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Addition of Velocities

  1. Feb 4, 2009 #1
    Additive velocities.

    If a car is traveling down a road at velocitiy u wrt the road and fires bullets in the forward and backward directions relative to the direction of travel at velocities v and -v respectively wrt the car,, then it is self evident that as measured in the coordinates of the road the bullet propelled with the forward momentum of the car is going to be faster than the speed of the bullet fired counter to that momentum

    But according to the relativity addition of velocities equation it appears that with a u =.8c and a v=2c this is not the case. I should make it clear that in this case I am refering to "bullets" with imaginary mass ie. tachyons

    The bullet propelled forward in the same direction as the source is slower than the bullet propelled counter to the motion of the source.

    In another context this would mean if two guns were mounted in opposite directions along a railroad track and fire as the middle of the train is coincident , that the bullet fired toward the rear of the train which is moving toward it would be slower than a bullet fired toward the front of the train which is moving away from it, as measured by observers on the train.

    This is completely counter to kinematics as we know it and certainly counter to logic so

    Should this be interpreted as applicable predictions of FTL physics ?
    Thanks
     
    Last edited: Feb 4, 2009
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  3. Feb 4, 2009 #2

    Doc Al

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    The speed of the bullets must be less than c--certainly not 2c.
     
  4. Feb 4, 2009 #3
    Of course you are right regarding bullets with real mass and I should have been more explicit.
    This is regarding hypothetical tachyons with imaginary mass and so subject to the ballistic additions formula or so I have been informed by others.

    Thanks
     
  5. Feb 4, 2009 #4

    robphy

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    If you really want to analyze the situation, it is probably best to draw a spacetime-diagram of the situation... and use unit-spacelike vectors for your tachyons. Then, rather than merely applying formulas that were derived for the usual real-mass case, re-derive the analogous formula for your tachyon... then apply and interpret.
     
  6. Feb 4, 2009 #5

    Fredrik

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    I did that in another thread, and got the result that the standard formula would hold. Austin0 has showed that that I was wrong just by plugging in a couple of numbers. I guess I should have done that myself. :smile: I'll give it some more thought.

    Edit: What I said in this post is wrong. See post #9.
     
    Last edited: Feb 5, 2009
  7. Feb 4, 2009 #6

    Doc Al

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    I haven't thought about tachyons in a while, but I'm pretty sure the standard velocity addition formula works just fine for all speeds.

    Of course one's relativistic intuition (based upon sub-light speeds) may be way off!
     
  8. Feb 4, 2009 #7

    JesseM

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    How is it "contrary to logic"? You really just mean it's counter to your intuitions which are based on the behavior of sublight objects. Regardless of whether tachyons exist, in the real world we could perform an analogue of this experiment using some phenomenon that can travel at 2c without actually transmitting matter or information FTL, like the red dot of a laser pointer. Do you agree that if we are changing the angle of the laser in such a way that the red dot is moving at 2c along the floor of a ship in the ship's own rest frame, moving from rear to nose, then if the ship is moving at 0.8c in your frame (in the direction the nose is pointing), you will measure the red dot of the laser pointer to move at only 1.077c?
     
    Last edited: Feb 4, 2009
  9. Feb 4, 2009 #8

    robphy

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    In the way I think about
    the velocity-composition formula, it arises from the hyperbolic-tangent of a sum of two rapidities (i.e. the Minkowski-angle intercepted by the two future unit-timelike vectors... where the Minkowski-angle is the arc-length on the Minkowski-circle [a hyperbola]).
    My interpretation relies on the angle-intercepted on the future-branch of the hyperbola. Note that a unit-spacelike vector does not intersect that future-branch... so there is no relative angle (as defined above) with respect to a timelike-vector.

    There may be another formulation in terms of slopes... but I would think it requires more work to define it properly.

    Can you post a link to your related post in the other thread?
     
  10. Feb 4, 2009 #9

    Fredrik

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    Yes, I just thought it through and realized that it does hold. I was only wrong when I thought I was wrong. Damn that intuition. :smile:

    Here's the derivation again (modified a bit from the other thread):

    Suppose that the velocity of frame F' in frame F is u, and the velocity of a particle in frame F' is v. Let's call velocity of that particle in frame F w. (Note that u must be <1, but we don't have to put that restriction on v or w).

