The space of $n$-dimensional relativistic velocities normalized to $c = 1$, $$\mathbb{ B} = \{ \mathbf{ v}\in\mathbb{ R}^n\colon \|\mathbf{ v}\| < 1\},$$ is naturally associated with Einstein velocity addition $\oplus_E$, which induces the rapidity metric $d_E$ on $\mathbb{ B}$ given by $d_E(\mathbf{ u}, \mathbf{ v}) = \tanh^{ -1}\|-\mathbf{ u}\oplus_E\mathbf{ v}\|$. This metric is also known as the Cayley-Klein metric. We give a complete description of the isometry group of $(\mathbb{ B}, d_E)$, along with its composition law.
Categories: math.MG math-ph math.MP