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The Physics of Length Contraction of physical objects and distances

  1. Jul 11, 2013 #1
    I've been trying for a long time to get an answer to the question, "What is the physics or mechanical explanation of contracting physical objects and the distances between them?" I understand that the phenomenon is theorized to depend on different observations from different frames of reference going at relativistic velocities and contracting in the direction of travel of the observing frame. But how can different observations of the same object or distance change (shorten) that physical object or distance?

    Example: Earth's diameter is nearly 8000 miles. Special relativity (SR) theorists claim that a frame approaching at .866c would measure its diameter (in the direction of the approach) to be about 4000 miles. But of course Earth's diameter would not actually physically shrink by half in that case, so how is the paradox resolved, given that all frames are said to be be "equally valid" in SR, and also that "length is not invariable" (it varies.) How is "proper length" (the 8000 mile diameter) reconciled with frame dependent length (4000 mile diameter) in this case?

    As for distance between objects, same question... Example: The astronomically determined distance to the Sun (ave. AU) is about 93 million miles. But a frame theoretically passing through our solar system at .866c would, according to SR, measure the AU to be about half that. Yet Earth obviously does not move 46.5 million miles closer to the Sun.

    I know that length contraction is the math reciprocal of "time dilation," but how can a slower ticking clock in that fly-by frame cut the actual distance to the Sun in half?

    Ps: If that clock showed only 4 minutes elapsed time for the Earth-Sun journey, would that mean that SR claims that the frame (future ship or whatever) made the journey twice as fast as light... which requires 8+ minutes to go the 93 million miles.

    I would very much appreciate help in resolving the above related time dilation and length contraction paradoxes.
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  3. Jul 11, 2013 #2


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    As far as I understand it, there is no such thing as "real length" of an object in SR. Coordinate systems that differ by a Lorentz transformation are as equally valid as are coordinate systems that differ by a rotation of coordinate axes or a displacement of the origin.

    Proper length of an object is its length measured in the object's own rest frame.
  4. Jul 11, 2013 #3


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    Because the observed length of an object, or the observed distance between two objects, depends on the observer, not just the objects.

    This is really no different conceptually than the fact that the same object can have a different apparent size when viewed from different angles. A coin, for example, has a very different apparent width when viewed face-on than when viewed edge-on. In spacetime, moving at different speeds corresponds to viewing things at different angles (I think this is what hilbert2's comment was getting at). The only added twist is that the "angle" involves time as well as space, which is why there is time dilation as well as length contraction.
  5. Jul 11, 2013 #4


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    It doesn't shrink. In that frame it was always an ellipsoid with a minor axis of about 4000 miles. There is no shrinking involved.
  6. Jul 11, 2013 #5
    I feel so bold approaching this from such a different angle (no pun intended lol)

    PeterDonis's answer is awesome, & Dalespam is right, "shrinking" is a misnomer, and contracting takes on a different meaning. In other words shrinking and contracting are not synonymous in this context. and it's ALL about how we are measuring & what is being measured, so the coin analogy is a wicked perspective...very visually "revealing".

    The universe is a causal system*, generally what this means is one thing causes another and in that order. Oddly it must be said that the future does not effect the present (or the past).

    Every observer must agree on this order of "happenings". Long and short of it is speed must have a limit for this to work, but results in observer dependent measures of time/length.

    What is neat is while the length of the planet changed, because of as you mentioned, the math reciprocal time dilation, really that spacetime calculation of "distance" across the planet hasn't changed at all. Causally everything is physically the same. But it absolutely is length contracted...[STRIKE]as in less length[/STRIKE].

    I find from this causal perspective it's better to think of time dilation as being "more time"; as in taking more time for the next "physical occurrence" due to a "contracted" length. all caused by particular angles of view through 4D of these "happenings" via comparative motion/speed.

