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On the interpretation of a spacetime diagram

  1. Jan 17, 2011 #1
    Below is a spacetime diagram which depicts the movement of a rod, four units long, relative to another rod of equal length. The red lines are the "moving" rod; the purple lines are the "stationary" rod. The gray lines are the world lines of the unit marks (0, 1, 2, 3, 4) on each rod.

    As I attempt to interpret the diagram, I am moved to ask a question which has been asked before on this forum, including by me. There is nothing new in the question itself--only my level of understanding has (hopefully) progressed.

    To set up the question, I stipulate that the stationary rod and the moving rod are identical, having been fabricated on the same high-precision equipment. The stationary rod was then fastened to a bench in the lab, and the moving rod was set in motion. As the moving rod passes over the resting rod, it is moving at a constant velocity of 0.6c.

    As can be seen in the diagram, instruments mounted on the moving rod will measure the resting rod to be 3.2 units long. Likewise, instruments mounted on the resting rod will measure the moving rod to be 3.2 units long.

    My question focuses on the stationary rod, because it has been minding its own business on the lab bench, and its dimensions are known to be the same as when it was fabricated. Knowing that the stationary rod is four units long, would we not have to conclude that the instruments on the moving rod are incorrect when they report its length to be 3.2 units?

  2. jcsd
  3. Jan 17, 2011 #2
    The whole point is that there's no meaning in saying that one rod is "really" moving and the other rod is "really" at rest. Think about the situation from the point of view of of the moving rod: "I've just been minding my business sitting in one place, I was fabricated by equipment which started moving at some point, and my dimensions are 4 units long, the same as when I was fabricated, but now a rod attached to a moving lab bench which just passed by me is claiming that I'm only 3.2 units long. But then again, it thinks it is 4 units long, even though I can clearly see that it's only 3.2 units long. So its measurement instruments are incorrect."
  4. Jan 17, 2011 #3


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    In the case of a relative speed of 0.6c, which each rod correctly measures, gamma is 1.25, which means that each one sees the other one's times as lengthened by a factor of 1.25 and sees the other's rod as being shortened by a factor 1/1.25 which equals 0.8. Let's say that light takes 1 unit of time to travel 1 unit of length (whatever you want that to be) so c equals 1 unit of length per 1 unit of time.

    There are many ways to illustrate how this is happening but one of the easiest, in the case where the two rods pass directly adjacent to each other is to time how long it takes for each rod to pass by the other one.

    So let's look at it from the standpoint of the "stationary" rod. It's clock/timer is running at the normal rate and so when the two rod's meet, it starts its timer. How long will it take for the other end of the moving rod to get to the leading edge of the stationary rod? Well, since the rod is 3.2 units long and it is traveling at 0.6 units of length per 1 unit of time, and since time is equal to distance divided by speed, the time will be 3.2 divided by 0.6 which equals 5.3333 units of time. That is what the stationary rod measures of how long it takes for the moving rod to pass a certain point. Now since length equals speed multiplied by time we get 0.6 times 5.3333 which equals 3.2. (Image that!)

    Now let's see how the moving rod measures the stationary rod's length. We're still examining this from the lab frame. When the two rods meet, the moving rod starts it's timer. How long will it take for the leading edge of the moving rod to get to the trailing edge of the stationary rod? Well, since the stationary rod is 4.0 units long and the moving rod is traveling at 0.6 units of length per 1 unit of time, and since time is equal to distance divided by speed, the time will be 4.0 divided by 0.6 which equals 6.6666 units of time (according to the stationary rod). Now to see what the moving rod measures for how long it takes for a point on the moving rod to pass the length of the stationary rod we have to divide by gamma so the result is 5.3333. Now since length equals speed multiplied by time, we get 0.6 multiplied by 5.3333 which equals 3.2.

    Please note that this is all done in the lab frame. It is also possible to transform the entire scenario into another frame, such as one in which the "moving" rod is at rest but this will result in the exact same calculation as was done for the "stationary" rod so it's rather unexciting.
    Last edited: Jan 17, 2011
  5. Jan 17, 2011 #4


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    On this thread I drew up some illustrations of two ruler/clock systems moving past each other, showing how each system measures the other rulers to be shrunk and the other clocks to be slowed-down and out-of-sync. You can see from the diagrams how the situation is completely symmetrical, and if you add to that the fact that all laws of physics work the same way in the coordinate systems defined by each ruler/clock system, you may get a better idea of why there is no basis for preferring one point of view over the other (including the fact that there's no basis for labeling either 'at rest' in an absolute sense).
  6. Jan 18, 2011 #5


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    Let me offer you a geometrically similar scenario and see what you think:

    Suppose we construct two identical parallel rods, each of unit length, with devices attached to measure projected lengths (e.g. A sliding T). Now we rotate one rod 45 deg and each rod measures the projected length of the other rod. Each rod finds the projection of the other rod to be .7.

