Relativistic composition of velocities -- beyond 1D

This will be a short entry.  The methods of menhirs presented in the previous blog allows one for a simple geometric construction of the composition of two velocities in more than 1 dimension.  Here it the result:

Move $\mathbf v_1$ and $\mathbf v_2$ by dragging their tips (red points) Exercise: Check out a number of configurations with $\mathbf v_1\bot \mathbf v_2$.  You should notice something interesting; check it against the first formula of the last blog.

The meaning: two consecutive boosts, first by velocity $\mathbf v_1$ denoted $B(\mathbf v_1)$, followed by boost $B(\mathbf v_2)$ can be replaced by a single boost along certain velocity denoted $\mathbf v_1 \oplus \mathbf v_2$ and called the composition of the two velocities. In the Galilean physics, we have simply $\mathbf v_1 \oplus \mathbf v_2 = \mathbf v_1 + \mathbf v_2$, and the case is closed.

In the relativity, the velocity $\mathbf v_1 \oplus \mathbf v_2$ is not anymore a linear sum of two vectors. Moreover, two boosts can be replaced by a single boost followed by a rotation. $$ B(\mathbf v_2) \ \circ \ B(\mathbf v_1) \quad = \quad R \ \circ \ B(\mathbf v_1 \oplus \mathbf v_2) $$ The applet above shows both $\mathbf v_1 \oplus \mathbf v_2$ as a vector and the rotation angle marked by an arc on the circle. Have fun exploring different configurations of the velocities.

To see better the difference between the relativistic case and simple sum of vectors (classical Galilean composition), play with the version of the applet on which the regular sum is shown too (red arrow)

Move $\mathbf v_1$ and $\mathbf v_2$ by dragging their tips (red points).

Exercise: check when the discrepancy between $\mathbf v_1+ \mathbf v_2$ and $\mathbf v_1\oplus\mathbf v_2$ is the greatest / smallest.

And that's it for now!  The explanation of  how and why it works and what does it have to do with the cromlech and menhirs will come soon.