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The speed composition vs the light aberration in SR

  1. Sep 18, 2014 #1

    quo

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    The Lorentz's transform:
    ##x' = k(x - vt), t' = k(t - vx)\ k = \gamma,\ and\ c = 1##

    I. The speed composition derivation:

    ##w' = dx'/dt' = \frac{dx - vdt}{dt - vdx}##
    and we divide everything by dt, and:
    ##w' = \frac{dx/dt - v}{1 - vdx/dt}##

    now we assume the dx/dt is some speed u, and the wanted formula is ready:
    ##w' = \frac{u - v}{1 - vu}##

    II. The second part - the relativistic light aberration

    ##\cos f = \frac{c_x}{c} = \frac{dx}{cdt} = \frac{dx}{dt}## (c = 1)
    thus the aberration is:

    cosf' = dx'/dt' = ... identical!

    we go, and on the stage: cosf' = (dx/dt - v)/(1 - vdx/dt)
    and now we don't assume dx/dt is a speed, but it's now just cosf, therefore:

    ##\cos f' = \frac{\cos f - v}{1 - v\cos f}##
    it's a correct result for the relativistic aberration.

    Very well, but I have one question: what is it in the SR the quantity dx/dt - a speed or a cosine (of a light ray)?
     
  2. jcsd
  3. Sep 18, 2014 #2

    Nugatory

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    Staff: Mentor

    If ##x## is the position of an object in a given frame, then dx/dt is the speed of that object in that frame.

    Depending on the trajectory of the object, both x and dx/dt may be more or less complicate functions of t.
     
  4. Sep 18, 2014 #3

    quo

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    This is not an answer, but my question, repeated in different form only.

    You told firstly the dx/dt is a speed, and further noticed only the speed can change is any way in time.

    The equations imply:
    u = cosf, or u = c cosf, in general, and this is rather wrong, thus: u <> c cosf, and this means one of two formulas must be wrong:
    the for the speed composition or for the aberration.
     
  5. Sep 18, 2014 #4

    Dale

    Staff: Mentor

    The speed composition formula you posted assumes colinear motion. The aberration formula does not. They are based on different assumptions, they can be incompatible without either being wrong.
     
    Last edited: Sep 18, 2014
  6. Sep 19, 2014 #5

    A.T.

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  7. Sep 19, 2014 #6

    quo

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    Where is an assumption about dx/dt, in the derivation?
    dx/dt = c_x - it's always true in SR.
     
  8. Sep 19, 2014 #7

    quo

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  9. Sep 19, 2014 #8

    Dale

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    Usually right at the beginning of the section where they derive the formula.

    Not always. Only for light. The velocity addition formula in particular is often used for slower than light objects.

    In any case, so what? That doesn't change my comment at all. If two equations are derived using different assumptions then they will often disagree. That does not imply that either is wrong.

    To find a true contradiction you must find a scenario where both equations assumptions are satisfied and they disagree. Here that means f=0, so your "contradictory" equations reduce to u=c and u=1, which are both true since you used units where c=1. They agree where they both apply.
     
    Last edited: Sep 19, 2014
  10. Sep 20, 2014 #9

    quo

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    Yes, because the c_x can be lower than c.
    I'm not trying to find a contradiction, but the... say: applicability of the Lorentz transform.
    And this example implies that it is probably a light transformation only.
     
  11. Sep 20, 2014 #10

    Dale

    Staff: Mentor

    The Lorentz transform is applicable to any pair of inertial reference frames in flat spacetime. It certainly is not limited to light only. I don't even know how a coordinate transform could in principle be limited to light only. If you are going to try to make conclusions like that then you need to provide some acceptable references, per the rules.

    The aberration formula is for light only. So anything you do using it is clearly going to apply only to light. Trying to make some claim about timelike (slower than light) objects using the aberration formula is nonsense.
     
  12. Sep 20, 2014 #11

    Nugatory

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    Staff: Mentor

    That is a standard mathematical technique, routinely used to transform intractable and messy equations into more easily solved forms. It works because rotating the frame is just a coordinate transformation, with no more physical significance than any other coordinate transformation. Once you have your result, you can invert the coordinate transformation if you want to see the results in the original coordinates.
     
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