    The stationary rod has been minding its own business on the lab bench, and its dimensions are known to be the same as when it was fabricated. Knowing that the stationary rod is 1 unit long, would we not have to conclude that the instruments on the rotated rod are incorrect when they report the projection of its length to be .7 units?
  7. Jan 18, 2011 #6
    Well, if we recognize that we are measuring a projected length, then there is no problem with either measurement. But in that case, we understand that the actual (or real) length of the rod is something other than the measured value.

    Perhaps this is the experiment which can (in principle) be conducted to determine whether length contraction is physically real.
  8. Jan 18, 2011 #7
    What about considering the situation with a focus on the 4-dimensional geometry. The situation described below is very similar to what you've set up with the two poles. However, to give the situation a definite physical character to it we use the well-known pole-in-the-barn example. My Special Relativity physics prof gave this one to us as a homework problem. The challenge was to use space-time diagrams to illustrate how a pole that was too long to fit in a barn in one coordinate system could be observed to fit easily in the barn in the view from another system.

    I think is helpful to first view the pole and the barn as two 4-dimensional objects. The pole flies toward the barn at relativistic speed. The space-time diagram is sketched as the pole (red 4-D object) moving to the left and the barn (blue 4-D object) moving to the right at the same relativistic speed with respect to a rest system (black coordinates). Again, we assert something more than just coordinate systems: We regard the two objects as literally 4-D objects--you construct the 4-D objects by extruding the easily visualized 3-D objects along their respective 4th dimensions. The impression of speed is a manifestation of the relative rotational orientations of the 4-D objects with respect to each other (and with respect to a photon world line).

    Please don't get into the side bar philosophical issue of whether I'm trying to float a literal interpretation of a static 4-D universe here. This is a pedagogical choice. Thus, for the sake of understanding the space-time diagrams, we at least temporarily abandon caution for the sake of getting the hang of cross-section views of 4-D objects.

    Once we accept this 4-D object view, we must then adapt another curious aspect of our 4-D world experience. Observers experience the laws of physics only by living in a continuous sequence of instantaneous 3-D cross-sections of the 4-D universe in which the following results: For any rotational angle of a given 4-D object's world line (angle of the world line of the 3-D extrusion along the 4th dimension world line), any instantaneous 3-D cross-section view of that 4-D object (representing a normal spatial axis) will be symmetrically oriented about a world line representing the world line of a photon of light. That is, the angle between the spatial dimension (axis, i.e., X1') and the photon world line is (for any observer) always the same as the angle between the observer's 4th dimension (i.e., world line direction--X4'). Thus, for two observers moving with respect to each other, their 4th dimension world lines will have different angular orientations with respect to a photon world line--and consequently their spatial axes will have different rotational axes as well. This accounts for the different cross-section views of 4-D objects.

    In the sketch below we have a red coordinate observer (view from the vantage point of the pole) and a blue observer (view from the viewpoint of the barn). The front and back doors are initially open. When the pole reaches the barn (blue guy's view) the blue guy waits until the pole is inside the barn, then he quickly closes both doors, then opens the back door just in time to let the pole fly on through. But, just for an instant the pole is INSIDE THE BARN WITH BOTH DOORS CLOSED.

    However, the red guy (moving with the pole) observes the front and back doors opening at different times, so that, in his cross-section view of the 4-D objects, the pole is never inside the barn with both doors closed (it better not be, since in his world the pole is too long to fit into the barn). Notice, for the red guy, the back door opens before the pole is completely inside the barn. This of course illustrates the length contraction as well as time dilation and differing impressions of simultanaeity.

    The point emphasized here is that the blue and red guys are each viewing dimensions of cross-sections views of objects that are in reality 4-dimensional. The 4-D objects are the reality. Our 3-D images are just particular cross-section views that depend on the realtive orientations with respect to photon world lines. It was important to use a SYMMETRIC space-time diagram in order to sketch the 4-D objects to the same scale on the computer screen.


    Last edited: Jan 18, 2011
  9. Jan 18, 2011 #8


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    Why would we conclude such a thing?
  10. Jan 18, 2011 #9


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    Geometrically, what would a non "projected", "actual" length be? Just the length measured along the axis parallel to the rod? I suppose you could similarly define the "actual" length or a rod in SR to be the length in the rest frame, and say that all other lengths are "projected". In both cases it's just a matter of definition, and there's no reason we must accept those definitions.
    What does "physically real" mean? I would normally interpret that phrase to mean something like "frame-invariant" but that doesn't really make sense here since obviously length is not frame-invariant.
  11. Jan 18, 2011 #10


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    Yes. I think that is the important thing, to recognize that any measurement of a length is a projection onto a plane of simultaneity.
  12. Jan 19, 2011 #11
    If I am the owner of a rod whose value is determined by its length, and I know that the rod is four units long, you can be sure that I will challenge any claim that the rod is in fact only 3.2 units long.
  13. Jan 19, 2011 #12
    It seems to me that for either definition to be meaningful (or "useful", or "an accurate model of reality"), all parties ought to be able to agree on the length of a particular rod if they adopt one definition or the other and measure accordingly.

