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Length contraction and time dilation in spacetime

  1. Oct 27, 2014 #1
    Observers that pass each other with a relative speed close to the speed of light will observe length contraction and time dilation at the other observer.

    In a spacetime diagram, this would be represented by two worldlines making an angle, right? Some textbooks suggest that some of the length is traded for time due to the different spacetime perspectives of the observers. Is there a way to visualize the length contraction and time dilation in such a spacetime diagram?

    However, I would think the mere difference in spacetime perspective cannot be the whole story, because e.g. in the twin paradox, there will be a real change in relative time passage. How does this work, does one of the observers skip time coordinates, or is the spacetime structure different for him?

    In this respect, I also read (e.g. Brian Greene) that all objects have a spacetime trajectory which equals c. Light is special, because the whole trajectory is in space and does not extend in time. In a spacetime diagram, I would think that this implies a vertical worldline for light, which would mean it could in fact not arrive at observers in the future. How come it actually does arrive at obervers in the future?
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  3. Oct 27, 2014 #2


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    In a space coordinate-time you don't see them directly, but in a space proper-time diagram you do:

    Yes, for non-inertial observers space time is distorted.

    He probably means "...does not extend in propertime".

    Because proper-time is the objects time, not the observers time (coordinate-time).
  4. Oct 27, 2014 #3

    Simon Bridge

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    It helps to set it up carefully ... have a read through:
    ... and the others in the series. There is a section on how to show time dilation and length contraction in space-time diagrams.

    It also covers the twin's paradox - both twin agrees that the accelerated twin is younger at the end of the journey but they disagree about how that happened.
  5. Oct 27, 2014 #4


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    Certainly. Let's start with one observer stationary in a space time diagram. Let's assume that he is in a car that is 6 feet long and that he is located at one end shown by the blue worldline in the following spacetime diagram:


    Note the dots that mark off equal increments of time at both ends of the car. I'm equating the speed of light to be one foot per nanosecond and the dots are one nsec apart just like the coordinate grid lines.

    Now we use the Lorentz Transformation process to convert all the coordinates of the events (marked by the dots) in this frame to a new frame moving at 60% of the speed of light to the left. This will make the observer and his car move to the right in this new diagram:


    At 60%c, gamma calculates to 1.25 which means the increments of time on each worldline are dilated to 1.25 nsec of coordinate time which you can easily see on this new diagram. It also means that the length of the car is contracted to the inverse of 1.25 or 80% of 6 feet which is 4.8 feet. Look at the blue line at the Coordinate Time of 4 nsec and you will see that the red line is at 4.8 feet at the same time.

    Now we can add in a second observer with his six-foot long car stationary in this same frame:


    And we can transform the coordinates of this frame to a frame that is moving at 60%c to the right so we get back to the rest frame of the first observer:


    So we have shown how in each observer's rest frame the other observer's time is dilated and his car is length contracted.

    I'm not sure what you mean by skipping time coordinates but I would say that the spacetime perspective does tell the whole store. You can add in any worldlines and use the Lorentz Transformation to see what they look like in any frame.

    I would suggest that you forget about Brian Greene. He only confuses people.

    Light is special in that it always propagates (by definition) along 45-degree lines in all spacetime diagrams. These diagonal lines also show what observers actually see and especially if they send out light signals between them. For example, the blue observer could send out a light signal at the time of -0.7 nsec and wait for a response back from the black observer which he gets at his time of -0.1 nsec:


    From this, he applies Einstein's convention that light took the same time to get to the observer as it took to get back to him. From this, he establishes that the light got there half way between sending and receiving which is at the time of -0.4 nsec. Since the speed of light is 1 foot per nsec, he can also conclude that the black observer was 3 feet away at that time since it took 6 nsecs for the round trip. Later, he observes that the black observer passes him at the time of 1 nsec and so now he can establish the speed of the black observer by noting that it took him 5 nsec to cover 3 feet which is 60%c.

    He also sees that the black observer's clock ticked 4 times during that 5 nsec interval so the Time Dilation factor is 1.25.