    It's remarkable how little spacetime has to do with time dilation / length contraction. :biggrin: To clarify I mean that spacetime really doesn't seem to have any effect on matter. Perhaps more diplomatically, spacetime seems physically elusive
    Last edited: Jul 11, 2013
  7. Jul 11, 2013 #6


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    Thanks for the kudos, but unfortunately I'm going to have to respond by pointing out some things in your post that are, at best, misleadingly phrased.

    This is ok, but one could also add that spacetime has a causal structure, determined by the light cones: two events can only be causally connected if they are in each other's light cones--the one that's in the other's past light cone is the "cause", and the one that's in the other's future light cone is the "effect". This gives the ordering you mention.

    Yes, but note that this only applies to events that can be causally connected, i.e., within the light cones. Events that are outside each other's light cones (i.e., spacelike separated) have no invariant ordering.

    Exactly right; it's a pity that you followed this immediately with...

    ...which, with the word "absolutely", gives absolutely (pun intended :wink:) the wrong impression. Length contraction is frame-dependent, not "absolute".

    Not sure what you mean by this; the structure of spacetime certainly does have plenty to do with why time dilation and length contraction are observed. Your clarification doesn't help much:

    This doesn't seem right, since the geometry of spacetime determines which curves are geodesics, which certainly does have an effect on matter.

    Measuring spacetime curvature is just measuring tidal gravity, which isn't elusive at all. Measuring the "effect of spacetime" on an object's motion is just measuring its proper acceleration (i.e., whether or not it is moving in a geodesic, and if it's not, how much it's not, so to speak), which also isn't elusive. So I'm not sure what you mean here either.
  8. Jul 11, 2013 #7


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    Doesn't that go without saying :wink:?
  9. Jul 11, 2013 #8
    I can't quite follow the light cone description or what it means to say they determine causal structure. Light-cone graphs / visuals are structured according to "rules". the units are not arbitrary, they are.. that is must be properly orientated. The elsewhere area is physically meaningless. A great example are the galaxies [STRIKE]fading[/STRIKE] breaking from causal connection due to spacetime expansion.
    Pulling them out of our "direct" future! (hows that for poetic lol)

    Peter of course when you read the term length contraction you understand what I mean by the "modifier" absolutely before the term length contracted when the question is "is it really contracted". So from a causal perspective it absolutely is length contracted. Of course as is the underling assumption for this discussion, this is a frame dependent magnitude, which we absolutely do not mean proper length. I re-read what I wrote, I took it to far saying "as in less length", that is flat out wrong..i had to strike through it lol .

    I was being tongue in cheek to a degree with spacetime not effecting anything, we measure the spacetime only between things.

    Mass/energy effects spacetime, not the other way around. I would guess GR presumes/calculates spacetime is perfect (straight lines we all agree are straight) when "empty/void", boo if it is mathematical nonsense.

    It is the "things" that seem to length contract, and dilate. We don't measure tidal gravity itself by measuring how it effects spacetime. Is tidal gravity a physical property? In other words your measurement doesn't explain what it is that's distorted. We presume it's spacetime that's distorted, and for good reason our clocks and rulers become comparatively distorted, but alas those are objects themselves and not some magic spacetime-interval-measuring-stick that isn't part of the continuum. And yes gravity is very evident. Note it's "fictitious" force label & how we have yet to detect anything physical about it. The most clear way for me to put it is spacetime is not a field, it's just geometry, kinematics. Other then that it leaves matter alone. Bottom line I truly think the "physical" nature of spacetime is perspective/opinion.

    Kinematics does explain proper acceleration, and not that I know certainly, but would be surprised to learn spacetime could also explain proper acceleration.

    Physically elusive refereed to my understanding of quantum gravity, standard model stuff.

    EDIT: reading about causality on wiki, it says that in GR acceleration is not considered an effect...GR ha, more like grrrrrr :smile:

    I don't know the definition well enough, is gravity part of kinematics?
    Last edited: Jul 11, 2013
  10. Jul 11, 2013 #9
    Yea, my confidence in his replies says it does.

