SR composition of velocities question

Discussion in 'Pseudoscience Archive' started by Jack_, Mar 13, 2010.

  1. Jack_ Banned Banned

    Messages:
    1,383
    Can anyone tell me the resultant vector direction of the velocity vector that results from the composition of velocities equation of SR?

    If you do not know the general one, then the orthogonal case is really the one am interested in.

    Thanks
     
  2. Google AdSense Guest Advertisement



    to hide all adverts.
  3. CptBork Valued Senior Member

    Messages:
    6,460
    Being the resident expert on Relativity you claim to be, I'm surprised you can't just use the Lorentz transformations and spend the requisite 5 minutes deriving the velocity composition equations on the back of a napkin.
     
  4. Google AdSense Guest Advertisement



    to hide all adverts.
  5. Jack_ Banned Banned

    Messages:
    1,383
    I have the magnitude of the velocity vector.

    That part is easy.

    I am looking for the direction.
     
  6. Google AdSense Guest Advertisement



    to hide all adverts.
  7. Jack_ Banned Banned

    Messages:
    1,383
    Anyway, this is what I get.

    θ = atan2( w[sub]η[/sub] / (γ ( ( w[sub]ξ[/sub] + v )) )
     
  8. Guest254 Valued Senior Member

    Messages:
    1,056
    The usual velocity addition is obtained by composing two Lorentz transformations which boost in the same direction. If you give the Lorentz transformations their matrix representations all you need to do is multiply the two matrices together. Hence the general result would still be multiplying two matrices together but their components will be of different forms.

    Let \(\Lambda^{a}_{b}\) be the boost from frame S with coords (t,x,y,z) to S' with coords (t',x',y',z') with v(S',S) = v. Then \(\tilde{\Lambda}^{a}_{b}\) be the boost from frame S' with coords (t',x',y',z') to S'' with coords (t'',x'',y'',z'') with speed v(S'',S') = u. The components of the transformations can be found in a book or online and by composing them (ie multiplying the 4x4 matrices together) you'll get the transformation for S to S''.

    The velocity v(S'',S) can then be computed by considering a rotation \(\hat{\Lambda}^{a}_{b}\) which sets the velocity to be parallel to one of the coordinate axes. That'll give you the angle and then you can read off the magnitude of the velocity.

    Its little more than multiplying 4x4 matrices together 3 or 4 times.
     
  9. Jack_ Banned Banned

    Messages:
    1,383
    Let me make sure.

    S is moving relative to S' and S' is moving relative to S''.
    Is this correct.

    Finally, given the point (x'',y'',z'',t'') what is the purpose of that final rotation to force the point to project onto an axis?

    If I have a point (x'',y'',z'',t''), I have the vector magnitude and direction.
     
  10. AlphaNumeric Fully ionized Registered Senior Member

    Messages:
    6,702
    Guest refers to coordinates, not points. The coordinates of S, S' and S'' are linked by Lorentz transformations and ytou know the one between S and S' and you know the one between S' and S'' then you just combine them together.

    I would imagine he suggested that because the resultant matrix formed by the two Lorentz transformations is pretty unpleasant and its not immediately clear how to extract the velocity. If you apply a general rotation you can work out the angle by seeing what rotate makes the velocity parallel to your axis of choice.

    As he says, its just multiplying matrices together.
     
  11. Jack_ Banned Banned

    Messages:
    1,383

    Thanks, I see why the final rotation is necessary.

    It is to bring the matrix multiplication parallel to the x-axis for standard form. This way the magnitude and direction are relative to a standard coordinate system.



    AN, Guest, thanks, have a nice day.
     
  12. Jack_ Banned Banned

    Messages:
    1,383
    Something is not making sense.

    http://www.fourmilab.ch/etexts/einstein/specrel/www/

    He has x = t(wx' + v) / ( 1 + v*wx'/c²)

    He has wx' = x'/t', wx' is one variable and not a multiplication.
    This is the x-direction speed of the particle moving in the K system which is moving relative to k.

    Substitute wx' = x'/t' in the above.


    x = t((x'/t') + v) / ( 1 + v*(x'/t')/c²)

    Mult by t'/t' and divide by t

    x/t = (x' + vt') / ( t' + vx'/c²)


    How does this work?
     
  13. Pete It's not rocket surgery Registered Senior Member

    Messages:
    10,167
    I'm not sure what you're asking, exactly, but this might help.

    This section is about the path of a particle described by parametric equations. In the reference frame k, the particle is moving at [wx'] in the x direction, and [wy'] in the y direction.

    You've found an expression for x/t, the particle's speed in the x-direction in reference frame K:
    x/t = (x' + vt') / ( t' + vx'/c²)

    I don't know that this equation signifies anything in that form, but undoing your last step:
    x/t = (x'/t' + v) / ( 1 + v(x'/t')/c²)

    We see that this is the relativistic addition of velocities equation:

    Please Register or Log in to view the hidden image!

     
  14. Jack_ Banned Banned

    Messages:
    1,383
    Thanks, I ended up creating an equality between LT and the composition of velocity equations. They are exactly the same thing but in a different format.

    I just never noticed that before.
     
  15. Pete It's not rocket surgery Registered Senior Member

    Messages:
    10,167
    Yes, all things in SR can be derived from the lorentz transforms. If ever something doesn't seem to make sense, going back to the transforms is the best way to resolve it.
     
  16. Jack_ Banned Banned

    Messages:
    1,383
    :roflmao:

    Words of wisdom.

    SR is an absolute truth.
     
  17. AlphaNumeric Fully ionized Registered Senior Member

    Messages:
    6,702
    It is a mathematically consistent and coherent structure. Whether or not it describes the real world is an entirely different question. You're attempting to prove it is mathematically inconsistent which it not something you'll manage to do since the mathematical soundness of geometry is already established. If you'd ever actually done a course on geometry or logic you'd know that you're trying to deny the demonstrated logical soundness of geometry. Its strange that you seem to be completely uncomprehending of such a pertinent point when it comes to special relativity yet you like to talk about transfinite or first order logic. You tried to claim that due to some decidability issue special relativity is wrong but if you knew sufficient amounts of mathematics to really grasp such concepts you'd not have made that claim in the first place.

    I suspect you fail to grasp just how intertwined special relativity's mathematical formalism is with other areas of mathematics. Group theory, manifolds, set theory, Lie theory. If special relativity is mathematically inconsistent then it would filter through into those areas yet their consistency and structures have been deeply examined.

    You appear to be naive about the topics you make claims on, both in terms of the quantitative results and the qualitative approaches/methodologies used. And yet you constantly try to convince all and sundry that you're well read on them. If you were you'd not be constantly posting "OMG I've got perfect proof [X] is wrong!".
     
  18. Pete It's not rocket surgery Registered Senior Member

    Messages:
    10,167
    :bugeye: Are you deliberately misunderstanding me?
     
  19. Jack_ Banned Banned

    Messages:
    1,383
    You are wrong.

    One light sphere does not morph into many.

    That is creationism.

    I am developed several proofs here against SR. You have not stopped any of them.

    Prove how one light sphere can be origined at two different positions in the space of one frame. You are permitted to use my point as an aid.


    I suspect you fail to grasp just how intertwined special relativity's mathematical formalism is with other areas of mathematics. Group theory, manifolds, set theory, Lie theory. If special relativity is mathematically inconsistent then it would filter through into those areas yet their consistency and structures have been deeply examined.

    Where in those theories does the origin of one sphere occupy two different points in one space?
     

Share This Page