    Finally, he can measure the length of the black observer's car by noting that the other end of the car passes him at the time of 9 nsec which means it took 8 nsecs to pass him and at 60%c this means the car is 4.8 feet long.

  6. Oct 27, 2014 #5
    Thank you for your replies. The diagrams nicely show how time dilation and length contraction result from Lorenz transformation. I also understand that Lorentz transformation is at the basis of "now slicing" which is often practiced in block universe models.

    However, I cannot get my head around what nature apparently does to make the Lorenz transformation work. As according to the first reply, spacetime distortion may be at the foundation?
  7. Oct 27, 2014 #6


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    The term "distortion" of space-time is extremely vague and has not been defined yet in this thread so until then I would avoid using the term. Space-time is an absolute entity and is completely independent of the observer (accelerating or inertial). However the 3+1 split of space-time into space + time is observer dependent. A 3+1 split of space-time is simply a foliation of space-time by a family of parallel spatial slices, one for each instant of time relative to a chosen observer such that the spatial slices are surfaces of simultaneity. Then if one inertial observer slices space-time into a family of spatial planes at different instants of time and we boost to the frame of another inertial observer then in this frame the previous observer's 3+1 split will be at a (fixed) angle relative to the 3+1 split of the new observer.

    In other words, the boost (Lorentz transformation) from one inertial observer to another simply alters (by a fixed angle in this case) the 3+1 split of space-time and as a result changes the set of events at any given instant that an observer considers simultaneous to their local clock. This transformation of simultaneous events under the Lorentz boost is what makes space and time, individually, relativized to an observer but space-time itself is, as already noted, an absolute entity. The same exact thing applies to non-inertial observers except their spatial slices can behave in more complicated manners e.g. they can be very complicated curved surfaces instead of just planes or they can be planes that sweep out and overlap with one another as a result, which is exactly what happens in the twin paradox with instantaneous turnaround.
  8. Oct 27, 2014 #7


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    That's not quite the right wording. Spacetime is the same for all observers, whether inertial or not. However, those coordinate systems in which a non-inertial observer is at rest or moving at a constant velocity will be distorted, in the sense that the trajectory of an object experiencing no forces will not be a straight line in those coordinates. (An example might be two people on a merry-go-round tossing a ball back and forth - the trajectory of the ball will be excruciatingly complicated using the natural but non-inertial coordinates in which they are at rest, but will be a perfectly ordinary Physics 101 problem to someone not on the merry-go-round).

    To actually distort space-time itself, you need a gravitational field. In the absence of gravity, space-time is flat and undistorted whether you're accelerating or not.
    Last edited: Oct 27, 2014
  9. Oct 27, 2014 #8


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    As I mentioned in a post above the notion of space-time distortion isn't applicable as long as gravity is not involved. The Lorentz transformations come from assuming that the speed of light is the same for all inertial observers. This assumption is a postulate that Einstein suggested in 1905; it is supported by reams of experimental evidence; and it follows from one natural interpretation of the laws of electricity and magnetism as discovered by Maxwell in 1861.

    There is only one space-time geometry consistent with that assumption, and it is one in which time and space are related according to the Lorentz transformations.
  10. Oct 27, 2014 #9


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    As others have pointed out, there's no spacetime distortion in Special Relativity so that can't be the foundation. The foundation of Special Relativity is its two postulates, the Principle of Relativity and the Constancy of the Propagation of Light. The Lorentz Transformation is derived from those two postulates.

    In your example, the first postulate says that since your scenario is symmetrical between the two inertial observers, what each observer experiences, observes and measures of the other observer has to also be symmetrical. Then the second postulate says that light propagates at c in any Inertial Reference Frame we want to construct. So let's take a look at what each observer sees going on with the other one. To do this, we will draw in light signals coming from one observer and going to the other observer. Here's the first one showing light signals going from the black observer to the blue observer:

    Note that in the bottom half of the diagram, the blue observer receives the signals from the black observer which were emitted at 1-nsec intervals at twice the rate of his own clock. Then, after the black observer passes him, he sees the signals coming in at half the rate of his own clock. Since this is an observation that the blue observer is making, it cannot be dependent or change when we transform to a different frame so let's go back to the rest frame of the black observer and see if that is the case:


    Indeed, we do see that in this frame, the blue observer first sees the signal coming in at twice the rate of his own clock and then half the rate after the black observer passes him.