    It's sufficient to prove the velocity addition formula for two frames F and F' that have a common origin, and for a particle moving as described by a world line that's a straight line through that origin. The slope of the particle's world line is 1/v in F' and 1/w in F. That means that the Lorentz transformation

    [tex]\gamma\begin{pmatrix}1 & u\\ u & 1\end{pmatrix}[/tex]

    must take [tex]\begin{pmatrix}1\\ v\end{pmatrix}[/tex] to a vector proportional to [tex]\begin{pmatrix}1\\ w\end{pmatrix}[/tex]. That's all we need to find w:

    [tex]\gamma\begin{pmatrix}1 & u\\ u & 1\end{pmatrix}\begin{pmatrix}1\\ v\end{pmatrix} =\gamma\begin{pmatrix}1+uv \\ u+v\end{pmatrix} =\gamma(1+uv)\begin{pmatrix}1 \\ \frac{u+v}{1+uv}\end{pmatrix}[/tex]

    So the standard formula must hold.

    [tex]w=\frac{u+v}{1+uv}[/tex]
     
  11. Feb 5, 2009 #10
    I was not questioning the formula itself , I had it from others that it was applicable as is.
    I was questioning the meaning of the results and whether these results could be accurate predictions of events in the real world if it should turn out that FTL is a possible reality.
    Thanks
     
    Last edited: Feb 5, 2009
  12. Feb 5, 2009 #11
    This is what I used . I was not suggesting that the derivation was incorrect.
    It is just that the results are food for thought as far as what they mean and how they should be interpreted.
    It seems to suggest to me that this use is not what was intended for the math that the results are mathematical aberrations stemming from the integral c factor.
    Thanks
     
  13. Feb 5, 2009 #12
    1) You do not find this completely counter to any logic??
    2) This is the speed as derived from the ballistic additions equation which is based on a particle having the initial momentum of the system so I dont really understand why the moving laser dot would have additive velocity in this way or be subject to the use of the equation..
    3) Be that as it may,, dont you find it illogical that your dot moving from the front to the back in this same situation would have a speed of 2c

    By the way where is the laser in this case????
     
  14. Feb 5, 2009 #13

    Fredrik

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    Austin, you don't seem to understand what the word "illogical" means. These results may be "unexepected" and "counter-intuitive" but definitely not "illogical". To be illogical, they would have to contradict other results obtained from the same axioms, or contradict themselves (like the statement "what I'm saying right now isn't true"), and they don't. The question of whether something is illogical or not is never a matter of opinion.
     
  15. Feb 5, 2009 #14

    Fredrik

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    You said that the result is incorrect, and if you had been right, it would have implied that the derivation is incorrect too (or that special relativity is logically inconsistent, which is much less likely). I can totally understand that you thought the results were unreasonable. I thought so too for a moment when I plugged in the numbers you suggested into the formula, but it's quite easy to see in a spacetime diagram that it doesn't hold in general that the "sum" of two velocities in the same direction is always greater than the larger velocity. The general rule is that the "sum" is always closer to c than the larger velocity. (And there's nothing illogical about that).
     
  16. Feb 5, 2009 #15

    JesseM

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    No. The only way anything can be "counter to logic" is if you can derive a logical contradiction from it. Where is the logical contradiction here?
    The velocity addition equation has nothing to do with ballistics or any other physical properties of the moving object, it's simply a matter of applying the Lorentz transform. For example, suppose the primed frame is moving at 0.5c relative to the unprimed frame, and some entity (a ballistic object, a laser dot, a sound wave in some material, anything) is measured to be moving at 0.5c in the primed frame. So if this entity passes the origin of the primed frame at t'=0, then we know at t'=20 it'll be at x'=10. So, we can ignore all the physical properties of this entity and just use the Lorentz transformation to figure out the corresponding coordinates in the unprimed frame...x'=0 and t'=0 of course corresponds to x=0 and t=0, while x'=10 and t'=20 correspond to:

