Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Relativistic velocity addition and signs

  1. Apr 27, 2014 #1
    Hello all,

    I have just covered a very brief module on special relativity as a part of my physics course. I have also done some extra reading mostly; Morrin's Classical Mechanics. While I found the book really illuminating in some aspects, I still feel that regardless of how hard I try there is something with relativity that prevents me form doing anything but the simplest questions. I was trying to pinpoint my problem and I think that a big part of it is velocity addition.

    I understand that the Galilean transformation would predict the $$V_{A}=V_{B}-V_{rel}$$ provided that A and B are two frames of reference. I also understand that we need to use the Lorentz transformation to get the velocity transformation in relativity;
    c \Delta T_A \\
    \Delta x_A\\
    \end{pmatrix} \begin{pmatrix}
    \gamma & \gamma \beta \\
    \gamma \beta & \gamma\\
    \end{pmatrix} = \begin{pmatrix}
    c \Delta T_B \\
    \Delta x_B\\

    Transforming the velocity u measured in frame to frame B;

    $$u = \dfrac{\Delta x_A}{\Delta t_A} = \dfrac{v_B + u_{rel}}{1+\dfrac{v_B u_{rel}}{c^2}}$$

    But as far as I understand we could equally reverse the frames A and B and simply transform the other way around which means we need the inverse of the transformation matrix; \begin{pmatrix}
    \gamma & - \gamma \beta \\
    - \gamma \beta & \gamma\\

    This will yield the formula;

    $$u = \dfrac{\Delta x_B}{\Delta t_B} = \dfrac{v_A - u_{rel}}{1-\dfrac{v_A u_{rel}}{c^2}}$$.

    However since the naming of frames is arbitrary, how do I know which of the two formula to use, the one with the all plus and the all minus signs. I have tried to look on the internet for the explanation of this, but I could not find anything. Also provided that I know which equation to use, how do I use it, what is the sign convention for the velocities?

    Thank you very much for all the help and sorry for the long post

    P.S. I would be also very grateful if someone could point me to some good and simple resources on relativistic dynamics especially collisions. Thanks again.
  2. jcsd
  3. Apr 27, 2014 #2

    Doc Al

    User Avatar

    Staff: Mentor

    For that very reason (sign issues) I prefer this version of the relativistic addition of velocities formula:
    [tex]V_{a/c} = \frac{V_{a/b} + V_{b/c}}{1 + (V_{a/b} V_{b/c})/c^2}[/tex]
  4. Apr 27, 2014 #3
    Actually, if the Galilean addition formula is to be consistent with what you are doing with the relativistic addition formula, then


    Then, when you're using the relativistic formula, you use the same sign for the relative velocity as you would with the corresponding Galilean situation.

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook