The isometry group of $n$-dimensional Einstein gyrogroup

Abstract

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.