2
$\begingroup$

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; $$ \begin{pmatrix} c \Delta T_A \\ \Delta x_A\\ \end{pmatrix} \begin{pmatrix} \gamma & \gamma \beta \\ \gamma \beta & \gamma\\ \end{pmatrix} = \begin{pmatrix} c \Delta T_B \\ \Delta x_B\\ \end{pmatrix} $$

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\\ \end{pmatrix}

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.

$\endgroup$
  • $\begingroup$ You need to be able to distinguish between the transformation with receding frames and that of approaching ones. They are connected by sign inversion of the velocity, as you noted. $\endgroup$ – auxsvr Apr 27 '14 at 23:07
  • 1
    $\begingroup$ You may find useful that in general, for (members of) three inertial systems $N$, $P$ and $Q$ holds: $$1 - (\beta_{P}[ N ] \, \, \beta_{P}[ Q ] \, \, \text{Cos}[\angle_{P}[N, Q]]) = \sqrt{ \frac{(1 - \beta^2_{P}[ N ]) \, (1 - \beta^2_{P}[ Q ])}{(1 - \beta^2_{N}[ Q ])}}.$$ $\endgroup$ – user12262 Apr 30 '14 at 14:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.