    Now let's see what the black observer measures of the blue observer's signals sent at 1-nsec intervals. First in the black observer's rest frame:


    We see that the black observer also sees the blue observer's clock going twice the rate of his own and then half the rate of his own when the blue observer passes him.

    And in the rest frame of the blue observer, the same thing holds true:


    The effect that I have been showing you is called Relativistic Doppler, in case you want to research some more on your own.

    Finally, I want to show you how the same timings and calculations that the blue observer made of the black observer's speed as depicted in the last diagram of my previous post (#4) are the same even in the rest frame of the black observer:


    The whole point of all these examples is to show that the Lorentz Transformation process maintains identical observations and measurements for each observer even though the coordinates of the events are completely different. So we should never give any precedence or special significance to the coordinates of any particular frame and that includes concepts of "now".
  11. Oct 27, 2014 #10


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    He it's nicely animated, starting at 1:10:

  12. Oct 28, 2014 #11
    It appears the Lorentz transformation in fact calculates which 3D reflection of the 4D world each observer sees. Relative velocity between two observers results in their 3D reflection, or "now slice" being tilted relative to each other.

    It would seem to me as if nature requires that the 3D reflection or now slice always makes a specific angle with the spacetime vector of an object, e.g. being perpendicular with the spacetime vector, thus explaining the tilting due to relative velocity. Does that make sense?
  13. Oct 29, 2014 #12


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    I tried to make it clear that an observer cannot see any "3D reflection", as you call it. Every observer sees exactly the same thing in every frame. No frame is preferred, not even the rest frame of an observer. That's the whole point of relativity, that all frames are equally valid.

    Again, observers don't have a "now slice". When we talk about an observer's rest frame, it's just a shorthand way of referring to the particular frame in which that observer is at rest. The observer doesn't own that frame and it is no more significant to that observer than any other frame, except that it may be the most convenient frame for him in which to construct a coordinate system long after he has the chance to see remote events. But after he finally does that, he can trivially transform to any other frame

    No. Nature doesn't have a "3D reflection" or a "now slice". Those are man-made artifacts that we create for the purpose of describing scenarios. A "now slice" is no more significant to nature that the "origin" of a coordinate system. What is significant is that the Lorentz Transformation is the correct basis for all the laws of nature.
  14. Oct 29, 2014 #13
    I can totally agree that no frame of reference or now slice can be preferred over the other, and that they are equally real. I believe Einstein once said that there is only now (stretching out over whole of time), each piece of it equally real.

    However I found multiple examples on the internet suggesting that Lorentz transformation in fact is about calculating "3D hypersurfaces" or "time slices" for particular observers or frames of references. I think you cannot deny that observers see a 3D world. If they want to change their frame of reference they have to accelerate. I think what you mean is that no matter what frame of reference, the order of events, i.e. causality is maintained?
  15. Oct 29, 2014 #14
    Special relativity theory was based on postulates that followed from observations; in other words, its foundation is observations. Consequently the "why" is left open, and from the start people have disagreed (you can learn a lot from reading old papers!).
    The only explanations that I'm aware of, historically, are stationary ether (impopular) and block universe (somewhat popular). If you search for "block universe" you can find many fruitless discussions here at Physicsforums and elsewhere; the problem is that we can't know for sure what is outside of our experience ("metaphysics"). The common physicist attitude is therefore "shut up and calculate". :)
  16. Oct 29, 2014 #15


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    No, they are equally unreal. Actually, unless you can define real (or unreal), the term doesn't mean anything.

    Reference, please?

    The world observers see is not the one that you are claiming they see. They cannot see the "3D hypersurface" or "time slice" or "now" that is calculated from the Lorentz Transformation. The Lorentz Transformation needs coordinates of distant events according to one frame before it can establish the coordinates of those events according to a second frame.