    And what's great is even if someone takes a point of contention with him, he sees them to the end...most often until they understand him. A teacher with patience is a great teacher!
    Last edited: Jul 11, 2013
  11. Jul 11, 2013 #10
    Just a quickie until I find more time to ask the forum to reply to the specifics of the apparent paradoxes as I presented them.
    PeterDonis answered my question,..." how can different observations of the same object or distance change (shorten) that physical object or distance?"... saying:

    "Because the observed length of an object, or the observed distance between two objects, depends on the observer, not just the objects."

    But my question is how the physical length of objects (and the distances between them) is supposed to become shorter as a result of differences in observation... if in fact a "real world" exists and has inherent, intrinsic properties, as naturally formed and distributed in space, independent of differences in observational perspectives, i.e., frames of reference.

    Or is it believed, with Einstein, that there is no "real world" independent of observation. In that case, what was the shape (diameter) of Earth and the distance to the Sun before any observers were present?
  12. Jul 11, 2013 #11
    The real (physical) world is FIRST about invariants. calculations with measures the whole (local:smile:) universe can agree on i.e. get the same magnitudes/results/determination. One of these fundamental physical attributes is causality, for the whole (local:smile:) universe.

    The physical important part is the actual "interactions" and spacetime interval between them.

    We are not blobs of mass-less EM or "overly" dense mass so must measure a spacetime interval with length and time separately to calculate the interval I think c is the "conversion" for this unit:tongue2:. Downside to living within the "speed is allowed" zone is it gives goofy measures of spacetime if there is comparative motion, length & time give different results, the interval is the same. Why? Because spacetime is physically even more simple than both length & time, causation. A concept nearly as simple as "things don't appear from nothing", but has remarkably deep consequences (differential aging).

    It's merely about intervals between happenings. What a kick in the walnuts to my ol' friends Simultaneity, ruler & clock :smile: So what's physically important is how "it all plays out" not specifically where & when, makes sense when everything is all moving around with their different points of view lol
    Last edited: Jul 11, 2013
  13. Jul 11, 2013 #12

    In the picture below two plates are moving to the right at speed 0.99999999999999 c.
    There are also two static observers.

    Code (Text):

    |     |

    The plates emit light, because they are radioactive. The emitted light is beamed to the right, because of the high velocity.

    The observers see both sides of the rightmost plate very well, because of the aforementioned beaming effect.
    Therefore we can conclude that the left plate does not see the other plate very well.
    The plate will say: "The other plate seems to be very far away."
  14. Jul 12, 2013 #13
    I'm not seeing the question being directly addressed, or I'm missing something...

    If I am touring the galaxy in my space craft and notice that a star is flattened, its planets are flattened, and their orbits appear to be flattened, that is going to strike me as interesting. When I park close to the system to take a look, I'll notice that the flattening is now absent. As I go on my way I'll see they appear flattened, again.

    In order to believe that the flattening is "real" (not sure the right word to describe existential vs illusory), I would seek an explanation... for the cause of what appears to be a deforming variation of the gravitational field of each planet and their star - a deformation that varies in intensity and in its axis of bilateral symmetry based on my craft's speed and line of approach or departure orientation.
    A revolving planet would appear to keep its plane of flatness normal to my line of sight in spite of its volume appearing to be displaced and squeezed in an amazing way to maintain the flatness orientation, likewise the star. I would expect both entities should be destroyed as well as the orbits, because I don't think Kepler's laws would be observed. From my perspective when in relative motion, it would seem like the laws going on around the star and its planets are "not holding good"...

    I would be suspicious that things seem to settle down and appear fine when I park and observe at relative rest, yet appear to attach unreasonable physical measures and dynamics when observed in relative motion.

