Relativistic velocity addition and signs

[asdfghhjkl]

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

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.

 

[Doc Al]

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?

For that very reason (sign issues) I prefer this version of the relativistic addition of velocities formula:
$$V_{a/c} = \frac{V_{a/b} + V_{b/c}}{1 + (V_{a/b} V_{b/c})/c^2}$$

 

[Chestermiller]

Actually, if the Galilean addition formula is to be consistent with what you are doing with the relativistic addition formula, then

V_A=V_B+V_rel

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.

Chet