    x = 1.1547*(10 + 0.5*20) = 23.094
    t = 1.1547*(20 + 0.5*10) = 28.8675

    So in the unprimed frame, the entity passes through (x=0, t=0) and (x=23.094, t=28.8675). So, just by the equation speed = (change in position)/(change in time), we conclude the entity's speed in the unprimed frame is 23.094/28.8675 = 0.8. And this is exactly what we'd have gotten if we applied the velocity addition formula: 0.5c + 0.5c/(1 + 0.5*0.5) = 0.8. So you see the derivation depends on no assumptions about the physical properties of the entity whose speed is measured in both frames, it just depends on the assumption that the coordinates of any event in these two inertial frames (including events on the worldline of the moving entity) are related by the Lorentz transformation.
    No, again it just follows from the Lorentz transformation, and there is nothing illogical about the idea that the coordinates of one frame are related to the coordinates of the other by the Lorentz transformation.
    Somewhere above the floor of the ship, pointing at the floor and changing its angle so that the laser dot moves along the floor. It's definitely possible to have laser dots move FTL in the real world, see here and here. Of course the laser dot can move FTL without any individual photons moving FTL, since the moving laser dot is really just a series of different photons hitting the floor at different positions. Still, it's certainly an entity where you can talk about its objective position at a given time (each clock in a given frame could have a photosensitive device and record the time when it's activated by the laser pointer shining over that clock).
     
    Last edited: Feb 5, 2009
  17. Feb 6, 2009 #16


    I did not say it was incorrect ,only that it was counter to kinematics and logic.
    In the case for all sub c velocities the results conform to both logic and kinematics as we know it.

    Speaking of logic perhaps you could explain the logic or physics behind the sum always being closer to c ? Why this is a logical expectation and prediction of real world events?

    To put it into another context: it is the exact same equation for inertial frames.
    SO do you feel that there is nothing counter to logic or kinematics to have an approach relative speed be slower than a parallel relative speed for systems with the same relative speeds wrt another frame???

    It would be one thing if Einstein and Lorentz specifically evolved a theory and kinematics for super c velocities , I do not beleive they did. I think that SR is rationally based on empirical reality and as such is totally valid for predicting real world occurences .
    I just dont think it can be applied to phenomena outside that world ,,,which phenomena, in fact, might be completely impossible .
    I also think that your view that the Lorentz structure is just a coordinate system and set of transform functions is incomplete. That would apply to Gallilean transforms applied to Cartesian coordinates where the transforms were purely mathematical translations with no physics implications whatsoever.
    Of course the Lorentz -Minkowski structure is a coordinate system and as such encompasses all spacetime.
    But the Lorentz math is also much more , it definitely has physics implications.
    In fact you could almost more correctly view it as an evolution of Newtons laws rather than an extension of Gallileo. The equations are statements of a higher theory of motion.
    With all kinds of physics involved, time dilation, contraction and clock desynchronization which would seem to be as "real" as any other physics we deal with. SO why should it be expected that this math would yield valid predictions for what is essentially unreal phenomena????
     
  18. Feb 6, 2009 #17
    I really like Jesse's analogue of the laser dot at 2c .
    It contains the whole question.

    If the ship is 20 lightsecs long in its frame then the dt=10 from back to front.

    Assuming the back of the ship is coincident with x'=0 .t'=0 at the beginning,,, moving .8c ==> +x'
    Can we not make some logical assumptions??
    1) At t'=0 the front of the ship is coincident with x'=12
    2) The dt=10 in the ship system time means dt'=6 system time in the primed frame
    3) At t'=6 the back of the ship will be observed coincident with x'=4.8
    4) At t'=6 the front of the ship will be coincident with x'=16.8

    Is this not straight forward??? To the observers in the primed frame this is just the passage of a ship at .8c through a certain time interval , whatever is happening with the laser dot or clocks inside the ship is irrelevant to their observations.
    So then logically the laser dot at the end of its journey inside the ship must be at the front of the ship at x'=16.8.
    SO in the coordinates of the primed frame ,,the dx' from the back to the front of the ship over dt'=6 is dx'=16.8 16.8/6 =2.8c

