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Simple realtivity: Length contraction

  1. Jul 23, 2010 #1
    the proper length is the length where we can measure the length, with one clock( not moving)
    and for someone looking at it as i moves it seems contracted.. how does this happen i do not under stand could someone explain

    thanks in advance
  2. jcsd
  3. Jul 23, 2010 #2
    Because there is a speed-limit at which information can be transmitted, the order in which we see events, or information emerge is dependent upon our relative speed with the system in question. This is famously known as the relativity of simultaneity.

    Thus is you are trying to measure a moving rod, like a meter stick, you have to measure both its front, and back simultaneously in order to get a correct measurement of length. Consider the opposite, if you have a pencil moving across a piece of paper and you use another pencil to mark where the front of the the moving pencil is first and then you mark the back end, in the time it took you to mark the front and the back the pencil has moved. thus your marks on the paper are shorter then when you were to measure the pencil not moving. If you measure the back first and then the front, your measurement is longer then the non-moving pencil length. Hence a moving frame of reference see's lengths which differ from non-moving ones.
  4. Jul 24, 2010 #3
    To understand stuff like the Lorentz contraction, you have to see the world as consisting of events and use the Lorentz transformations to find where events in one frame are located in another frame. If you don't do this, you'll end up creating "paradoxes" where none exist. Events that occur at the same time in one frame, don't in another, is called the relativity of simultaneity which is extremely important in resolving these "paradoxes".

    So with the length contraction problem, you define 2 events as being where the ends of a moving rod are in a frame at the *same* time t. Transform these two events to where the rod is stationary and you'll find they now occur at different times, and a space difference gamma times that measured in the other frame.

    Have a go at seeing if you can derive how a moving length measured in a stationary frame is transformed to a frame where it's static.
  5. Jul 28, 2010 #4
    Are you saying that moving objects don't physically contract (at the atomic level), they just appear to contract when viewed from a stationary frame?
  6. Jul 28, 2010 #5
    Not sure what you have in mind by `physical contraction'. Length contraction is a genuine physical fact, but it's not to be thought of as some kind of deforming pressure that moving objects experience.

    In relativity, velocity is relative, so nothing is *really* moving. Set a rod in motion, it's Lorentz contracted relative to my frame. Set MYSELF in motion, the rod is Lorentz contracted to precisely the same degree, relative to my frame.
  7. Jul 28, 2010 #6
    How can an objects length contract without being deformend at the atomic level?
  8. Jul 28, 2010 #7
    Still not sure what you have in mind or what's been said that's puzzling you. Care to expand?

    In the moving frame, the rod is smaller, and the distance between atoms is smaller. Even the atoms are smaller. It's quite general.
  9. Jul 28, 2010 #8
    That's a good question! I was wondering the same thing myself.

    I don't pretend to understand quantum physics too much but I thought that electrons in an atom can only have certain have discreet 'orbits'. These orbits wouldn't be able to physically 'contract' nor would the space between atoms without affecting the forces holding them together. Isn't that right?
  10. Jul 28, 2010 #9
    Good question. I'm no authority, but here's my overall take:

    Yes, in a marriage of quantum mechanics and relativity, you'll find that in every reference frame, the electrons can only have a discrete orbits. However, the shape of those orbits is dependent on the frame.

    Some subtleties (i.e. covering myself here):

    Firstly, so far, the focus has been on special relativity, and I've only been discussing the consequences of special relativity. Here, nothing much is assumed about what keeps rods stable or what keeps clocks ticking - they're just postulated as ideal devices that measure lengths and time.

    Indeed, I think that standard quantum mechanics isn't relativistic, and that you have to modify the theory somewhat to get it relativistic - I think that leads to quantum field theory, but I'm not sure.

    But even if the discreet results hold in Quantum Field theory, moving objects will still appear to contract. Again, I emphasise that such Lorentz contractions *mustn't* be thought of as akin to a kind of mysterious force or pressure that deforms objects, a pressure caused by the aether wind. Providing we're talking about inertial motion, there's a deformation whether the rod is moving with respect to me, or whether I leave the rod alone and am moving with respect to the rod.
  11. Jul 28, 2010 #10


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    Do you see http://www.tpub.com/content/draftsman/14276/css/14276_311.htm"contracted circle?
    That's the same (or at least a very similar) mechanism as in SR.
    The circle is not deformed, you just see it from a different perspective, rotated. Instead of a third spatial dimension, things are rotated in the "time" direction in SR. You can draw diagrams of such rotations (called Lorentz transformations) to see how they really work.

    To be exact, this description applies best to time dilation. Length contraction is a bit different, it's rather a "slicing" (of a cylinder in this case) than a projection of a circle.
    But it's all geometry and not forces.
    Last edited by a moderator: Apr 25, 2017
  12. Jul 28, 2010 #11
    What I don't understand is the physics that causes the rod to contract when it is in motion.
    If the atoms of the rod are smaller, are the components that define the atom (electrons, quarks, etc.) smaller as well?
    In the moving frame, which of the four forces act on the rod to compress it?
  13. Jul 28, 2010 #12
    No stresses or forces act upon the rod (in SR). Every object is observed to be smaller in the direction of motion when observed from a frame of reference moving relative to it. If you apply this to a collection of objects the space between then appears to get smaller in the same proportion.