    The heart of the question is; what lines of thought encompass all this without attributing the flattening and otherwise to "illusion", and how is the potentially infinite variety of observable deformations (frames wrt) reconciled? That is, what are the assumptions that must be made (and those that must be discarded) for the "normal" and the flattened views to be understood as coexistent without contradiction?
  15. Jul 12, 2013 #14


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    There is no contradiction between the physics in different frames, because all laws of physics are Lorentz covariant. If a process obeys laws of physics in one frame, then it does in all others, too.
  16. Jul 12, 2013 #15


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    There are "inherent, intrinsic properties, as naturally formed ... independent of differences in observational perspectives", but length is not one of those properties. Those intrinsic properties are called "invariants".
  17. Jul 12, 2013 #16
    It cuts the measured moving distance in half, thanks to a difference in clock synchronisation. The essential point which Einstein stressed in his introduction in 1911, is that clock synchronisation is necessary for the measurement of moving lengths (in fact you can do without, but whatever you do still includes an assumption about distant time).
    This is a matter of not mixing up reference systems: a light ray still goes faster, and will arrive before the ship - and for the light the trip even takes even zero time! (if light could observe and establish a valid or useful reference frame, which it can't). However, it is true that in that sense one is not hindered by the light speed limit, as our "time" is slowed down; as Einstein remarked in 1905, because of that effect the speed of light plays physically the same role as infinite speed (in classical physics).

    It only depends on the observer in the sense that the observer is free to choose the synchronisation according to convention and his/her choice of inertial reference system. That should not be confounded (as often happens!) with QM effects of "observation" (interaction with detector).
    Last edited: Jul 12, 2013
  18. Jul 12, 2013 #17


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    Yes - pre-relativity, length was an invariant, independent of the observer. Post-relativity, length is no longer an invariant, is is not a property of the object alone but depends on the object and the observer. If you have an object, but don't know who is observing it, you don't know it's length.

    What is invariant, observer independent, and a property of the object is the Lorentz interval. As applied to length, this would be the "proper length".
  19. Jul 12, 2013 #18


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    They determine causal structure because no causal influence can travel faster than light.

    If you pick any event, call it event E, the past light cone of E is the set of all events that can send a causal signal to E; obviously this set is bounded by the set of incoming light rays at E, hence the term "light cone".

    Similarly, the future light cone of E is the set of all events that E can send a causal signal to; obviously this set is bounded by the set of outgoing light rays at E.

    I'm not talking about visualizations or graphs or spacetime diagrams. The light cone structure of spacetime exists independently of all those things; it's an invariant part of the underlying geometry of spacetime.

    Huh? There are certainly events there, and those events can emit causal influences that we can eventually detect. (Note that the "elsewhere" area is *not* the same as the region beyond the cosmological horizon--see below.) Similarly, we could have emitted causal influences in our past that are now being detected at events in our "elsewhere" region. The fact that there can't be a causal connection between a particular event's "elsewhere" region and that particular event in no way implies that the "elsewhere" region is physically meaningless.

    As noted above, this is *not* the same as the galaxies being in our "elsewhere" region. There are (presumably--see below) events on the Andromeda galaxy's worldline in our "elsewhere" region that we will someday receive causal signals from; we just have to wait for a couple million years (by Earth clocks). I say "presumably" only because it is in principle possible that something could have happened to destroy the Andromeda galaxy after the last light signals we have currently detected from it were emitted; but would you care to give any odds?

    OTOH, galaxies that are moving beyond our cosmological horizon are no longer able to send causal signals to us at all--not even if we wait for an infinite amount of time by Earth clocks. This is why the term "horizon" is used, and why it's a very different thing from just being spacelike separated.

    Sorry, I'm still not buying it. What does "from a causal perspective" mean? What specific causal properties of the object are different when it's length contracted, compared to when it's not? (Btw, this is a trick question: the correct answer is "none of them". That's why I'm objecting. Length, or length contraction, is not a causal property; it can't be since it's frame-dependent, and causality is not.)

