I'll explain my method just in case someone is interested, and because I often find mistakes when I try to explain things.
First the 1+1-dimensional case. F and F' are two inertial frames with the same origin. Let the line
\tau\mapsto\tau\begin{pmatrix}1\\v\end{pmatrix}
be the world line of a mass M in the coordinates of F'. The idea is to pick any point (t' x')
T on that world line and do a Lorentz transformation to F. The velocity we seek is x/t.
\begin{pmatrix}t\\x\end{pmatrix}=\Lambda_{FF'}\begin{pmatrix}1\\v\end{pmatrix} = \gamma(u)\begin{pmatrix}1 & u\\ u & 1\end{pmatrix}\begin{pmatrix}1\\v\end{pmatrix} =\gamma(u)\begin{pmatrix}1+uv\\ u+v\end{pmatrix} \implies w=\frac{u+v}{1+uv}
Now the 2+1-dimensional case. We use the same basic assumptions, and the same idea. (I just have to add that F and F' are also rotated the same way, now that there are two spatial dimensions). We're going to find our "w" by using an appropriate Lorentz transformation \Lambda_{FF'} to transform the vector (1 v
x v
x)
T from F' to F.
The "trick" I'm using is to do this Lorentz transformation in three steps. I'm introducing two new frames, G and G'. G' has the same velocity as F', but is rotated so that its x-axis is parallel with u. G is rotated the same way as G', but has the same velocity as F.
\Lambda_{FF'}=\Lambda_{FG}\Lambda_{GG'}\Lambda_{G'F'}
So instead of applying the complicated Lorentz transformation, I'm doing a rotation by an angle \theta=\arctan (u_y/u_x), then a boost in the x direction, and finally the inverse of the original rotation. I'll save myself some typing by writing the above equation as
\Lambda_{FF'}=R^T\Lambda R
We have
R=\begin{pmatrix}1 & 0 & 0\\ 0 & \cos\theta & \sin\theta\\ 0 & -\sin\theta & \cos\theta \end{pmatrix} =\begin{pmatrix}1 & 0 & 0\\ 0 & \frac{u_x}{u} & \frac{u_y}{u}\\ 0 & -\frac{u_y}{u} & \frac{u_x}{u} \end{pmatrix}
and
\Lambda=\gamma(u)\begin{pmatrix}1 & u & 0\\ u & 1 & 0\\ 0 & 0 & 1/\gamma(u) \end{pmatrix}
All we have to do is calculate
\begin{pmatrix}t \\ x \\ y\end{pmatrix}=R^T\Lambda R\begin{pmatrix}1 \\ v_x \\ v_y\end{pmatrix}
and then we get our "w" the same way as before: w
x=x/t, w
y=y/t. I might post some intermediate results later, but right now I have to go out and get some food.
Is there something wrong with the \LaTeX? It's a bit hard to read. Is the text smaller than before or something?
DaleSpam, do you think you could break your long equation into two lines, so that it doesn't make every line of text longer than the browser window?