  14. Jul 28, 2010 #13
    Well, it seems that I have received two contradictory answers rergarding the contracting rod. One answer implies that a deformation of the rod takes place at the atomic level and the other asserts that the rod only appears to get smaller.
    Can both answers be correct?
  15. Jul 28, 2010 #14
    What's the contradiction? I think there's agreement, in a stationary reference frame, moving rod A contracts. In a stationary reference frame, the spaces between the atoms of the rod contract. In a stationary reference frame, the atoms themselves contract. It's just that these statements about distance and length - and time and simultaneity too - have to be relativised to a reference to make sense.

    Do you want to ask: does it *really* contract or is it just an illusion? If the *really* question is a demand to know, independent of all frames, whether the rod contracts, then, according to relativity, it's a bad question. There is no frame-independent answer to questions about length. In this respect, length turns out to be like velocity - there's no absolute answer to the question of an object's velocity, independent of a frame of reference.

    On the other hand, I wouldn't say it's an illusion either: that too seems to suppose that there's a question about how things really are vs how they appear.
  16. Jul 28, 2010 #15
    There is, as someone has already mentioned, a limit to the speed at which information can be transfered. In this case "reality" is affected by this limit too. "Reality" is what we can measure or observe. If we can't measure it then it doesn't exist. So the universal speed limit directly affects our experience of reality. So therefore, when considering length contraction of a moving rod, the rod shrinks in overall length. The parts that make up the rod all shrink, and the space between those parts is reduced also.

    We know this to be true because we can mathematically prove that we could not possibly measure the length to be greater (due to the information speed limit). Therefore we can never experience, in any way, the length to be greater. Therefore the length is not greater. It REALLY has contracted.
  17. Jul 28, 2010 #16
    I am not sure if this will help you at all, but spacetime diagrams really helped me to conceptualize length contraction. Be patient with my crude diagrams, they should get the concept across even though they aren't pretty (or even totally straight).

    Because light travels at a set speed in all frames, c, when you draw a spacetime diagram of light it must travel at a 45 degree angle (assuming you are using a system of natural units). So let's say you have one frame, A', that is moving with velocity v in the x direction relative to frame A, we will label the axis in A’ as x’ and t’. If you shine a light at time t0 in frame A' and it reflects from a mirror at event a and returns to the source at some later time t1 it will look like this:


    But what will it look like in frame A? Well, remember light has to travel at 45 degree angles no matter what frame you are looking at (c is constant in all frames), as long as it is an inertial frame. Frame A' is moving with some velocity v relative to A so first imagine what an object moving relative to frame A with constant velocity would look like, it would be a line with angle < 45 degrees. Because A’ is moving relative to A as you move along the t’ axis (or at any point where x=constant) you are traveling some distance in frame A. So the t’ axis of frame A’ appears to be rotated by an angle relative to frame A’s t axis (the angle is related to the velocity A’ travels relative to A, and is pretty simple to work out geometrically). What about the x-axis though? Well, we label the times t0 and t1 on our rotated t’ axis, but light can only travel at 45 degrees in all frames so we draw lines that make 45 degree angles from t0 and t1 events and where they intersect is where the mirror in frame A’ appears in frame A’s coordinates. So the reflection event in A’ will look like this:


    How does this relate to length contraction? Let’s look at a rod that is at rest in the A’ frame, and draw a spacetime diagram of it in frame A. If we measure the length (proper length) in frame A’ we get a larger length then we do in Frame A:


    Using this geometrical picture (and the concept of the spacetime interval ∆s2) you can derive the formula for length contraction, but more importantly you can visualize and understand that length contraction is a geometric effect that arises because light moves the same velocity in all inertial frames.
    Last edited: Jul 28, 2010
  18. Jul 29, 2010 #17
    hold on from what i heard they do not shrink physically. is it true?

    for ex take car to be 10m and a tunnel to be 5m. car moving at say .75c so, the person standing beside the tunnel will see the car as say like 4m. say the person beside the tunnel can shut both side of the tunnel after he thinks that 4m car is in side. he would cut off the rest of the 6m. correct. because of time dilation correct?
  19. Jul 30, 2010 #18
    To hit the question from another angle - B is moving away and the rod in B is contracting in A's frame. But it's not contracting in B's frame, is it? Please correct me if I am wrong.

    So do we have the rod contracting for A and not B? Does this mean that it is only the adjustment for light to travel between frames that makes the contraction or is it really contracting?
  20. Jul 30, 2010 #19


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    You're right.
    The problem is defining what you mean by "really contracting". It is true that if in frame A you make two simultaneous measurements of the position of the front and back of the rod, and you also make simultaneous measurements of the position of the front and back in frame B, then the distance between front and back in frame A (as measured by a ruler at rest in frame A) is shorter than the distance between front and back in frame B (as measured by a ruler at rest in frame B). But because of the relativity of simultaneity different frames disagree on what it means for two measurements to be "simultaneous", and each frame is using its own ruler to measure the distance between the positions of the two measurements.

    You could take a look at the diagrams I gave in this thread showing how two ruler/clock systems moving alongside each measure the other ruler to be shrunk, and the other clocks to be slowed down and out-of-sync...
  21. Jul 30, 2010 #20


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    It is really contracted. This is why we usually insist on the fact that the measurements are made by local observers. There is no effect due to the finite speed of propagation of light.
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