    Maybe an example might help: Let's say an observer sees a distant star explode. Does he see it according to the "now" that you keep talking about? I don't think you would say that. He sees it at his "now" long after it happened according to his rest frame. After he sees it, he can go back and insert the coordinates of the event into his rest frame many years ago and he can show how the light from the explosion propagated at the speed of light to him so that he actually sees it when he did.

    Or think about this, of all the billions of stars that we can now see in our night sky, some of them have already exploded and no longer exist in the "now" of our rest frame. We can't see what is going on "now" at distant locations, not even on our own star. Isn't that obvious?
  17. Oct 29, 2014 #16


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    The part of Einstein's second postulate that light propagates at c is not based on observation.

    Did you see the new Relativity FAQ on this subject at the top of the relativity forum page? Let's not invite closure of this thread.[/PLAIN] [Broken]
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  18. Oct 29, 2014 #17
    Sorry, perhaps my phrasing was a bit too compact! Thus I'll elaborate. His theory was indeed, as he mentioned in 1905, based on Maxwell's theory; however there was a good motivation for choosing Maxwell's theory over other theories of light propagation such as those by Stokes or Ritz. He clarified in 1907 that the second postulate was indirectly based on observation as follows:

    "We [...] assume that the clocks can be adjusted in such a way that the propagation velocity of every light ray in vacuum - measured by means of these clocks - becomes everywhere equal to a universal constant c, provided that the coordinate system is not accelerated. [..]
    It is by no means self-evident that the assumption made here, which we will call the "principle of the constancy of the velocity of light", is actually realized in nature, but - at least for a coordinate system in a certain state of motion - it is made plausible by the confirmations of the Lorentz theory [1895], which is based on the assumption of an ether that is absolutely at rest, through experiment [by Fizeau]".
    In the second footnote which I abbreviated he clarifies that he in particular was thinking of the experiment of Fizeau. (Note: I made a small improvement in translation here, the German original has plural "confirmations" and not "confirmation").

    But let's not distract too much from the main topic. I have nothing to add to the old historical discussion here: https://www.physicsforums.com/threads/some-remarks-on-einsteins-1907-paper-on-relativity.575526/
    Has the FAQ recently changed? Anyway, my reply was exactly meant to prevent this thread from getting lost in philosophical discussions, as such are not appropriate for this forum (although I see that even this year a new long philosophical discussion took place; I trust that the OP will be sufficiently informed by reading those old threads ;) ).
  19. Oct 29, 2014 #18


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    This FAQ, which I assume is what ghwellsjr was referring to, was published about a month ago:

    https://www.physicsforums.com/threads/what-is-the-pfs-policy-on-lorentz-ether-theory-and-block-universe.772224/ [Broken]

    Your instinct to not distract from the main topic is a good one. :)
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  20. Oct 29, 2014 #19


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    Maxwell's theory doesn't lead directly to Einstein's second postulate, in fact, Maxwell believed that it indicated that ether could be detected.

    I don't see any difference between what Einstein said in 1907 and what he said in 1905. Both times, he states that we postulate or assume that light propagates at c and we adjust our clocks to make it happen--not the other way around. He doesn't even claim that this "is actually realized in nature". Nature doesn't let us know how light propagates. We can't figure it out by observation or measurement. We impose on nature the propagation speed of light, not because we must, but because we can.

    [/PLAIN] [Broken]
    Einstein's second postulate is an important topic in this thread. I have brought it up at least a couple times. I have tried to emphasize that the postulates are the foundation of Special Relativity (post #9) and that light propagates at c by definition (post #4) so I don't think it is a distraction to point these things out once more.

    He should read the Relativity FAQ. That should be enough.
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  21. Oct 29, 2014 #20


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    Strictly speaking, this is true for the one-way speed of light only, because measuring it requires a clock synchronization convention. We can measure the round-trip speed of light by experiment without such a convention. (We can also measure a lot of other things about how light propagates; I assume you didn't mean "how light propagates" as generally as that phrase actually means when taken literally.)
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