    Really? There's no spacetime inside the Earth? Or inside your body? Or inside an atom? If so, then our theories of the internal structure of these objects need some serious revision, since they all assume there *is* spacetime inside the objects.

    Huh? Spacetime does affect mass/energy, by telling it how to move (in John Wheeler's phrase). I don't understand what you're getting at here.

    Sure we do. Take two rocks and hold them high above the Earth at slightly different altitudes. Then drop them in free fall (we're assuming we're well above the atmosphere so there is no air resistance). They are initially at rest relative to each other, but they don't stay at relative rest; their separation increases with time. This is a direct measurement of tidal gravity (one component of it, anyway) using its effect on spacetime--specifically, on neighboring geodesics of spacetime.

    Sure. Why not?

    You appear to be confusing tidal gravity with length contraction and time dilation. They're not the same. In my description of a direct measurement of tidal gravity, above, nothing at all was distorted except the neighboring geodesics described by the motion of the two rocks--i.e., spacetime. If you wanted to confirm that nothing else was distorted, put very sensitive strain gauges and accelerometers on the rocks and verify that they read zero throughout the experiment.

    Really? I guess you've never fallen down. :wink: Also, tidal gravity is part of "gravity", and I described above how to detect something physical about it.

    I think you are confused about a number of things; see above.

    Huh? I don't understand the distinction between "kinematics" and "spacetime" that you're making here. Proper acceleration is just a measure of how far a particular worldline departs from being a geodesic, which is a property of spacetime. So I would say that "kinematics" and "spacetime" are really different ways of describing the same thing, or at least aspects of the same thing (at least, as you appear to be using the term "kinematics" here--see below).

    I would say it depends on what part of "gravity" you are referring to, but on the whole, I would consider it part of dynamics, since it involves the Einstein Field Equation, which goes beyond kinematics as the term is usually used. However, I think all of these terms get used differently by different authors, so it's hard to be sure.

    My personal usage goes something like this: "kinematics" refers to properties that can be defined and studied without asking any questions about causes. If you tell me an object is moving on a geodesic, that tells me certain kinematic properties of its motion that I can use without knowing why it's traveling on a geodesic, or why that particular worldline is a geodesic instead of some other one (i.e., why spacetime in that region has the particular geometry it has). "Dynamics" refers to properties that you have to have some knowledge of causes to define and study. For example, the Einstein Field Equation is a dynamic equation, since you have to know the stress-energy tensor, which means you have to know what kind of matter/energy you're dealing with, how it works, what it's made of, what interactions it has, etc., etc. But the boundaries of these terms are fuzzy, and I prefer not to use them if I can avoid it.
  20. Jul 12, 2013 #19


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    Your question is really about how to measure the distance between two objects or two points on the same object. If you are at rest with respect to the object, then it's pretty easy--you just get out your ruler and measure it. Of course, nowadays the cool way to measure a distance is with at Laser Rangefinder. The principle of such a device is actually much better for really long distances and allows the observer to make the measurement without actually traversing the distance.

    So let's consider the process of measuring a distance as great as the diameter of the Earth. To make it easier to illustrate, I'm going to assume that the object is simply a very long rod with reflectors at both ends and the observer at the near end. I'm going to assume that the speed of light is 1 foot per nanosecond or 1 million feet per millisecond. I'll place the length of the rod at 40 million feet which is close to 8000 miles.

    To illustrate the process, I have drawn a spacetime diagram for the Inertial Reference Frame (IRF) of the observer at the near end of the rod in blue and the reflector at the far end of the rod in red. The dots show the Proper Time for each observer/object:


    At the Coordinate Time of 0, the observer points his Laser Rangefinder at the reflector at the other end of the rod and pulls the trigger. A light signal, depicted by the thin blue line traverses the distance to the red reflector which hits it at the Coordinate Time of 40 msecs. The return signal, depicted in red, also takes 40 msecs and arrives back at the observer at the Coordinate Time of 80 msecs. Since the roundtrip time for the signal took 80 msecs, the Laser Rangerfinder divides that time by 2 to get the assumed time that it took for the light to propagate from the observer to the reflector and multiplies that result by one million feet per msec to get 40 million feet as the length of the rod.

