster than the train's front clock from stationary frame of reference ?

    im asking because i don't see how else the train's rear clock would read farther into the future than the train front clock. Either the rear clock would appear to speed up or the front clock appear to slow down or both.

    2nd question: if yes to above, will the train's rear clock tick faster than stationary clock from stationary frame of reference during acceleration?
  2. jcsd
  3. Nov 6, 2006 #2
    Please have a critical look at
    arXiv.org > physics > physics/0609118

    Physics, abstract

    Radar echo, Doppler Effect and Radar detection in the uniformly accelerated reference frame

    Subj-class: General Physics

    The uniformly accelerated reference frame described by Hamilton, Desloge and Philpott involves the observers who perform the hyperbolic motion with constant proper acceleration gi. They start to move from different distances measured from the origin O of the inertial reference frame K(XOY), along its OX axis with zero initial velocity. Equipped with clocks and light sources they are engaged with each other in Radar echo, Doppler Effect and Radar detection experiments. They are also engaged in the same experiments with an inertial observer at rest in K(XOY) and located at its origin O. We derive formulas that account for the experiments mentioned above. We study also the landing conditions of the accelerating observers on a uniformly moving platform.
  4. Nov 7, 2006 #3


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    Given that a train approximates "Born rigid motion" which can be roughlly described as keeping the distance between the head of the train and the tail of the train constant, and that there is some inertial "station frame" that represents the station that the train is pulling away from, the following remarks are true:

    1) The tail of the train has a higher coordinate acceleration than the head of the train as viewed in the station frame

    2) The tail of the train also has has a higher proper accelration as read out on an acclerometer travelling with the train.

    3) The head and tail of the train appear, as a result, to get closer to each other in the station frame. The statement about the "distance being the same" applies not to the station frame, but the "train frame".

    3) The point of view of the train is NOT an inertial frame, but an accelerated coordinate system. This is the point of view in which the head of the train has a clock which appears to tick faster than the tail.

    See http://www.mathpages.com/home/kmath422/kmath422.htm for more details about born rigid motion, or any of the several recent posts on the forum talking about this topic.

    It is very helpful to draw a space-time diagram for the situation, because there are several different coordinate systems and attendent notions of simultaneity in this problem.
  5. Nov 7, 2006 #4

    Regarding 1 & 3, I'm good there.

    Regarding 2, I don't see why the train would note any accelrational differential along its length. The train doesn't appear to change at all per the passengers as it transitions frames. From the embankment, what you say would makes sense to me. But from the proper vantage of the train, I don't understand your point there?

    Regarding 4, the 1st sentence looks good. However it seems that during acceleration, clocks drop out of sync only per the embankment vantage, not the train vantage. Or did I misunderstand your point there maybe?

  6. Nov 7, 2006 #5


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    The "proper acceleration", which one feels, can be derived from the coordinate aacceleration via a formula. The key point is that when the velocity is zero, the two concepts are the same. I.e. in a frame in which you start out at rest, your coordinate acceleration is your proper acceleration.

    Given that you've already accepted the point that the coordinate accelerations of the two trains are not equal, you should therfore already be convinced that the proper accelerations are not equal either, for the proper acceleration is the same as the coordinate acceleration when the trains are not moving, and you've already accepted that the coordinate acceleration of the two trains are not equal.

    The formula for correcting coordinate acceleration into proper accelration when one is moving is given at


    One can also see that the accelerations are different by looking at the space-time diagram (more below).

    It's very difficult to accurately describe the situation in words. I really don't know what you mean by "drop out of sync". What exact experiment do these words represent? I cant tell. Describing an experiment in exacting detail would be time consuming, and probably not all that enlightening.

    Thats why I strongly recommend looking at a space-time diagram, such as the one at http://www.mathpages.com/home/kmath422/Image5456.gif. (This is an image from the page on Born rigid motion webpage, BTW). If you don't understand space-time diagrams, now is a great time to learn :-)

    A space-time diagram is just a diagram of a collection of "events". Every different event in space-time has its own unique point. For a problem with 1 spatial and 1 time dimension, a space-time diagram can conveniently be drawn as a 2 d diagram. Several different space-time diagrams may represent the same physical situation, because the labels (coordinates) assigned to specific events in space-time depend on the point of view (POV) of the observer.

    In this particular space-time diagram the POV of the diagram is the station frame. The red line segment on this diagram represents two points on the train "at the same time" as judged in the non-inertial train frame. The path of the train on the diagram can be seen to be a hyperbola (the mathematical analysis justifies this).

    The horizontal red line segment represents the train "at rest" at the station.

    The next horizontal red line segment above the first one represents the two ends of the train a short time later after the train has pulled away from the station. These two points are simultaneous in the "train frame", but you can see they are not simultaneous in the station frame.

    Lines of simultaneity for the station frame would be represented on the diagram by horizontal lines.

    A line of simultaneity for an observer is just a set of points that are regarded as happening "at the same time".

    You can see that the lines of simultaneity for the "train frame" all pass through one point, a "pivot point". This is a consequence of the hyperbolic motion. To really see why this happens, you'd need to do some math.

    Finally, as far as acceleration goes. You can see that the hyperbola on the left has a greater second derivative (curvature) than the hyperbola on the right. This means that the train on the left is accelerating harder than the one on the right.

    At t=0 in the station frame, both frames have not yet started moving, so that their "felt" accelerations will be equal to their coordinate accelerations. Because we demand that the trains stay the same distance away from each other, the train on the right must accelerate less to maintain this distance.

    This is related to Bell's spaceship paradox http://en.wikipedia.org/wiki/Bell's_spaceship_paradox
  7. Nov 7, 2006 #6
    OK Pervect. I know Minkowski worldlines fairly well, so I'll crawl in my hole eventually here and think on it. The web links look good too. Thanx much, highly appreciated. I'll come back & bug you if I get stumped anywhere.

  8. Nov 13, 2006 #7


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