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How is velocity defined?

  1. Aug 5, 2008 #1
    I found the following in a locked thread (https://www.physicsforums.com/showthread.php?t=16514), and it perfectly illustrates how the discussion about these things usually starts (it's always the same):

    - What if two objects (A and B) move away from each other at the speed of 0.7c relative to some third observer C..? Do they (regarding to observer C) have a relative speed of 1.4c?

    This is called the "closing speed" regarding to Wikipedia. The answer then usually contains something about "you obviously don't understand this stuff", and in the next second you get a standard school-answer proving that the one giving the answer doesn't understand it either. Finally someone says: "the actual relative speed is 0.94c".

    Now my question is, regarding to who is the speed 0.94c? How is this velocity defined? Remember that there is no absolute motion. So which observer (A, B or C) will measure this speed? A and B will never be able to do any measurements on each other whatsoever, since no light from A will ever reach B and vice versa. (Why? Because the "light cone" says so, or because A actually moves at 1.4c relative to B? Or are both these answers equivalent?) So my guess would be C!? But C measures 1.4c, doesn't he/she?

    My actual guess would be A or B. We could let them collide with each other and measure the released energy, right? No. The energy released in the collision would be the same as if they actually travelled at 1.4c relative to each other. Hmm.. Any suggestions?

    And finally about light cones... They are just a workaround to say the forbidden words: "A and B both move at 1.4c relative to each other". They do so by saying that "the space between A and B expands while the light travels between them", which is exactly the same thing. The problem occurs when the light from A actually hits B, in which case B measures the speed of the light to the everlasting constant c. But this is another topic. It is about Lorentz transformation.
    Last edited: Aug 5, 2008
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  3. Aug 5, 2008 #2


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    A and B will both measure the speed of eachother to be 0.94c (assuming that's the number obtained when using the velocity addition formula). This means that when in A you will observe B to have the speed 0.94c and vice versa.
    You say A and B cannot measure this speed because the light from A will never reach B and vice versa, but that's not true. Because of the fact that they go 0.94c according to eachother, the light will reach the other ship. This might cause some confusion as to what C will observe, because how can he see the light from A reach B when they are travelling at 0.7c? Well you must remember that for C the speed of light is still c so any light emitted of A, will travel towards C with c and further on to B and have no problem "catching up" since B is only going with 0.7c according to C.
  4. Aug 5, 2008 #3


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    Inherent in every SR problem is the inertial frame of the observer. Velocites are measured in the frame or co-ordinate system of the observer. The observer frame is assumed to be stationary.
  5. Aug 5, 2008 #4


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    Usually "relative speed" is used to mean the speed of one in the other's rest frame, but it is true that in the frame of C, the distance between them increases at a rate of 1.4c...as you say below, this is called the "closing speed".
    It's just semantics really, depends how you define "relative speed". Certainly in the rest frame of A, B's speed is 0.94c, and vice versa in the frame of B.
    Each observer can define speed in their frame using a ruler which is at rest in their frame, and which has clocks attached to different markings, with the clocks "synchronized" in this frame using the Einstein synchronization convention (which is based on the assumption that light moves at the same speed in both directions in this frame, so if you set off a flash at the midpoint of two clocks, they are defined as synchronized if they both read the same time when the light reaches them). So for example if a ship passes the 0-light-year mark on my ruler when the clock there reads 0 years, and later the ship passes the 7-light-year mark when the clock at that mark reads 10 years, then the ship is moving at 0.7c in my frame. The reason the closing speed between A and B in C's frame is different from the speed of B in A's frame (unlike in Newtonian mechanics where they'd always be identical) is that rulers and clocks in motion relative to one another or distorted by length contraction and time dilation, and also if different observers all use the Einstein synchronization convention for clocks in their own frame, they will find that the clocks of other observers are out-of-sync in this frame.
    No, light from A can definitely get to B. In C's frame B's velocity is only 0.7c while the velocity of the signal from A is 1c, so the signal will catch up to B eventually.
    Who says that? That doesn't sound like any explanation of relativity I've ever heard. In C's frame, according to C's rulers and clocks, the distance between A and B does increase at a rate of 1.4 light years/year, but in A's frame, according to A's rulers and clocks, the distance between A and B increases at a rate of 0.94 light years/year. Both frames agree that the light from an event on A's worldline will eventually catch up with B, meaning that the event on A's worldline does lie in the past light cone of any event on B's worldline after the signal reaches him.
  6. Aug 5, 2008 #5
    Nizzeberra, you might find the concept of rapidity more appealing to you than velocity. Rapidities, unlike velocites, are additive.
  7. Aug 5, 2008 #6


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    The relative speed (as defined by the explanations used below) between two observers is a Lorentz-invariant quantity.
    All inertial-observers will determine this quantity to be 0.94c (assuming you measure it correctly).