    But now let's use the Lorentz Transformation process to convert the coordinates of all the events (the dots) from the original IRF to one moving at 0.866c:


    Now in this frame, we can see that the observer makes the same measurement because he assumes that the light takes the same time to get to the reflector as it takes to get back. Of course, in this IRF, this is not true, but the observer has no way of knowing that.

    Note that at the Coordinate Time of 0, the distance between the observer and the rod is only 20 million feet. It doesn't matter which IRF we use to depict the scenario, the observer in the scenario sees the same things, makes the same measurements and calculates the same results.

    Now let's see what happens to another observer who is traveling at 0.866c. How does he determine the length of the rod? Here's another diagram like the first one but with the added observer depicted in black:


    Now since he is moving with respect to the rod, he can't simply do what the first observer did, instead he must make two measurements to each end of the rod and apply them at a point in time somewhere along his trajectory between the two ends of the rod and he does this by assuming again that the light takes the same time to get to the reflector as it does to get back.

    In practice, the observer would be continually sending out laser signals from his rangefinder and making continuous measurements but I only show one for each reflector that happen to apply at the same point in time. Looking at the diagram, we can see that he sent out a signal at his time of zero to the far end of the rod and another one at his time of 1.5 msecs to the near end of the rod. He receives the echo from the near end at his time of 20 msecs so he takes the difference and divides by 2 to get a distance of 9.25 msecs times 1 million feet per msec or 9.25 million feet applied at the average of the two measurements or 10.75 msecs.

    Meanwhile, the first signal is returning and arrives at his time of 21.5 msecs. Since this was sent out at his time of 0, this represents a distance of 10.75 million feet applied at his time of 10.75 msecs.

    Now he can add those two distances together, 9.25 plus 10.75 to arrive at the length of the rod at 20 million feet. Now we can plainly see that in this IRF, the length of the rod is 40 million feet and yet he measures it to be half that by virtue of the fact that he assumes that his laser light traverses towards the reflectors in the same amount of time as the return.

    Now there is another way that the observer can verify that the length of the rod is 20 feet and not 40 feet and that is by measuring the speed of the rod relative to himself. He uses the same measurements as before and can calculate the speed from either one of them. From his first measurement, he determines that the rod moved away from him at a rate of 9.25 million feet in 10.75 msecs or 0.856c, close enough for eye balling. For his second measurement, he has to wait until he reaches the far end of the rod and then he uses the time from the measurement until he reaches the end of the rod which is 23-10.75 or 12.25 msec so he calculates a speed of 10.75 divided by 12.25 or 0.878c, again close enough. We can average these to get 0.867c which is right on.

    Now he can use the calculated speed along with his total time for the rod to pass him to determine the length of the rod. This is 23 times 0.867 which is 19.9 feet, in agreement with the other measurement.

    In the next post, I'm going to show the same scenario for this second observer but in different IRF's to show again that different IRF's indicate the same observations for any observer.

    Attached Files:

    Last edited: Jul 13, 2013
  21. Jul 12, 2013 #20


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    Now I take the last IRF diagram from the previous post and use the Lorentz Transformation process to calculate the coordinates for a frame moving at 0.866c with respect to the first one. This puts the "moving" observer at rest and "validates" all of his measurements:


    But since it is hard to see details in this diagram, I show another one zoomed in on the important activity:


    You should confirm that the explanation of the radar signals and the calculations apply just as well to this IRF as it does to the first one.

    And just to make the point even clearer, I transform to another IRF in which both observers are traveling away from each other at the same speed, 0.577c, but in opposite directions:


    Attached Files:

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