    So any calculations that derive a different relative velocity or a different x' location or t' for the end of the exercise would seem to create the kind of logical contradiction you are talking about , no???
    Would require that the laser dot, merely by intersecting the floor of the ship, somehow transported parts of the the ship through spacetime.
    Thanks
     
  19. Feb 6, 2009 #18

    Fredrik

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    Yes you did. You just didn't understand that, because you have misunderstood the words "logic", "logical" and "illogical". If my result had been illogical (which is exactly what you said it was), it would have implied that the derivation is incorrect or that special relativity is logically inconsistent. The latter is extremely unlikely since just about all of mathematics would fall with it, so to claim that my result is illogical is to claim that the derivation is incorrect.

    It's logical because it follows from the axiom that says that events are represented by points in Minkowski space. That axiom is a part of the definition of special relativity. It's not illogical because it doesn't contradict itself, that axiom, or anything else that we have a valid reason to assume.

    It's hard to explain in words why the sum is always closer to c, but if you draw a spacetime diagram, you should see it immediately.

    It doesn't matter what I feel. Logic doesn't care about feelings. I did feel that the result was very strange at first, as you can see in my first post in this thread. But when I actually used logic instead of intuition, I saw that it wasn't. The result may be counter-intuitive, but as Jesse and I have been saying, it can't be illogical unless it contradicts something.

    SR says that events are represented by points in Minkowski space. This implies that the motion of a particle with constant velocity v in some inertial frame is represented by a line with slope 1/v in a spacetime diagram. So what you're saying is essentially that it's impossible to draw a line that makes a greater angle than 45° with the vertical, on a piece of paper.


    Austin, you didn't ask about the velocity addition formula of a still undiscovered (and probably nonexistent) theory that agrees with SR about speeds <c and includes a description of tachyons that don't violate causality. You asked about the velocity addition formula of special relativity. At least that's what you forced us to assume, since we can't possibly give you an answer from a theory that hasn't been discovered and probably doesn't exist.

    I'm not saying (and neither is anyone else) that if tachyons exist in the real world, they must necessarily behave as predicted by SR. If they exist, then maybe they behave in a different way. However, that would imply that SR is wrong even in situations where gravity can be ignored. So we don't consider this to be very likely, given the enormous amount of experimental evidence that's consistent with the predictions of SR. If SR is completely wrong about a velocity of 1.0001c, and I mean absolutely insanely wrong (which is what you're suggesting), don't you think it would also be a little wrong about a velocity of 0.9999c?

    I'll leave that part of the discussion to Jesse.
     
  20. Feb 6, 2009 #19

    JesseM

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    Yes.
    No, for any individual clock on the ship to tick forward by 10 seconds will take a greater time in the primed frame of the outside observer rather than a shorter time since the ticks of moving clocks appear lengthened (in this case, any individual clock on the ship takes 10/0.6 = 16.666... seconds in the primed frame to tick forward by 10 seconds of proper time), and also you're forgetting the relativity of simultaneity here. Imagine clocks at the front and back of the ship--it's true that if laser dot passes the back clock when it reads t=0 then the laser dot will pass the front clock when it reads t=10. But in the primed observer's frame, the two clocks are out-of-sync by (20)*(0.8c)/c^2 = 16 seconds, so at t'=0 in the primed frame when the laser dot is next to the back clock on the ship, the front clock reads t=-16 seconds. So the front clock will have to tick 26 seconds forward before it reads t=10, and since its ticks are extended by a factor of 1.666... in the primed frame, this will take a time of 26*1.666... = 43.333... seconds in the primed frame. And at t'=43.333..., the front of the ship will be at position x' = (43.333...)*(0.8c) + 12 = 46.666... So, this shows that in the primed frame the speed of the laser dot was 46.666.../43.333... = 1.0769 light-seconds/second, which is what you'd get plugging into the velocity addition formula: (2c + 0.8c)/(1 + 2*0.8) = 2.8c/2.6 = 1.0769c.
     
    Last edited: Feb 6, 2009
  21. Feb 7, 2009 #20

    Thanks for the links
     
    Last edited: Feb 7, 2009
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