    If length is not frame invariant, why would time be frame invariant? In the Lorentz transform, distance and time are interdependent and inseparable. The invariant is (of course) the spacetime interval, which is a combination of distance and time.
  14. Jan 19, 2011 #13


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    Well, rest length is something all parties can agree on, but then so is length in some agreed-upon frame.
    Proper time along a timelike path through spacetime is frame-invariant, as is proper distance along a spacelike path through spacetime (which could be imagined as the path of a faster-than-light particle). Both are found by integrating the infinitesimal spacetime interval (possible multiplied by or divided by c to make sure the answer has units of time in the first case and distance in the second case) along the path.
  15. Jan 19, 2011 #14
    Equally, if you measure the length of the rod moving relative to you as 3.2 units and the observer co-moving with the rod will measure his rod to be 4 units long, so he will challenge your claim his rod is 3.2 units long.

    While this interpretation sort of solves the problem, it also sweeps the problem under the carpet. Imagine we have a horizontal rod that is one metre long. We place a light above it and its shadow is one metre long. Now we tilt the rod and the shadow is less than one metre long. The shadow is the projected length, but we understand that the length of the rod has not actually changed and is still one metre long. Is length contraction effectively just measuring the shadow of the moving rod? Well consider this thought experiment and tell me what you think. We have a circular railway track that is ten units long and banked like a "wall of death" so that a train can move at high speed on it. On the track we place ten self powered railway carriages each one unit long and connect them to each other all the way around except at one point. The carriages are accelerated to say 0.8c and due to length contraction the total length of the train is 6 units long. With some careful timing, we can now fit another 6.6666 carriages on the track so that we have a total of 16.6666 carriages, each with a proper length of one unit (as measured by an observer on the carriage) circulating on the 10 unit length track comfortably fitting end to end. If the speed is increased, we can fit further carriages on the track and by continuing this process we can fit as many carriages on the track as we like without any change in the proper length of the track or the carriages. Would you agree in this case that we are not just "measuring shadows"?

    P.S. Edited to correct the number of extra carriages from 4 to 6.6666
    Last edited: Jan 19, 2011
  16. Jan 19, 2011 #15
    Movement around a circular track does not satisfy the conditions of special relativity. I'm not sure that is a fatal problem, but in any case the experiment can be imagined on a straight track, with a gap between two sets of cars.

    On first look, I think this is the same as the pole-in-the-barn scenario. If so, then the answer is that you can't fit more cars in the gap as the speed increases, because (roughly speaking) the gap isn't on the track all at once (that is, at the same time).

    Suppose we accept your premise. There are two observers of the train, moving at different speeds relative to the train. The one observer will gauge that he can put two extra cars in the gap; the other will see room for three extra cars. Which one is right?
  17. Jan 19, 2011 #16
    Yes, but this means that length (and thus time) cannot be measured directly. Which poses a number of practical problems as speeds approach c. Those practical problems leave open the possibility that the theory is inaccurate at speeds very near c.
  18. Jan 19, 2011 #17


    Staff: Mentor

    Yes, that is why invariants like the spacetime interval are so important and useful in modern physics.

    That is always a possibility that we cannot exclude. All we can say is that it is accurate at all speeds we have tested so we have no scientific reason to believe that it is inaccurate at higher speeds.
  19. Jan 19, 2011 #18
    It does satisfy the conditions of Special Relativity. SR can handle acceleration in flat space which is what we are discussing in the case of the relativistic train on the circular track. All observers will agree on the number of extra cars that can fit on the circular track.
    Last edited: Jan 19, 2011
  20. Jan 19, 2011 #19


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    Why not? What does it mean to measure them "directly"? (why doesn't measurement or proper time along a worldline by an atomic clock moving along that worldline count, for example?) Do you think length and time can be "measured directly" in classical Newtonian physics, and if so what's the difference?
  21. Jan 19, 2011 #20

    I have two questions for you:

    1) Are you thinking of your two poles as static 4-dimensional objects? That is, objects that really don't move at all--for those objects it would be hard to define time--they are frozen in place as time flows quite independantly. "But, who or what is doing the moving?", you may ask. Don't worry about it. We just don't talk about that. For pedagogical purposes, you might play like there is some 3-dimensional consciousness that flies along the world line of a 4-dimensional object, "observing" a sequence of 3-dimensional images. Herman Weyl once made a comment something like, "...the observer crawls along his world line..." Actually, it should be more like, "...the observer flies along his world line at 186,000 mi/sec." So, I just wanted to ask if this is the kind of model for your poles that you envision as you pose your question.

    2) Are you considering observers moving along the 4th dimension worldlines as having different 3-D cross-section views of 4-D objects?

    (Refer to my post #7 on page 1)
    Last edited: Jan 19, 2011
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