    [The relative spatial-velocity of A wrt B is
    spatial-(according to B)-component of the (unit)-4-velocity of A
    divided by the
    temporal-(according to B)-component of the (unit)-4-velocity of A.
    ...and similarly for B wrt A.

    These spacelike vectors are generally different (since one is purely spatial to B, and the other to A)... i.e. they don't point along the same line in spacetime. (Note they do lie on the same plane with the vectors formed by A and B... which is usually used to say they point in opposite directions along the same spatial axis.) However, their magnitudes are equal... hence we can call these "THEIR RELATIVE SPEED".

    Note also that all observers will agree on what the spatial-and-temporal-components of A's 4-velocity according to B....
    since (using * for dot-product with +--- convention) the "temporal piece of A's 4-velocity wrt B" is (A*B)B and the spatial piece is A-(A*B)B... perfectly good 4-vector expressions.

    Using rapidities,
    the relative-rapidity of A wrt B is the signed-Minkowski-angle from B to A,
    and similarly for B wrt A. Geometrically, you can see they have opposite sign, but the same magnitude. The relative velocities are determined by tanh(rapidity), also with opposite sign but the same magnitude (aka THEIR RELATIVE SPEED).

    ...probably more than you wanted to know.
    Last edited: Aug 5, 2008
  8. Aug 18, 2008 #7
    First I want to say: Thank you guys for all your answers! They are all really interesting and finally gave me some peace in mind. I have never been able to discuss relativity with anybody, and the theory is really twisted. I even think quantum mechanics make more sense! =)

    But I still think (despite your answers) that the theory of relativity is quite inconsistent. Probably it's just me who doesn't understand it. But that's why I write this in the first place, right?

    First of all, the expansion of the universe has been proven to be accelerating. This can be interpreted as if the space itself is expanding. The more space there is between two galaxies, the faster the expansion goes. If this is the case, then if there is enough space between two galaxies the distance between them will increase at superluminal speeds - which violates relativity. Can someone please patch this hole up for me?

    Another inconsistensy can be found in any schoolbook about relativity. In the beginning all books usually have the following thought experiment, describing the effect of length contraction.

    It is about a train wagon going through a tunnel which has the exact same length as the wagon (when standing still). When the front of the wagon passes the end of the tunnel, a flashlight goes off. The same happens when the back passes the entrance of the tunnel. Both lights should go off exactly at the same time (using Newtonian physics). But regarding to an observer on the wagon the flashlight ahead (at the end of the tunnel) goes first.

    This is explained as "since the wagon moves towards the light at the end of the tunnel, this light hits the observer first". Do you see the problem? Is the light from the two flashlights moving at the constant speed c or not? This experiment is supposed to show why length contraction occurs - but it does so by violating relativity, doesn't it? I'm confused.

    / Nizzeberra
  9. Aug 18, 2008 #8

    Doc Al

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    I'll leave it to others to give you a good explanation. All I'll say is that special relativity limits things from moving through space at superluminal speeds. To understand the metric expansion of space itself, one must use general relativity.

    Right. The wagon frame and the tunnel frame observers will disagree as to when the flashes went off and which one went off first. To understand this, you need to apply your knowledge of length contraction.
    Not clear to me why this would be used to explain the different sequence of flashes between the two frames in this example of train through tunnel. It's much easier than that. Perhaps you are mixing this up with a different thought experiment--Einstein's train--that is used to argue that simultaneity is frame dependent?

    In that thought experiment, a moving train passes a station. At the exact same moment--according to the station observers--lightning strikes the front and rear of the train. Those flashes are (eventually) seen by an observer riding in the middle of the train. Which flash does he see first?

    If you look at things from the station's viewpoint you can easily figure it out. From the viewpoint of the station, the train is moving towards one flash and away from the other. So, of course, the light from the front of the train reaches the middle of the train first. It has to, since we have agreed that the speed of light is a constant for all frames--and that includes the station frame.

    Of course, from the train observer's viewpoint, light also travels at speed c with respect to the train. Since the light from the front of the train reaches him first, he is forced to conclude that the lightning struck the front of the train first.
  10. Aug 18, 2008 #9


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    SR is completely self-consistent. In the end it is nothing more than Minkowski geometry, which is just computing intervals using hyperbolas instead of circles.
  11. Aug 18, 2008 #10


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    There are lots of threads about this already. This is the one where I figured it out myself (towards the end of the thread).

    I'm confused by your description of this scenario, so I can only say that all these things are much easier to understand when you know how to draw spacetime diagrams. (Learning that is easy, so I highly recommend it).
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